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Minor revisions 21/2/07.\n", StyleBox["(This is retained in ", FontSize->12], StyleBox["MathSource", FontSize->12, FontSlant->"Italic"], StyleBox["/6198/ for continuity, as the source for\nmuch of ", FontSize->12], StyleBox["HoopAlg&Physics.nb", FontSize->12, FontSlant->"Italic"], StyleBox[" and ", FontSize->12], StyleBox["HoopSup.nb.", FontSize->12, FontSlant->"Italic"], StyleBox[")\n", FontSize->12] }], "Subsubtitle", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["1. Introduction", "Section", PageWidth->WindowWidth, FontFamily->"Times New Roman"], Cell[TextData[{ "\tHoops are vector algebras that ", StyleBox["conserve the symbolic determinant factors of their Cayley \ multiplication tables as symmetries or \"sizes\". Symmetry conservation \ leads, via Noether's theorem, to forces and to particles", PageWidth->WindowWidth], ". Consequently, Hoops subsume and unify all the algebras (including Real, \ Complex, Quaternion, Octonion, Clifford and Wedge) relevant to physics. They \ may provide a new paradigm for particle physics, based on vector \ multiplication by finite Moufang loops as well as coordinate transformations \ using continuous groups. Using experimental mathematics, the author has found \ many analogies between hoop algebras and particle physics, but (as a 77 yr. \ old chemical engineer) lacks the skills and time to take the subject much \ further.\n\tThe Hoops concept has been implemented in ", StyleBox["MathSource/6198/", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], ". The Mathematica package ", StyleBox["Hoops.m", FontSlant->"Italic"], " provides a databank of over 80 named Hoops (with 5 counter-example \ non-hoop Loops), three main procedures ", StyleBox["hoopTimes, hoopInverse, hoopPower", FontSlant->"Italic"], " and a few subsidiary procedures. Each hoop is defined by a Cayley index \ table (a preferred isomorph) and its \"shape\" (the distinct factors of the \ symbolic Cayley table). Multiplication, division, and (in many cases) \ conversion from Cartesian vector {a,b,...} to polar {r,\[Theta]} form are \ provided." }], "Text"], Cell[TextData[{ "\tThe figures are self-explanatory output from a ", StyleBox["Mathematica", FontSlant->"Italic"], " session (with results of calculations appearing in bold-face) intended to \ be comprehensible readers with no knowledge of ", StyleBox["Mathematica", FontSlant->"Italic"], ". They are based on the ", StyleBox["Hoops.m", FontSlant->"Italic"], " package. ", StyleBox["Mathematica", FontSlant->"Italic"], " users can test them with different data, by installing ", StyleBox["Hoops.nb", FontSlant->"Italic"], " and running it (to create the ", StyleBox["Hoops.m", FontSlant->"Italic"], " package) in the ", StyleBox["Addons\\ExtraPackages", FontSlant->"Italic"], " directory, and then using the following instruction:-." }], "Text", PageWidth->WindowWidth], Cell[BoxData[ \(Quit[]\)], "Input"], Cell[BoxData[ \(<< "\"\)], "Input", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["2. Hoop Concepts.", "Section", FontFamily->"Times New Roman"], Cell[TextData[StyleBox["Summary.\n\tSymmetry-conserving \"Hoop Algebras\" \ involve new mathematical concepts, based on unsigned \ \[OpenCurlyDoubleQuote]Primal numbers\[CloseCurlyDoubleQuote]. Hoop \ multiplication tables are \[OpenCurlyDoubleQuote]folded\ \[CloseCurlyDoubleQuote] from finite groups and Moufang loops; this \ introduces \"generalized signs\". Hoops multiply, divide, and add \"vecs\" \ (generalized vectors). Their \"Frobenius\" symmetry-conservation property and \ \"Moufang\" partial-fraction-division property make them relevant to physics. \ A few hoops have stable multi-phase de-Broglie-like \"orbits\" and \ Planck-like fundamental areas; this may explain particle stability. Every vec \ has a multiplicative inverse, so Hoop multiplication and division use the \ same \"hoopTimes\" operation; two vecs produce a third vec and two (often \ null) \"remainder\" vecs carrying different symmetries. This is analogous to \ symmetry-conserving particle interactions. Do not confuse hoops with the \ traditional use of groups as coordinate transformation matrices for a single \ vector. Hoop tables are not restricted to complex matrix generators, and many \ complex operations are degenerate cases of more general hoop operations.", FontSlant->"Italic"]], "Text"], Cell[TextData[{ "\tThe development of hoops involves some neglected and some new \ mathematical concepts. The latter are introduced in \ \[OpenCurlyDoubleQuote]quotation marks\[CloseCurlyDoubleQuote]:- \n(1) \ \"Primal\" unsigned continuous numbers (the half-line, the union of 0 & ", Cell[BoxData[ \(TraditionalForm\`\(\[DoubleStruckCapitalR]\^+\)\)]], ") can be developed from set theory, without introducing integers and reals \ (with negation and subtraction), by using a continuity axiom on Landau's [1] \ unsigned rational numbers ", Cell[BoxData[ \(TraditionalForm\`\(\[DoubleStruckCapitalQ]\^+\)\)]], ". Only the most rigorous mathematicians acknowledge that integers are \ equivalence relations on pairs of natural numbers; others conflate (unary) \ negation with (binary) subtraction by working in fields and implicitly \ assuming that minus one exists and is unique. Some physical concepts (time? \ mass?) should, perhaps, have primal (rather than real) measures.\n(2) \ Algebraic loops are ", StyleBox["m\[Times]m", FontSlant->"Italic"], " multiplication tables for ", StyleBox["m", FontSlant->"Italic"], "-element sets of coefficients or operators that define \"vecs\" \ (generalized vectors, which are vectors in some algebras) such as A = {", Cell[BoxData[ \(TraditionalForm\`\(\(a\_1\)\(,\)\)\)]], Cell[BoxData[ \(TraditionalForm\`a\_2\)]], ",...", Cell[BoxData[ \(TraditionalForm\`a\_m\)]], "},{", StyleBox["a,b,...,m", FontSlant->"Italic"], "}, or ", Cell[BoxData[ \(TraditionalForm\`a\_1\)]], Cell[BoxData[ \(TraditionalForm\`d\_1\)]], "+", Cell[BoxData[ \(TraditionalForm\`a\_2\)]], Cell[BoxData[ \(TraditionalForm\`d\_2\)]], "+...+", Cell[BoxData[ FormBox[ RowBox[{\(a\_m\), FormBox[\(d\_m\), "TraditionalForm"]}], TraditionalForm]]], ". Loops can be developed without introducing negation, so their \ coefficients are unsigned. Moufang loops have the Moufang division property ( \ ", StyleBox["zx.yz=(z.xy)z ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[") (so ", FontWeight->"Plain"], StyleBox["every vector has a multiplicative inverse)", PageWidth->WindowWidth], StyleBox["; groups ", FontWeight->"Plain"], "are associative Moufang loops with (", StyleBox["x(yz)=(xy)z", FontSlant->"Italic"], ") .\n(3) Frobenius showed that all groups conserve their determinants when \ they are used as vector multiplication tables; Det[A] Det[B] = \ \[PlusMinus]Det[AB], where Det is the determinant of the inverse \ multiplication table mapped with the appropriate vector. A few \ non-associative Moufang loops (including octonions and split-octonions) also \ have this \"Frobenius conservation\" property.\n(4) \"Generalized signs\" are \ orthogonal primitive roots of unity ", StyleBox["s", FontSlant->"Italic", FontVariations->{"Shadow"->True}], StyleBox["j", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], " (with ", StyleBox["s", FontSlant->"Italic", FontVariations->{"Shadow"->True}], StyleBox["r", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], " =1) describing ", StyleBox["r", FontSlant->"Italic"], "-directional spaces or sub-spaces. Describing roots of unity as ", StyleBox["cyclotomic numbers", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " distorts their properties by projecting them onto the complex field. \ Reals, with signs {+.-}, are an ", StyleBox["r", FontSlant->"Italic"], "=2 case. Complex numbers are one case of ", StyleBox["r", FontSlant->"Italic"], "=4, with signs {+,\[ImaginaryI],-,-\[ImaginaryI]}. ", StyleBox["r", FontSlant->"Italic"], "=3 gives \"terplex\" signs {+, \[DoubleStruckCapitalJ], ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalJ]\^2\)]], "} with ternary symmetries relevant to quarks.\n(5) \"Directors\" are sets \ of primal coefficients with associated \"directions\", i.e. the ", Cell[BoxData[ \(TraditionalForm\`d\_i\)]], "'s in (2). ", StyleBox["r", FontSlant->"Italic"], "-fold equivalence relationships on ", StyleBox["m", FontSlant->"Italic"], "-element directors \"fold\" them to \"signed vecs\", generalized vectors \ with ", StyleBox["m/r", FontSlant->"Italic"], " associated dimensions and ", StyleBox["r-", FontSlant->"Italic"], "signed coefficients. Real vecs are the result of 2-folding, ", Cell[BoxData[ FormBox[ RowBox[{\(r\_i\), "~", FormBox[\(a\_i\), "TraditionalForm"]}], TraditionalForm]], PageWidth->WindowWidth], "-", Cell[BoxData[ FormBox[ RowBox[{\(a\_\(i + m/2\)\), FormBox["", "TraditionalForm"]}], TraditionalForm]], PageWidth->WindowWidth], " with the signs {+, -}; this is the special case \"Minus 1 exists and is \ unique\" that allows negation and subtraction to be conflated. (The \ \"directions\" concept is due to Sir W.R.Hamilton. He related the eight \ directions of the quaternion group to the four dimensions of the quaternion \ algebra, and the sixteen directions of the octonion table to the eight \ dimensions of the octonion algebra.)\n(6) ", StyleBox["Folding is a key concept because it relates generalized signs to \ loop symmetries. ", PageWidth->WindowWidth], "An equivalence relationship on a conservative ", StyleBox["m\[Times]m", FontSlant->"Italic"], "-element loop with ", StyleBox["r", FontSlant->"Italic"], "-fold symmetry \"folds\" it to an ", StyleBox["(m/r)\[Times](m/r)", FontSlant->"Italic"], " multiplication table. I call such tables \"Hoop Algebra\". In many cases \ they are not loops, because some products are \"signed elements\" that are \ not members of the defining set. E.g. Quaternions have four elements \ {e,i,j,ij} with i.j=ij, but j.i=-ij. \n", StyleBox["Figure 1 shows \[OpenCurlyDoubleQuote]protoloops\ \[CloseCurlyDoubleQuote] (preferred Cayley table isomorphs) for three Moufang \ loops (the C4C2, quaternion, and C9 groups). The top left corner is repeated \ in the tables; it also occurs with offsets of 3, 4, or 6, showing the \ symmetry. These sub-tables are then shown as folded tables, with the indices \ replaced by the usual complex or quaternion symbols, or (in the C9J algebra) \ by signed indices, providing a new sign ", PageWidth->WindowWidth], StyleBox["J (", PageWidth->WindowWidth, FontVariations->{"Shadow"->True}], StyleBox["with", PageWidth->WindowWidth], StyleBox[" J", PageWidth->WindowWidth, FontVariations->{"Shadow"->True}], StyleBox["3", PageWidth->WindowWidth, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["=1) so that indices 4 & 5 become ", PageWidth->WindowWidth], StyleBox["J", PageWidth->WindowWidth, FontVariations->{"Shadow"->True}], StyleBox["1 and", PageWidth->WindowWidth], StyleBox[" J", PageWidth->WindowWidth, FontVariations->{"Shadow"->True}], StyleBox["2. Note that the C4C2 table can be folded a second time, to give \ ", PageWidth->WindowWidth], StyleBox["the complex algebra table with \[ImaginaryI] \[ImaginaryI] = -1. \ The Pauli-\[Sigma] algebra is also shown; it is 4-folded from a 16-element \ group ", PageWidth->WindowWidth, FontFamily->"Times New Roman"], StyleBox["(not shown) that can be created by dot-multiplying ", PageWidth->WindowWidth], StyleBox["the Pauli-\[Sigma] matrices", PageWidth->WindowWidth, FontFamily->"Times New Roman"], StyleBox[".", PageWidth->WindowWidth, FontFamily->"Times New Roman Greek"] }], "Text"], Cell[TextData[{ StyleBox["Figure 1.", FontFamily->"Courier New", FontSize->9, FontWeight->"Bold"], StyleBox[" Two- Three- & four-fold folding of Moufang Loops,\n the \ resulting signed algebras, and the Pauli-", FontFamily->"Courier New", FontSize->9], StyleBox["\[OAcute]", FontFamily->"Courier New Greek", FontSize->9], StyleBox[" algebra.\n C4C2 Group Quaternion group C9 group\n", FontFamily->"Courier New", 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StyleBox["2", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[")", FontFamily->"Courier New", FontSize->9], StyleBox["2\n", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["hoopTimes[{1,2,3,4},{a,b,c,d}]=\n\ {a-2b-3c-4d,2a+b-4c+3d,3a+4b+c-2d,4a-3b+2c+d}", FontFamily->"Courier New", FontSize->9], StyleBox["\n", FontFamily->"Courier New", FontSize->9, FontWeight->"Bold"], StyleBox["C9J", FontFamily->"Times New Roman", FontSize->9, FontWeight->"Bold"], StyleBox[" Factors", FontFamily->"Times New Roman", FontSize->9], StyleBox[" ", FontFamily->"Times New Roman", FontSize->9, FontWeight->"Bold", FontVariations->{"Shadow"->True}], StyleBox[" -c", FontFamily->"Courier New", FontSize->9], StyleBox["3", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["-b", FontFamily->"Courier New", FontSize->9], StyleBox["3", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["J", FontFamily->"Courier New", FontSize->9, FontVariations->{"Shadow"->True}], StyleBox["+3abc", FontFamily->"Courier New", FontSize->9], StyleBox["J", FontFamily->"Courier New", FontSize->9, FontVariations->{"Shadow"->True}], StyleBox["-a", FontFamily->"Courier New", FontSize->9], StyleBox["3", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["J", FontFamily->"Courier New", FontSize->9, FontVariations->{"Shadow"->True}], StyleBox["2", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\nhoopTimes[{1,2,3},{a,b,c}]={a+3bJ+2cJ,2a+b+3cJ,3a+2b+c}\n", FontFamily->"Courier New", FontSize->9], StyleBox["Pauli-\[Sigma] ", FontFamily->"Times New Roman", FontSize->9, FontWeight->"Bold"], StyleBox[" Factors", FontFamily->"Times New Roman", FontSize->9], StyleBox[" ", FontFamily->"Times New Roman", FontSize->9, FontWeight->"Bold", FontVariations->{"Shadow"->True}], StyleBox[" ", FontFamily->"Courier New", FontSize->9, FontWeight->"Bold"], StyleBox["(a", FontFamily->"Courier New", FontSize->9], StyleBox["2", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["-b", FontFamily->"Courier New", FontSize->9], StyleBox["2", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["-c", FontFamily->"Courier New", FontSize->9], StyleBox["2", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["-d", FontFamily->"Courier New", FontSize->9], StyleBox["2", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[")", FontFamily->"Courier New", FontSize->9], StyleBox["2", FontFamily->"Courier New", FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["\n", FontFamily->"Courier New", FontSize->9, FontWeight->"Bold"], StyleBox["hoopTimes[{1,2,3,4},{a,b,c,d}]=\n\ {a+2b+3c+4d,2a+b+4\[ImaginaryI]c-3\[ImaginaryI]d,3a-4\[ImaginaryI]b+c+2\ \[ImaginaryI]d,4a+3\[ImaginaryI]b-2\[ImaginaryI]c+d}", FontFamily->"Courier New", FontSize->9] }], "Text", Editable->False], Cell[TextData[{ "\tThe three C4 sizes collapse to the single conserved size ", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`b\^2\)]], " of Complex algebra. Quaternion and Pauli-\[Sigma] algebras conserve their \ single (but repeated) factors. C9J conserves ", Cell[BoxData[ \(TraditionalForm\`a\^3\)]], "+", Cell[BoxData[ \(TraditionalForm\`b\^3\)]], "J-3", StyleBox["abc", FontSlant->"Italic"], "J+", Cell[BoxData[ \(TraditionalForm\`c\^3\)]], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["J", FontSlant->"Plain"], "2"], TraditionalForm]]], ", a conjugate of the determinant factor. (Conjugate conservation is the \ general rule, but conjugation has no effect on most sizes.)\n(8) The Moufang \ property ensures that every vec has a multiplicative inverse in each hoop of \ the same length, merging multiplication and division into one procedure. The \ inverse Ai solves Ai.A={1,0,..} (the \[OpenCurlyDoubleQuote]unit\ \[CloseCurlyDoubleQuote]). The determinant is a divisor for each element of \ the inverse, so multiple sizes provide denominators for partial-fraction \ formulations of the inverse.", StyleBox[" The un-factored determinant becomes zero if one or more size \ becomes zero; other factors remain finite.\n\tFigure 3 shows the symbolic C3 \ table, the product of two symbolic vectors, and the two determinant factors. \ The inverse is then shown; each term splits into two partial fractions. Then \ two numeric vectors A & B are multiplied; the sizes are shown as sA, etc. ", PageWidth->WindowWidth, FontFamily->"Times New Roman"], "(Mathematica users can repeat the calculations with their own input data \ in", StyleBox[" hoopDemo.nb", FontSlant->"Italic"], ".) ", StyleBox["The sizes of the product AB can be seen to be the products of the \ sizes of A & B. Ai is a left inverse, so ", PageWidth->WindowWidth, FontFamily->"Times New Roman"], StyleBox["hoopTimes[Ai,AB]", PageWidth->WindowWidth, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" recovers B. ", PageWidth->WindowWidth, FontFamily->"Times New Roman"] }], "Text"], Cell[TextData[{ StyleBox["Figure 3.", FontWeight->"Bold"], " C3 Product, Shape, Inverse, Division." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {hoopTbl /. {1 \[Rule] a, 2 \[Rule] b, 3 \[Rule] c, 4 -> d, 5 \[Rule] e, 6 \[Rule] f} // tf, \[IndentingNewLine]"\<\nhoopTimes[{a,b,c},{d,e,f}=\n\>", hoopTimes[{a, b, c}, {d, e, f}], \[IndentingNewLine]"\<\nshape =\n\>", sh, "\<\nInverse =\n\>", \[IndentingNewLine]Apart[ hoopInverse[{a, b, c, d, e, f}]]}\)], "Input", CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b", "c"}, {"b", "c", "a"}, {"c", "a", "b"} }], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nhoopTimes[{a,b,c},{d,e,f}=\\n\"\>", ",", \({a\ d + c\ e + b\ f, b\ d + a\ e + c\ f, c\ d + b\ e + a\ f}\), ",", "\<\"\\nshape =\\n\"\>", ",", \({a + b + c, 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + \ c)\)\^2)\)}\), ",", "\<\"\\nInverse =\\n\"\>", ",", \({1\/\(3\ \((a + b + c)\)\) + \(2\ a - b - c\)\/\(3\ \((a\^2 - \ a\ b + b\^2 - a\ c - b\ c + c\^2)\)\), 1\/\(3\ \((a + b + c)\)\) + \(\(-a\) - b + 2\ c\)\/\(3\ \((a\^2 - a\ \ b + b\^2 - a\ c - b\ c + c\^2)\)\), 1\/\(3\ \((a + b + c)\)\) + \(\(-a\) + 2\ b - c\)\/\(3\ \((a\^2 - a\ \ b + b\^2 - a\ c - b\ c + c\^2)\)\)}\)}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({Flatten[{"\", A = {3, 1, \(-2\)}, "\", sh /. as[A]}], \[IndentingNewLine]Flatten[{"\", B = {1, 0, 2}, "\", sh /. as[B]}], \n Flatten[{"\", AB = hoopTimes[A, B], "\", sh /. as[AB]}], \[IndentingNewLine]Flatten[{"\", Ai = hoopInverse[A], "\", sh /. as[Ai]}], \[IndentingNewLine]Flatten[{"\", AiAB = hoopTimes[Ai, AB], "\", sh /. as[AiAB]}]} // tf\)], "Input", CellOpen->False, FontSize->9], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\<\"If A=\"\>", "3", "1", \(-2\), "\<\"sA=\"\>", "2", "19"}, {"\<\"and B=\"\>", "1", "0", "2", "\<\"sB=\"\>", "3", "3"}, {"\<\"AB=\"\>", "5", \(-3\), "4", "\<\"sAB=\"\>", "6", "57"}, {"\<\"Ai=\"\>", \(11\/38\), \(1\/38\), \(7\/38\), "\<\"sAi=\"\>", \ \(1\/2\), \(1\/19\)}, {"\<\"AiAB=B\"\>", "1", "0", "2", "\<\"sAiAB=\"\>", "3", "3"} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output", FontSize->9] }, Open ]], Cell["\<\ (9) Vecs with \"zero-sizes\" exist in sub-algebras where these sizes are \ constrained to zero. Their use in multiplication and division \"projects\" \ the result into the sub-algebras and \"ejects\" remainders to maintain size \ conservation. This eliminates division-by-zero; zeroed sizes are \ \"factored-out\" by projection into the constrained sub-algebra.\ \>", "Text"], Cell[TextData[{ StyleBox["Figure 4.", FontWeight->"Bold"], " If A has a zero size in C3, so do AB and Ai; \n AB/A only \ become B after adding a remainder." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({Flatten[{"\", A0 = {2, \(-1\), \(-1\)}, "\", sh /. as[A0]}], \[IndentingNewLine]Flatten[{"\", B = {1, 0, 2}, "\", sh /. as[B]}], \[IndentingNewLine]Flatten[{"\", AB = hoopTimes[A0, B], "\", sh /. as[AB]}], Flatten[{"\", Rleft = Rl, "\", sh /. as[Rleft]}], \[IndentingNewLine]Flatten[{"\", Rright = Rr, "\", sh /. as[Rr]}], \[IndentingNewLine]Flatten[{"\", Ai = hoopInverse[A0], "\", sh /. as[Ai]}], \[IndentingNewLine]Flatten[{"\", AiAB = hoopTimes[Ai, AB], "\", sh /. as[ AiAB]}], \[IndentingNewLine]Flatten[{"\", R = AiAB + Rright, "\", sh /. as[R]}]} // tf\)], "Input", CellOpen->False], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\<\"If A=\"\>", "2", \(-1\), \(-1\), "\<\"sA=\"\>", "0", "9"}, {"\<\"and B=\"\>", "1", "0", "2", "\<\"sB=\"\>", "3", "3"}, {"\<\"AB=\"\>", "0", \(-3\), "3", "\<\"sAB=\"\>", "0", "27"}, {"\<\"Rleft=\"\>", "0", "0", "0", "\<\"sRleft=\"\>", "0", "0"}, {"\<\"Rright=\"\>", "1", "1", "1", "\<\"sRright=\"\>", "3", "0"}, {"\<\"Ai=\"\>", \(2\/9\), \(-\(1\/9\)\), \(-\(1\/9\)\), "\<\"sAi=\ \"\>", "0", \(1\/9\)}, {"\<\"AB/A (=AiAB)=\"\>", "0", \(-1\), "1", "\<\"sAiAB=\"\>", "0", "3"}, {"\<\"AiAB+Rright=B\"\>", "1", "0", "2", "\<\"s=\"\>", "3", "3"} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output", FontSize->9] }, Open ]], Cell["\<\ \tNote the analogy between integer division, I/J=K+R where JK+R=I, with Hoop \ operations, where A.B=C+Rleft+Rright and C/B+Rleft=A, C/A+Rright=B. Rleft \ carries the sizes that are zero in C but not in A; Rright carries the sizes \ that are zero in C but not in B. Remainders ensure that conservation is \ maintained in hoop multiplication (as well as division) when some sizes are \ zero. There is an obvious analogy with particle interactions, where there may \ be several products with different symmetries. \tDo not confuse loop operations, acting on two vecs to give a new vec and \ two (possibly null) remainder vecs, with the use of matrices to transform a \ single vector to different coordinates in the \[OpenCurlyDoubleQuote]standard \ model\[CloseCurlyDoubleQuote] of particle physics.\ \>", "Text"], Cell[TextData[{ "(10) Some hoops have quadratic sizes that provide polar-Cartesian dual \ formulations and allow the definition of continuous multi-phase sinusoidal \ \"orbits", StyleBox["\", as shown in Example 5", PageWidth->WindowWidth], ". Quadratic sizes release a degree of freedom that is taken up by an \ angle", StyleBox[", which is a hidden variable for the Cartesian form. \n\t", PageWidth->WindowWidth, FontFamily->"Times New Roman"], "Abelian duals have angles that add on vector multiplication, leading to \ powers and roots. ", StyleBox["Figure 5 shows that the polar form for the C3 vec {a,b,c} \ consists of expressions for two sizes {\[Alpha],", PageWidth->WindowWidth], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^2\)]], StyleBox["} and an angle \[Sigma]. The reversion from polar to cartesian \ form has \[Alpha] as an offset from zero and three equispaced phases. It then \ demonstrates that ", PageWidth->WindowWidth], StyleBox["vector[polar[A]=A", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox[", angles add on C3 multiplication, and taking a square root \ halves the angle and gives sizes that are square roots of the original sizes. \ C3 has a prototypical ternary (3-phase) form polar form. Figure 5 goes on to \ show that C4 has the prototypical 4-phase. This folds to the (asymmetric) \ {Cos[\[Theta]],Sin[\[Theta]]} pair. All known polar duals are related to \ these two prototypes.", PageWidth->WindowWidth] }], "Text", PageWidth->WindowWidth], Cell["Figure 5. C3 & C4 Polar Duals, Reversions, Powers & Roots.", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(0 Use["\"]; "\"\), "\[IndentingNewLine]", \(topol\), "\[IndentingNewLine]", \("\"\), "\[IndentingNewLine]", \(tovec\), "\[IndentingNewLine]", \({Flatten[{"\", A = {2. , \(-3. \), 5. }, "\", topol /. as[A]}], Flatten[{"\", tovec /. as[ topol /. as[ A]], "\<\>", "\<\>", "\<\>", "\<\>"}], \ \[IndentingNewLine]Flatten[{"\", B = {1, \(-5\), 6. }, "\", topol /. as[B]}], \[IndentingNewLine]Flatten[{"\", AB = hoopTimes[A, B], "\", topol /. as[AB]}], Flatten[{\*"\"\<\!\(\@A\)=\>\"", RootA = hoopPower[A], \*"\"\\"", topol /. as[RootA]}]} // tf\), "\[IndentingNewLine]", \(Use["\"]; "\"\), "\n", \(topol\), "\n", \("\"\), "\n", \(tovec\), "\n", \({Flatten[{"\", A = {2. , \(-3. \), 5. , 1. }, "\", topol /. as[A]}], Flatten[{"\", tovec /. as[ topol /. as[ A]], "\<=A\>", "\<\>", "\<\>", "\<\>", "\<\>"}], \ \[IndentingNewLine]Flatten[{"\", B = {\(-1\), 5, \(-6. \), 3. }, "\", topol /. as[B]}], \[IndentingNewLine]Flatten[{"\", AB = hoopTimes[A, B], "\", topol /. as[AB]}], Flatten[{\*"\"\<\!\(\@A\)=\>\"", RootA = hoopPower[A], \*"\"\\"", topol /. as[RootA]}]} // tf\)}], "Input", CellOpen->False], Cell[BoxData[ \("Polar C3"\)], "Output"], Cell[BoxData[ \({a + b + c, 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2)\), ArcTan[2\ a - b - c, \(-\@3\)\ \((b - c)\)]}\)], "Output"], Cell[BoxData[ \("Vector C3"\)], "Output"], Cell[BoxData[ \({1\/3\ \((a + 2\ \@b\ Cos[c])\), 1\/3\ \((a + 2\ \@b\ Cos[c + \(2\ \[Pi]\)\/3])\), 1\/3\ \((a + 2\ \@b\ Cos[c - \(2\ \[Pi]\)\/3])\)}\)], "Output"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\<\"If A=\"\>", "2.`", \(-3.`\), "5.`", "\<\"polarA=\"\>", "4.`", "49.`", "1.4274487578895312`"}, {"\<\"Vec(polar(A))=\"\>", "2.`", \(-3.0000000000000004`\), "5.`", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"and B=\"\>", "1", \(-5\), "6.`", "\<\"polarB=\"\>", "2.`", "91.`", "1.518358056045878`"}, {"\<\"AB=\"\>", \(-41.`\), "17.`", "32.`", "\<\"polarAB=\"\>", "8.`", "4459.`", "2.9458068139354094`"}, {"\<\"\\!\\(\\@A\\)=\"\>", "2.`", \(-1.`\), "1.0000000000000002`", "\<\"polar\\!\\(\\@A\\)=\"\>", "2.`", "7.`", "0.7137243789447656`"} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output"], Cell[BoxData[ \("Polar C4"\)], "Output"], Cell[BoxData[ \({a + b + c + d, a - b + c - d, \((a - c)\)\^2 + \((b - d)\)\^2, ArcTan[a - c, b - d]}\)], "Output"], Cell[BoxData[ \("Vector C4"\)], "Output"], Cell[BoxData[ \({1\/4\ \((a + b + 2\ \@c\ Cos[d])\), 1\/4\ \((a - b + 2\ \@c\ Sin[d])\), 1\/4\ \((a + b - 2\ \@c\ Cos[d])\), 1\/4\ \((a - b - 2\ \@c\ Sin[d])\)}\)], "Output"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\<\"If A=\"\>", "2.`", \(-3.`\), "5.`", "1.`", "\<\"polarA=\"\>", "5.`", "9.`", "25.`", \(-2.214297435588181`\)}, {"\<\"Vec(polar(A))=\"\>", "2.0000000000000004`", \(-3.0000000000000004`\), "5.`", "1.0000000000000004`", "\<\"=A\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>"}, {"\<\"and B=\"\>", \(-1\), "5", \(-6.`\), "3.`", "\<\"polarB=\"\>", "1.`", \(-15.`\), "29.`", "0.3805063771123649`"}, {"\<\"AB=\"\>", \(-36.`\), "22.`", \(-29.`\), "48.`", "\<\"polarAB=\"\>", "5.`", \(-135.`\), "725.`", \(-1.8337910584758161`\)}, {"\<\"\\!\\(\\@A\\)=\"\>", "1.8090169943749475`", \(-1.1909830056250525`\), "0.8090169943749473`", "0.8090169943749475`", "\<\"polar\\!\\(\\@A\\)=\"\>", "2.23606797749979`", "3.`", "5.`", \(-1.1071487177940904`\)} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Note that the linear sizes are \[OpenCurlyDoubleQuote]offsets\ \[CloseCurlyDoubleQuote] that displace the centres of rotation for the \ sinusoidal terms. They are conserved linear symmetries. Pairs of linear sizes \ can be combined to give conserved differences of squares, from which \ hyperbolic duals can be constructed. Non-abelian hoops have repeated roots; \ their duals are discussed later.\n\n(11) A few Hoops have stable orbits \ resembling multi-phase de Broglie waves; they define fundamental sizes \ resembling Planck areas and", StyleBox[" may determine which particles are stable. They have some \ elements (e.g. \[Zeta] in figure 6) that are neither real nor complex but act \ as quantum operators. This is discussed later.\n", PageWidth->WindowWidth, FontFamily->"Times New Roman"], "\n(12) Four \"Dozal\" multiplication rules (for vecs with 12 elements) may \ correspond to four types of interaction between particles, with different \ sizes corresponding to different conserved properties and remainders \ corresponding to emitted particles. C3K and C3C4 are abelian, Q12 & D3C2 are \ non abelian. The non-abelian rules have \"pseudo-polar duals\" with degrees \ of freedom that may introduce uncertainty. They are indirect compositions of \ C2 with (bosonic) abelian groups that appear to be \"supersymmetric\" to the \ (fermionic) larger groups. They have more symmetries (and more particle \ types) than the bosons. Supersymmetry is not one-to-one.\n\tFigure 6 shows \ the polar and vector forms for C3K. ", "This combines the two square roots of unity (Study numbers) of the Klein \ group K, with the cube root of unity of Terplex numbers. ", "Each line defines a single element of one form as a function of the \ elements of the other. Orbits are \[OpenCurlyDoubleQuote]unital\ \[CloseCurlyDoubleQuote]; their non-zero sizes multiply to one so that \ repeated products are unital.", StyleBox[" Different orbits are obtained by constraining some of the sizes \ {\[Alpha],\[Beta],\[Gamma],\[Delta],\[Zeta],\[Lambda],\[Eta],\[Kappa]} to be \ zero whilst the rest multiply to one. Note that \[Zeta] has three phases ", FontFamily->"Times New Roman"], "(repeated four times)", StyleBox[", whilst {\[Lambda],\[Eta],\[Kappa]} each have six; different \ combinations can give twelve phases. The C3C4 orbits (not shown) are closely \ related to those for C3K, but include a four-phase system in which \[Gamma] \ and \[Delta] are replaced by ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\^2\)]], StyleBox[" and \[Sigma]. The author conjectures that this relates to \ hadrons with quarks as 3, 6, and 12-phase orbits, and leptons as 4-phase \ orbits", FontFamily->"Times New Roman"], ".", StyleBox[" Orbits properties include spin, chirality and polarisation.", FontFamily->"Times New Roman"] }], "Text"], Cell["Figure 6. C3K Polar form and general orbit.", "Text", FontWeight->"Bold"], Cell[BoxData[ \({a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]a + b + c + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, 1\/2\ \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h \ - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\), ArcTan[2\ a - b - c + 2\ d - e - f + 2\ g - h - i + 2\ j - k - l, \(-\@3\)\ \((b - c + e - f + h - i + k - l)\)], 1\/2\ \((\((a - b - d + e + g - h - j + k)\)\^2 + \((\(-a\) + c + d - f \ - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\), ArcTan[2\ a - b - c - 2\ d + e + f + 2\ g - h - i - 2\ j + k + l, \(-\@3\)\ \((b - c - e + f + h - i - k + l)\)], 1\/2\ \((\((a - b + d - e - g + h - j + k)\)\^2 + \((\(-a\) + c - d + f \ + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\), ArcTan[2\ a - b - c + 2\ d - e - f - 2\ g + h + i - 2\ j + k + l, \(-\@3\)\ \((b - c + e - f - h + i - k + l)\)], 1\/2\ \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - c - e + f - h \ + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g - i - j + l)\)\^2)\), ArcTan[2\ a - b - c - 2\ d + e + f - 2\ g + h + i + 2\ j - k - l, \(-\@3\)\ \((b - c - e + f - h + i + k - l)\)]}\)], "Input", FontSize->9], Cell[BoxData[ \(\(\(\ \ \ \ \ \)\({\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ \ Cos[\[Tau]]\ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ + \[Eta]\ Cos[\ \[Chi]]\ \ \ \ \ \ + \[Kappa]\ Cos[\[Psi]], \[Alpha] + \[Beta] + \[Gamma] + \ \[Delta] + \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Lambda]\ Cos[\[Phi] + \ \[Omega]] + \[Eta]\ Cos[\[Chi] + \[Omega]] + \[Kappa]\ Cos[\[Psi] + \ \[Omega]], \ \[IndentingNewLine]\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \ \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ Cos[\[Phi] - \[Omega]] + \[Eta]\ \ Cos[\[Chi] - \[Omega]] + \[Kappa]\ Cos[\[Psi] - \[Omega]], \ \[IndentingNewLine]\[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ Cos[\ \[Tau]]\ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ + \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ \ - \[Kappa]\ Cos[\[Psi]], \ \[Alpha] - \[Beta] + \[Gamma] - \ \[Delta] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Lambda]\ Cos[\[Phi] + \ \[Omega]] + \[Eta]\ Cos[\[Chi] + \[Omega]] - \[Kappa]\ Cos[\[Psi] + \ \[Omega]], \ \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] - \ \[Omega]] - \[Lambda]\ Cos[\[Phi] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \ \[Omega]] - \[Kappa]\ Cos[\[Psi] - \[Omega]], \ \[Alpha] + \[Beta] - \[Gamma] \ - \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ - \[Kappa]\ Cos[\[Psi]], \ \[Alpha] + \ \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Lambda]\ \ Cos[\[Phi] + \[Omega]] - \[Eta]\ Cos[\[Chi] + \[Omega]] - \[Kappa]\ \ Cos[\[Psi] + \[Omega]], \ \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ \ Cos[\[Tau] - \[Omega]] + \[Lambda]\ Cos[\[Phi] - \[Omega]] - \[Eta]\ Cos[\ \[Chi] - \[Omega]] - \[Kappa]\ Cos[\[Psi] - \[Omega]], \ \[Alpha] - \[Beta] - \ \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ - \[Lambda]\ \ Cos[\[Phi]]\ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ + \[Kappa]\ \ Cos[\[Psi]], \ \[Alpha] - \[Beta] - \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau] \ + \[Omega]] - \[Lambda]\ Cos[\[Phi] + \[Omega]] - \[Eta]\ Cos[\[Chi] + \ \[Omega]] + \[Kappa]\ Cos[\[Psi] + \[Omega]], \ \[Alpha] - \[Beta] - \[Gamma] \ + \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ Cos[\[Phi] - \ \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]] + \[Kappa]\ Cos[\[Psi] - \ \[Omega]]}/12;\)\)\)], "Input", FontSize->9], Cell["\<\ \tThe following table (taken from GroupLoopHoop.nb, and omitted from the \ \"Word\" version) defines 39 functions that are conserved in one or more of \ the Dozal operations. The author has attempted to relate them to particle \ properties, without success. \ \>", "Text"], Cell[BoxData[ \(\(sh12[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}] := Module[{ab = a - b - g + h, ag = a - b + g - h, bc = b - c - h + i, bh = b - c + h - i, \[IndentingNewLine]ca = a - c - g + i, ci = a - c + g - i, de = d - e - j + k, dj = d - e + j - k, \[IndentingNewLine]ef = e - f - k + l, ek = e - f + k - l, fd = d - f - j + l, fl = d - f + j - l, f1 = a - b - g + h + \((d + e - 2\ f - j - k + 2 l)\)/\@3, \[IndentingNewLine]f2 = \((a + b - 2 c - g - h + 2 i)\)/\@3 - d + e + j - k, \[IndentingNewLine]f3 = a - b - g + h - \((d + e - 2\ f - j - k + 2 l)\)/\@3, \[IndentingNewLine]f4 = \((a + b - 2 c - g - h + 2 i)\)/\@3 + d - e - j + k, abef, agek}, \[IndentingNewLine]abef = 3 \((ab\ ef - bc\ de)\)\^2; agek = 3 \((ag\ ek - bh\ dj)\)\^2; \[IndentingNewLine]{o1 = a + b + c + d + e + f + g + h + i + j + k + l, o2 = a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]o3 = a + b + c + d + e + f - g - h - i - j - k - l, o4 = a + b + c - d - e - f - g - h - i + j + k + l, \[IndentingNewLine] (*5*) l22a = 3 \((f1\^2 + f2\^2)\)/4, (*6*) l22b = 3 \((f3\^2 + f4\^2)\)/4, \[IndentingNewLine] (*7*) p22 = \((a + b + c + g + h + i)\)\^2 + \((d + e + f + j + k + \ l)\)\^2, (*8*) q22 = \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - \ l)\)\^2, (*9*) r22 = \((a + b + c + g + h + i)\)\^2 - \((d + e + f + j + k + \ l)\)\^2, (*10*) s22 = \((a + b + c - g - h - i)\)\^2 - \((d + e + f - j - k - \ l)\)\^2, \[IndentingNewLine] (*11*) p23 = \((\((ag + dj)\)\^2 + \((bh + ek)\)\^2 + \((ci + \ fl)\)\^2)\)/2, (*12*) q23 = \((\((ag - dj)\)\^2 + \((bh - ek)\)\^2 + \((ci - \ fl)\)\^2)\)/2, \[IndentingNewLine] (*13*) r23 = \((\((ab + de)\)\^2 + \((bc + ef)\)\^2 + \((ca + \ fd)\)\^2)\)/2, (*14*) s23 = \((\((ab - de)\)\^2 + \((bc - ef)\)\^2 + \((ca - \ fd)\)\^2)\)/2, \[IndentingNewLine] (*15*) q24 = \((a + b + c)\)\^2 + \((d + e + f)\)\^2 + \((g + h + i)\)\ \^2 + \((j + k + l)\)\^2, \[IndentingNewLine] (*16*) s24 = \((a + b + c)\)\^2 - \((d + e + f)\)\^2 - \((g + h + i)\)\ \^2 - \((j + k + l)\)\^2, \[IndentingNewLine] (*17*) p26 = \((ag\^2 + bh\^2 + ci\^2 - dj\^2 - fl\^2 - ek\^2)\)/ 2, (*18*) q26 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)/ 2, \[IndentingNewLine] (*19*) r26 = \((ag\^2 + bh\^2 + ci\^2 + dj\^2 + fl\^2 + ek\^2)\)/ 2, (*20*) s26 = \((ab\^2 + bc\^2 + ca\^2 + de\^2 + ef\^2 + fd\^2)\)/ 2, \n\ \ \ \ \ \ \ \ (*21*) \ l3 = \((a + d - g - j)\)\ \((a - d + g - j)\)\ \((a - d - g + j)\) + \((b + e - h - k)\)\ \((b - e + h - k)\)\ \((b - e - h + k)\) + \[IndentingNewLine]\((c + f - \ i\ - l)\)\ \((c - f + \ i\ - l)\)\ \((c - f - \ i\ + l)\) - \((a - d + g - j)\)\ \((b - e - h + k)\)\ \((c + f - \ i\ - l)\) - \[IndentingNewLine]\((a - d - g + j)\)\ \((b + e - h - k)\)\ \((c - f + i - l)\) - \((a + d - g - j)\)\ \((b - e + h - k)\)\ \((c - f - \ i\ + l)\), \[IndentingNewLine] (*22*) l4p = p26\^2 + agek, \[IndentingNewLine] (*23*) l4q = q26\^2 + abef, \[IndentingNewLine] (*24*) l4r = r26\^2 - 3 agek, \[IndentingNewLine] (*25*) l4s = l22a\ l22b (*s26\^2 - abef*) , (*26*) L4Q = \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((a - c)\)\^2 + \ \((d - e)\)\^2 + \((e - f)\)\^2 + \((d - f)\)\^2 + \((g - h)\)\^2 + \ \[IndentingNewLine]\((h - i)\)\^2 + \((g - i)\)\^2 + \((j - k)\)\^2 + \((k - \ l)\)\^2 + \((j - l)\)\^2)\)^2/4 - 3\ \((\((a\ \((f - e)\) + b\ \((d - f)\) + c\ \((e - \ d)\))\)\^2 + \((a\ \((i - h)\) + b\ \((g - i)\) + c\ \((h - g)\))\)\^2 + \((d\ \ \((h - i)\) + e\ \((i - g)\) + f\ \((g - h)\))\)\^2 + \((g\ \((k - l)\) + h\ \ \((l - j)\) + i\ \((j - k)\))\)\^2 + \[IndentingNewLine]\((\((b - c)\)\^2 + \ \((e - f)\)\^2)\)\ j\^2 + \((\((a - c)\)\^2 + \((d - f)\)\^2)\)\ k\^2 + \ \((\((a - b)\)\^2 + \((d - e)\)\^2)\)\ l\^2)\) + \[IndentingNewLine]6\ \((\((\ \((c - a)\)\ \((c - b)\) + \((f - d)\)\ \((f - e)\))\)\ j\ k + \((\((b - a)\)\ \((b - c)\) + \((e - d)\)\ \((e - f)\))\)\ j\ l + \[IndentingNewLine]\((\((a \ - b)\)\ \((a - c)\) + \((d - e)\)\ \((d - f)\))\)\ k\ l)\), \[IndentingNewLine] (*27*) L4\[Sigma] = \ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((c - a)\)\ \^2 - \((d - e)\)\^2 - \((e - f)\)\^2 - \((f - d)\)\^2 - \((g - h)\)\^2 - \ \((h - i)\)\^2 - \((i - g)\)\^2 - \((j - k)\)\^2 - \((k - l)\)\^2 - \((l - j)\ \)\^2)\)^2/4 + 3\ \((\((b\ d - c\ d - a\ e + c\ e + a\ f - b\ f)\)\^2 + \((b\ \ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^2 - \((e\ g - f\ g - d\ h + f\ h + \ d\ i - e\ i)\)\^2 + \((b\ j - c\ j - a\ k + c\ k + a\ l - b\ l)\)\^2 - \ \((\(-e\)\ j + f\ j + d\ k - f\ k - d\ l + e\ l)\)\^2 - \((h\ j - i\ j - g\ k \ + i\ k + g\ l - h\ l)\)\^2)\), \[IndentingNewLine] (*28*) L4a3 = \((\((ag + dj)\)\^2 + \((bh + ek)\)\^2 + \((ci + \ fl)\)\^2)\)\^2/4 + agek, (*29*) L4b3 = \((\((ag - dj)\)\^2 + \((bh - ek)\)\^2 + \((ci - \ fl)\)\^2)\)\^2/4 + agek, (*30*) L4c3 = \((\((ab + de)\)\^2 + \((bc + ef)\)\^2 + \((ca + \ fd)\)\^2)\)\^2/4 - abef, (*31*) L4d3 = \((\((ab - de)\)\^2 + \((bc - ef)\)\^2 + \((ca - \ fd)\)\^2)\)\^2/4 - abef, (*32*) L4a6 = \((ag\^2 + bh\^2 + ci\^2 - dj\^2 - fl\^2 - ek\^2)\)\^2/ 4 + agek, (*33*) L4b6 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)\^2/ 4 + abef, (*34*) L4c6 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)\^2/ 4 - abef, \n\t (*35*) L4d6 = \ \((ag\^2 + bh\^2 + ci\^2 + dj\^2 + fl\^2 + ek\^2)\)\^2/ 4 - agek, \[IndentingNewLine] (*36*) L49 = \((\((ab\^2 - de\^2)\)\^2 + \((ab\^2 - ef\^2)\)\^2 + \ \((ab\^2 - fd\^2)\)\^2 + \((bc\^2 - de\^2)\)\^2 + \((bc\^2 - ef\^2)\)\^2 + \ \((bc\^2 - fd\^2)\)\^2 + \((ca\^2 - de\^2)\)\^2 + \((ca\^2 - ef\^2)\)\^2 + \ \((ca\^2 - fd\^2)\)\^2)\), \[IndentingNewLine] (*37*) L49a = \((\((ag\^2 - dj\^2)\)\^2 + \((ag\^2 - ek\^2)\)\^2 + \ \((ag\^2 - fl\^2)\)\^2 + \((bh\^2 - dj\^2)\)\^2 + \((bh\^2 - ek\^2)\)\^2 + \ \((bh\^2 - fl\^2)\)\^2 + \((ci\^2 - dj\^2)\)\^2 + \((ci\^2 - ek\^2)\)\^2 + \ \((ci\^2 - fl\^2)\)\^2)\), \[IndentingNewLine] (*38*) L49b = \((\((ab\^2 - dj\^2)\)\^2 + \((ab\^2 - ek\^2)\)\^2 + \ \((ab\^2 - fl\^2)\)\^2 + \((bc\^2 - dj\^2)\)\^2 + \((bc\^2 - ek\^2)\)\^2 + \ \((bc\^2 - fl\^2)\)\^2 + \((ca\^2 - dj\^2)\)\^2 + \((ca\^2 - ek\^2)\)\^2 + \ \((ca\^2 - fl\^2)\)\^2)\), \[IndentingNewLine] (*39*) L49c = \((\((ag\^2 - de\^2)\)\^2 + \((ag\^2 - ef\^2)\)\^2 + \ \((ag\^2 - fd\^2)\)\^2 + \((bh\^2 - de\^2)\)\^2 + \((bh\^2 - ef\^2)\)\^2 + \ \((bh\^2 - fd\^2)\)\^2 + \((ci\^2 - de\^2)\)\^2 + \((ci\^2 - ef\^2)\)\^2 + \ \((ci\^2 - fd\^2)\)\^2)\)}];\)\)], "Input", PageWidth->WindowWidth, CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, PageBreakWithin->Automatic, FontSize->9], Cell[TextData[{ "(13) The real division algebras \[DoubleStruckCapitalR], \ \[DoubleStruckCapitalC], \[DoubleStruckCapitalH], \[DoubleStruckCapitalO] are \ degenerate (monosized) hoops without divisors of zero because their sizes are \ positive, being the sums of the squared elements. These hoops operate \ correctly if their coefficients are complex, but then have divisors of zero. \ This gives the light cone in the complex quaternion case.\n\n(14) Hoops \ subsume, but do not invalidate, complex mathematics. Complex functions are \ specializations of more general functions. Many standard mathematical \ operations are specialisations of more general hoop operations; many \ \"obvious mathematical truths\" are special cases restricted to the \"real \ algebras without divisors of zero\". E.g. repeated products and powers (", StyleBox["AA", FontSlant->"Italic"], " and ", Cell[BoxData[ \(TraditionalForm\`A\^2\)]], ") may differ; vector division is defined; both multiplication and division \ of vectors may leave remainders." }], "Text"], Cell[TextData[{ "\tThe author believes that \"Truth transcends Proof\", and so relies on \ demonstrations of the existence and properties of concepts, even though he \ cannot prove their consistency. This document demonstrates the properties of \ many Hoop algebras, using a ", StyleBox["Mathematica", FontSlant->"Italic"], " package ", StyleBox["hoops.m", FontSlant->"Italic"], " and instructions with output that is meant to be self-explanatory to \ non-", StyleBox["Mathematica", FontSlant->"Italic"], " users. A separate notebook ", StyleBox["hoops.nb", FontSlant->"Italic"], " provides the package as a databank of over 80 Hoops (with 5 \ counter-example non-hoop Loops), three main procedures ", StyleBox["hoopTimes, hoopInverse, hoopPower", FontSlant->"Italic"], ", and a few subsidiary procedures. The package is written in elementary ", StyleBox["Mathematica", FontSlant->"Italic"], ", and includes Usage and (* comments *) explaining the details, so it \ should be readily comprehensible and easy to transcribe into other \ computer-aided-mathematics systems for independent verification. A larger \ package (creation and properties of groups, loops, & hoops) is available in \ [2], which also provides a glossary.\nPlease send any comments to me at ", StyleBox["rhberesford@btinternet.com", FontSlant->"Italic"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["2. Demonstrations of Hoop Properties.", "Section", FontFamily->"Times New Roman"], Cell["\<\ Summary. \tA set of examples demonstrates multiplication and division of vecs in \ different hoop algebras, the \"folding\" of loops to signed algebras with \ conserved sizes and remainders, polar-duals with powers and roots, \ non-Abelian algebras with rotated roots, and the unification of many algebras \ by the Hoop concept. Then \"Orbits\" are introduced, as unital dual \ sub-algebras with relevance to physics.\ \>", "Text", PageWidth->WindowWidth, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[TextData[{ "2.1 ", StyleBox["S", FontSlant->"Italic"], "electing Hoop Algebras." }], "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tA specific Hoop is selected by ", StyleBox["Use[\"H\"]", FontSlant->"Italic"], ", where ", StyleBox["\"H\"", FontSlant->"Italic"], " is any Hoop name. This assigns values to the variables {", StyleBox["mm, nn, hoopTbl, sh, topol, tovec, gi, gp, plex", FontSlant->"Italic"], "}. The Cayley table is put into ", StyleBox["hoopTbl", FontSlant->"Italic"], ". ", StyleBox["sh", FontSlant->"Italic"], " is the shape. ", StyleBox["topol", FontSlant->"Italic"], " is the polar form (if known); it defines angles to go with each quadratic \ size in ", StyleBox["sh", FontSlant->"Italic"], ". ", StyleBox["tovec", FontSlant->"Italic"], " is the reversion from polar to vector form. This data allows the \ procedures ", StyleBox["hoopTimes, hoopInverse,", FontSlant->"Italic"], " &", StyleBox[" hoopPower", FontSlant->"Italic"], " to calculate vector products, inverses, powers, and roots. (Vector \ addition is element-by-element for all hoops with the same number of \ elements.)\n\tExample 1 shows the index table and the symbolic form for the \ \"C3\" algebra, and then gives the shape, the polar form and the vector \ reversion. (Index tables will be shown in later examples if they have not \ been shown previously and are not too large; later input instructions will be \ hidden in \"closed cells\" because they may be incomprehensible to non-", StyleBox["Mathematica", FontSlant->"Italic"], " users.) Please ignore the commas that separate explanatory messages and \ the calculated results." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\( (*Example\ 1*) \), RowBox[{"{", RowBox[{\("\" // bf\), " ", ",", "\"\\"", ",", \(Use["\"]; hoopTbl // tf\), ",", "\"\<\\nThe symbolic form is \>\"", ",", \(s = hoopTbl /. Table[ii \[Rule] alph\[LeftDoubleBracket]ii\[RightDoubleBracket], {ii, mm}]; s // tf\), ",", "\"\\"", ",", \(Det[s]\), ",", RowBox[{\(s =. \), ";", "\"\<\\nthis factorises into the shape \!\(\* StyleBox[\"sh\",\nFontSlant->\"Italic\"]\)\>\""}], ",", "sh", ",", "\"\<\\ntopol is\>\"", ",", "topol", ",", "\"\<\\ntovec is\>\"", ",", "tovec", ",", "\[IndentingNewLine]", \("\<\nExample 1b\>" // bf\), ",", "\"\< C3 Table and properties.\\nThe C3 index table is\>\"", ",", \(Use["\"]; hoopTbl // tf\), ",", "\"\<\\nThe symbolic form is \>\"", ",", \(s = hoopTbl /. Table[ii \[Rule] alph\[LeftDoubleBracket]ii\[RightDoubleBracket], {ii, mm}]; s // tf\), ",", "\"\\"", ",", \(Det[s]\), ",", RowBox[{\(s =. \), ";", "\"\<\\nthis factorises into the shape \!\(\* StyleBox[\"sh\",\nFontSlant->\"Italic\"]\)\>\""}], ",", "sh", ",", "\"\<\\ntopol is\>\"", ",", "topol", ",", "\"\<\\ntovec is\>\"", ",", "tovec", ",", "\[IndentingNewLine]", \("\<\nExample 1c\>" // bf\), " ", ",", "\"\< C4 Table and properties.\\nThe C4 index table is\>\"", ",", \(Use["\"]; hoopTbl // tf\), ",", "\"\<\\nThe symbolic form is \>\"", ",", \(s = hoopTbl /. Table[ii \[Rule] alph\[LeftDoubleBracket]ii\[RightDoubleBracket], {ii, mm}]; s // tf\), ",", "\"\\"", ",", \(Det[s]\), ",", RowBox[{\(s =. \), ";", "\"\<\\nthis factorises into the shape \!\(\* StyleBox[\"sh\",\nFontSlant->\"Italic\"]\)\>\""}], ",", "sh", ",", "\"\<\\ntopol is\>\"", ",", "topol", ",", "\"\<\\ntovec is\>\"", ",", "tovec"}], "}"}]}]], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ StyleBox["\<\"Example 1a\"\>", FontWeight->"Bold"], (StyleForm[ #, FontWeight -> "Bold"]&)], ",", "\<\"C2 Table and properties.\\nThe C2 index table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2"}, {"2", "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nThe symbolic form is \"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b"}, {"b", "a"} }], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"with determinant\"\>", ",", \(a\^2 - b\^2\), ",", "\<\"\\nthis factorises into the shape \\!\\(\\* StyleBox[\\\"sh\ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\"\>", ",", \({a - b, a + b}\), ",", "\<\"\\ntopol is\"\>", ",", \({a - b, a + b}\), ",", "\<\"\\ntovec is\"\>", ",", \({\(a + b\)\/2, 1\/2\ \((\(-a\) + b)\)}\), ",", TagBox[ StyleBox["\<\"\\nExample 1b\"\>", FontWeight->"Bold"], (StyleForm[ #, FontWeight -> "Bold"]&)], ",", "\<\" C3 Table and properties.\\nThe C3 index table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3"}, {"2", "3", "1"}, {"3", "1", "2"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nThe symbolic form is \"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b", "c"}, {"b", "c", "a"}, {"c", "a", "b"} }], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"with determinant\"\>", ",", \(\(-a\^3\) - b\^3 + 3\ a\ b\ c - c\^3\), ",", "\<\"\\nthis factorises into the shape \\!\\(\\* StyleBox[\\\"sh\ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\"\>", ",", \({a + b + c, 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + \ c)\)\^2)\)}\), ",", "\<\"\\ntopol is\"\>", ",", \({a + b + c, 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2)\), ArcTan[2\ a - b - c, \(-\@3\)\ \((b - c)\)]}\), ",", "\<\"\\ntovec is\"\>", ",", \({1\/3\ \((a + 2\ \@b\ Cos[c])\), 1\/3\ \((a + 2\ \@b\ Cos[c + \(2\ \[Pi]\)\/3])\), 1\/3\ \((a + 2\ \@b\ Cos[c - \(2\ \[Pi]\)\/3])\)}\), ",", TagBox[ StyleBox["\<\"\\nExample 1c\"\>", FontWeight->"Bold"], (StyleForm[ #, FontWeight -> "Bold"]&)], ",", "\<\" C4 Table and properties.\\nThe C4 index table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4"}, {"2", "3", "4", "1"}, {"3", "4", "1", "2"}, {"4", "1", "2", "3"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nThe symbolic form is \"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b", "c", "d"}, {"b", "c", "d", "a"}, {"c", "d", "a", "b"}, {"d", "a", "b", "c"} }], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"with determinant\"\>", ",", \(\(-a\^4\) + b\^4 - 4\ a\ b\^2\ c + 2\ a\^2\ c\^2 - c\^4 + 4\ a\^2\ b\ d + 4\ b\ c\^2\ d - 2\ b\^2\ d\^2 - 4\ a\ c\ d\^2 + d\^4\), ",", "\<\"\\nthis factorises into the shape \\!\\(\\* \ StyleBox[\\\"sh\\\",\\nFontSlant->\\\"Italic\\\"]\\)\"\>", ",", \({a + b + c + d, a - b + c - d, \((a - c)\)\^2 + \((b - d)\)\^2}\), ",", "\<\"\\ntopol is\"\>", ",", \({a + b + c + d, a - b + c - d, \((a - c)\)\^2 + \((b - d)\)\^2, ArcTan[a - c, b - d]}\), ",", "\<\"\\ntovec is\"\>", ",", \({1\/4\ \((a + b + 2\ \@c\ Cos[d])\), 1\/4\ \((a - b + 2\ \@c\ Sin[d])\), 1\/4\ \((a + b - 2\ \@c\ Cos[d])\), 1\/4\ \((a - b - 2\ \@c\ Sin[d])\)}\)}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.2 Vector multiplication and division.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tGeneralized vectors (abbreviated to \"vecs\") are lists of ", StyleBox["m", FontSlant->"Italic"], " elements ", StyleBox["A=", FontSlant->"Italic"], "{", Cell[BoxData[ \(TraditionalForm\`\(\(a\_1\)\(,\)\)\)]], Cell[BoxData[ \(TraditionalForm\`a\_2\)]], ",...", Cell[BoxData[ \(TraditionalForm\`a\_m\)]], "}. Their components can be numbers (integer, real, complex, or unsigned), \ operators, or symbols; the \"standard form\" uses symbols from a table called \ ", StyleBox["alph", FontSlant->"Italic"], ", i.e. A={", StyleBox["a,b,c, ", FontSlant->"Italic"], "...}. Vectors can also be thought of as summations of coefficients with \ associated dimensions, signs or directions, A = ", Cell[BoxData[ \(TraditionalForm\`a\_1\)]], Cell[BoxData[ \(TraditionalForm\`d\_1\)]], " +", Cell[BoxData[ \(TraditionalForm\`a\_2\)]], Cell[BoxData[ \(TraditionalForm\`d\_2\)]], " +...+ ", Cell[BoxData[ FormBox[ RowBox[{\(a\_m\), FormBox[\(d\_m\), "TraditionalForm"]}], TraditionalForm]]], ".\n\tGeneralized Hoop multiplication is effected by ", StyleBox["hoopTimes[A,B]", FontSlant->"Italic"], ". The elements of the product ", StyleBox["AB ", FontSlant->"Italic"], "are the sum of each ", StyleBox["a\[LeftDoubleBracket]k\[RightDoubleBracket] \ b\[LeftDoubleBracket]l\[RightDoubleBracket] Sign[hoopTbl[[k,l]]] ", FontSlant->"Italic"], "where Table[[k,l]] contains the (possibly signed) index ", StyleBox["j", FontSlant->"Italic"], ".", StyleBox[" ", FontSlant->"Italic"], "The Moufang property ensures that every vec has a multiplicative inverse; \ division of B by A is ", StyleBox["hoopTimes[hoopInverse[A],B].", FontSlant->"Italic"], "\n\tHoops are \"conservative\" - the real factors of the symbolic \ determinant are \"size\" functions that are conserved on multiplication, \ size[AB] = \[PlusMinus] size[A]size[B]. The sizes are supplied in ", StyleBox["sh", FontSlant->"Italic"], ". The inverse has been found by Cramer's method, so the determinant is a \ denominator of the inverse; multiple sizes allow inverses to split into \ partial fractions.\n\tExample 2 demonstrates multiplication and shape \ conservation in the C2 and C3 algebras; both have two sizes. Only the output \ is shown; commas separate a description of the input from the resulting \ output. Note that the sizes of AB are the product of the sizes of A and B. \ (Section 2.8 shows that this is untrue for many loops.)" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\<\nThe C2 table is\>", hoopTbl // tf, "\", sh, "\<\nIf A = \>", A = {3, 4}, "\<\nand B = \>", B = {x, y}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nShape of A = \>", sh /. \ as[A], "\<\nShape of B = \>", sh /. \ as[B], "\<\nShape of AB = \>", sh /. \ as[AB], Use["\"]; "\<\n\nChange to C3.\nIf A = \>", A = {3, 4, 1}, "\<\nand B = \>", B = {5, 1, 2}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nShape of A = \>", sh /. \ as[A], "\<\nShape of B = \>", sh /. \ as[B], "\<\nShape of AB = \>", sh /. \ as[AB]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 2. C2 & C3 multiplication & size conservation\"\>", ",", "\<\"\\nThe C2 table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2"}, {"2", "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"with shape\"\>", ",", \({a - b, a + b}\), ",", "\<\"\\nIf A = \"\>", ",", \({3, 4}\), ",", "\<\"\\nand B = \"\>", ",", \({x, y}\), ",", "\<\"\\nAB = \"\>", ",", \({3\ x + 4\ y, 4\ x + 3\ y}\), ",", "\<\"\\nShape of A = \"\>", ",", \({\(-1\), 7}\), ",", "\<\"\\nShape of B = \"\>", ",", \({x - y, x + y}\), ",", "\<\"\\nShape of AB = \"\>", ",", \({\(-x\) + y, 7\ x + 7\ y}\), ",", "\<\"\\n\\nChange to C3.\\nIf A = \"\>", ",", \({3, 4, 1}\), ",", "\<\"\\nand B = \"\>", ",", \({5, 1, 2}\), ",", "\<\"\\nAB = \"\>", ",", \({24, 25, 15}\), ",", "\<\"\\nShape of A = \"\>", ",", \({8, 7}\), ",", "\<\"\\nShape of B = \"\>", ",", \({8, 13}\), ",", "\<\"\\nShape of AB = \"\>", ",", \({64, 91}\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Example 3 calculates the inverses of A & B and shows that their sizes are \ inverses of the sizes of A & B. Then ", StyleBox["hoopTimes", FontSlant->"Italic"], " is used to show that Ainverse.AB recovers B, whilst AB.Binverse recovers \ A. (In general, the inverses are \"Left-inverses\"; C3 is abelian so the \ distinction is irrelevant here.)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\<\nAinverse = \>", Ai = hoopInverse[A], "\<\nShape of Ainverse = \>", sh /. \ as[Ai], "\<\nBinverse = \>", Bi = hoopInverse[B], "\<\nShape of Binverse = \>", sh /. \ as[Bi], "\<\nAinverse.AB recovers B\>", hoopTimes[Ai, AB], "\<\nAB.Ainverse recovers A\>", hoopTimes[AB, Bi]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ \({"Example 3. C3 division", "\nAinverse = ", {5\/56, \(-\(11\/56\)\), 13\/56}, "\nShape of Ainverse = ", {1\/8, 1\/7}, "\nBinverse = ", {23\/104, \(-\(1\/104\)\), \(-\(9\/104\)\)}, "\nShape of Binverse = ", {1\/8, 1\/13}, "\nAinverse.AB recovers B", {5, 1, 2}, "\nAB.Ainverse recovers A", {3, 4, 1}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.3 \"Folding\" Vectors and Loops.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tA key concept in Hoop algebra is the \"Folding\" of a vector via an \ equivalence relationship \"~\" such as ", StyleBox["r[[i]]=a[[i]]-a[[i+m/2]] ~ {a[[i]],a[[i+m/2]]}", FontSlant->"Italic"], ", which creates a single real number by the \"2-folding\" of pairs of \ unsigned numbers and introduces the sign \"-\". Similarly, ", StyleBox["t[[i]] ~ a[[i]]+J a[[i+m/3]]+JJ a[[i+2m/3]]", FontSlant->"Italic"], " folds three unsigned numbers to a \"terplex\" number (with signs ", StyleBox["J ", FontSlant->"Italic"], "& ", StyleBox["JJ", FontSlant->"Italic"], ").\nOne way to fold four unsigned numbers is ", StyleBox["c[[i]] ~ a[[i]]+", FontSlant->"Italic"], StyleBox["i ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["a[[i+m/4]]-a[[i+2m/4]]-", FontSlant->"Italic"], StyleBox["i ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["a[[i+3m/4]].", FontSlant->"Italic"], " This creates a complex number and the signs ", StyleBox["i ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["&", FontSlant->"Italic"], " ", StyleBox["-", FontSlant->"Italic"], StyleBox["i", FontWeight->"Bold", FontSlant->"Italic"], ".\nThe multiplication tables also fold; an ", StyleBox["m\[Times]m", FontSlant->"Italic"], " table with ", StyleBox["r", FontSlant->"Italic"], "-fold symmetry can undergo ", StyleBox["r", FontSlant->"Italic"], "-folding to an ", StyleBox["(m/r)\[Times](m/r)", FontSlant->"Italic"], " table. Following Sir William Hamilton, sets ", StyleBox["m", FontSlant->"Italic"], " of unsigned numbers are \"directors\" with \"", StyleBox["m ", FontSlant->"Italic"], "directions\"; they fold to vectors with \"", StyleBox["m/r", FontSlant->"Italic"], " dimensions\". \n\tExample 4 shows that \"C4\" folds to complex algebra. \ It demonstrates multiplication, division (multiplication by an inverse), \ raising to a power, and extracting a root. In each case, the result is shown \ to fold to the corresponding complex result:-" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Use["\"]\), ";", RowBox[{"{", RowBox[{ "\"\\"", ",", \(hoopTbl // tf\), ",", "\"\\"", ",", \({{1, i, \(-1\), \(-i\)}, {i, \(-1\), \(-i\), 1}, {\(-1\), \(-i\), 1, i}, {\(-i\), 1, i, \(-1\)}} // tf\), ",", "\"\<\\nMultiplication using\!\(\* StyleBox[\" \",\nFontWeight->\"Plain\"]\)\!\(\* StyleBox[\"hoopTimes\",\nFontWeight->\"Plain\",\n\ FontSlant->\"Italic\"]\):-\\n If A =\>\"", ",", \(A = {3. , 1. , 1. , 2. }\), ",", "\"\\"", ",", \(B = {4. , 7. , 2. , 3. }\), ",", "\"\<\\nmultiplication \!\(\* StyleBox[\"gives\",\nFontSlant->\"Plain\"]\) \\n AB =\>\"", ",", \(AB = hoopTimes[A, B]\), ",", "\[IndentingNewLine]", "\"\<\\nThis 'folds' to the product of 3-1 + \ (1-2)\[ImaginaryI]\\nand 4-2 + (7-3)\[ImaginaryI] to give 31-23 + (32-26)\ \[ImaginaryI],\\ni.e. (2-\[ImaginaryI])(2+4\[ImaginaryI])=\>\"", " ", ",", \(\((2 - \[ImaginaryI])\) \((2 + 4 \[ImaginaryI])\)\), ",", "\"\<\\n\\nDivision using \!\(\* StyleBox[\"hoopInverse\",\nFontWeight->\"Plain\",\nFontSlant->\"Italic\"]\):-\ \\nThe inverse of A is\>\"", ",", \(Ainv = hoopInverse[{3, 1, 1, 2}]\), ",", "\"\<\\nAinv.A is the unit\>\"", ",", \(hoopTimes[Ainv, A]\), ",", "\"\<\\nand Ainv.AB recovers B\>\"", ",", \(Simplify[hoopTimes[Ainv, AB]]\), ",", "\"\<\\n\\nPowers and Roots:-\\nhoopPower\!\(\* StyleBox[\"[\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"A\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"]\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"gives\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"the\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"cube\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\) \\nA3=\>\"", ",", \(A3 = hoopPower[A, 3]\), ",", "\"\\"", ",", \(\((2 - \[ImaginaryI])\)^3\), ",", "\"\<\\n\!\(\* StyleBox[\"hoopPower\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"[\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"A3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"/\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"]\",\nFontSlant->\"Italic\"]\) gives the 2/3 power of A3\>\"", ",", \(hoopPower[A3, 2/3]\), ",", "\"\\"", ",", \(AA = hoopPower[A, 2]\), ",", "\"\<\\nA^3 is the same as the repeated product AAA\>\"", ",", \(AAA = hoopTimes[A, hoopTimes[A, A]]\)}], "}"}]}]], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 4. C4 folds to Complex algebra.\\nThe table \"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4"}, {"2", "3", "4", "1"}, {"3", "4", "1", "2"}, {"4", "1", "2", "3"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"can be written as\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "i", \(-1\), \(-i\)}, {"i", \(-1\), \(-i\), "1"}, {\(-1\), \(-i\), "1", "i"}, {\(-i\), "1", "i", \(-1\)} }], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nMultiplication using\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontWeight->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"hoopTimes\\\",\\nFontWeight->\\\"Plain\\\",\\nFontSlant->\\\"\ Italic\\\"]\\):-\\n If A =\"\>", ",", \({3.`, 1.`, 1.`, 2.`}\), ",", "\<\"and B =\"\>", ",", \({4.`, 7.`, 2.`, 3.`}\), ",", "\<\"\\nmultiplication \\!\\(\\* \ StyleBox[\\\"gives\\\",\\nFontSlant->\\\"Plain\\\"]\\) \\n AB =\"\>", ",", \({31.`, 32.`, 23.`, 26.`}\), ",", "\<\"\\nThis 'folds' to the product of 3-1 + \ (1-2)\[ImaginaryI]\\nand 4-2 + (7-3)\[ImaginaryI] to give 31-23 + (32-26)\ \[ImaginaryI],\\ni.e. (2-\[ImaginaryI])(2+4\[ImaginaryI])=\"\>", ",", \(8 + 6\ \[ImaginaryI]\), ",", "\<\"\\n\\nDivision using \\!\\(\\* \ StyleBox[\\\"hoopInverse\\\",\\nFontWeight->\\\"Plain\\\",\\nFontSlant->\\\"\ Italic\\\"]\\):-\\nThe inverse of A is\"\>", ",", \({17\/35, \(-\(4\/35\)\), 3\/35, \(-\(11\/35\)\)}\), ",", "\<\"\\nAinv.A is the unit\"\>", ",", \({0.9999999999999998`, 0.`, 0.`, 0.`}\), ",", "\<\"\\nand Ainv.AB recovers B\"\>", ",", \({4.`, 6.999999999999999`, 2.`, 3.0000000000000018`}\), ",", "\<\"\\n\\nPowers and Roots:-\\nhoopPower\\!\\(\\* \ StyleBox[\\\"[\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"A\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\",\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"3\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"]\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \\\ \",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"gives\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"the\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"cube\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\",\ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\) \\nA3=\"\>", ",", \({87.`, 80.`, 85.`, 91.`}\), ",", "\<\"which folds to (2-\[ImaginaryI])^3=\"\>", ",", \(2 - 11\ \[ImaginaryI]\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"hoopPower\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"[\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"A3\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\",\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"2\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"/\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"3\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"]\\\",\\nFontSlant->\\\"Italic\\\"]\\) gives the 2/3 power of A3\ \"\>", ",", \({13.999999999999996`, 9.999999999999996`, 10.999999999999996`, 13.999999999999996`}\), ",", "\<\"which folds to 3 - 4i and matches the repeated product AA\"\ \>", ",", \({14.`, 10.`, 11.`, 14.`}\), ",", "\<\"\\nA^3 is the same as the repeated product AAA\"\>", ",", \({87.`, 80.`, 85.`, 91.`}\)}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.4 Powers & repeated products may differ.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tExample 4 checked whether ", Cell[BoxData[ \(TraditionalForm\`A\^2\)]], " = ", StyleBox["AA", FontSlant->"Italic"], " and ", Cell[BoxData[ \(TraditionalForm\`A\^3\)]], " = ", StyleBox["AAA", FontSlant->"Italic"], ". Powers and repeated products may differ for two reasons, negative sizes \ and \"angle wrap-round\". If a size is negative for A it alternates in sign \ for repeated products but the power should be a continuous function of the \ exponent, as in Example 5." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Use["\"]\), ";", RowBox[{"{", RowBox[{ "\"\\"", ",", \(A = {1. , 1. , \(-1. \), 0. }\), ",", "\[IndentingNewLine]", "\"\< with shape \>\"", ",", \(sh /. as[A]\), ",", "\"\<\\n\!\(\* StyleBox[\"hoopPower\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"[\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"A\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"]\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"gives\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"the\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"cube\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\\\ \)]\)A3= \>\"", ",", \(A3 = hoopPower[A, 3]\), ",", "\"\<\\nwhich folds to 2+11i.\\n\!\(\* StyleBox[\"hoopPower\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"[\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"A3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"2\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"/\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"]\",\nFontSlant->\"Italic\"]\) gives the 2/3 power of A3\>\"", ",", \(hoopPower[A3, 2/3]\), ",", "\"\<\\nwhich folds to 3+4i and matches A^2\>\"", ",", \(A2 = hoopPower[A, 2]\), ",", "\[IndentingNewLine]", "\"\<\\nwith shape\>\"", ",", \(sh /. as[A2]\), ",", "\"\<\\nwhilst the repeated product AA\>\"", ",", \(AA = hoopTimes[\ A, A]\), ",", "\[IndentingNewLine]", "\"\<\\nhas the second size positive\>\"", ",", \(sh /. as[AA]\), ",", "\"\<\\nA^3 is the same as the repeated product AAA\>\"", ",", \(AAA = hoopTimes[A, hoopTimes[A, A]]\), ",", "\"\<\\nThe second size of A^2 is negative, of AA is \ positive:-\\nPolar A =\>\"", ",", \(topol /. \ as[A]\), ",", "\"\<\\nPolar A^2 =\>\"", ",", \(topol /. \ as[A2]\), ",", "\"\<\\nPolar AA =\>\"", ",", \(topol /. \ as[AA]\), ",", "\"\<\\nPolar AAA =\>\"", ",", \(topol /. \ as[A3]\), ",", "\"\<\\n\\n A^1.99\>\"", ",", \(hoopPower[A, 1.99]\), ",", "\"\<\\n& A^2.01\>\"", ",", \(hoopPower[A, 2.01]\), ",", "\"\<\\n bracket A^2\\nwhilst AA =\>\"", ",", "AA"}], "}"}]}]], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 5. C4 with negative size., so AA differs from \ A^2\\nIf A =\"\>", ",", \({1.`, 1.`, \(-1.`\), 0.`}\), ",", "\<\" with shape \"\>", ",", \({1.`, \(-1.`\), 5.`}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"hoopPower\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"[\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"A\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\",\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"3\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"]\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \\\ \",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"gives\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"the\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"cube\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\",\ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\\\ \\)]\\)A3= \"\>", ",", \({1.0000000000000007`, 6.`, \(-1.0000000000000007`\), \(-5.`\)}\), ",", "\<\"\\nwhich folds to 2+11i.\\n\\!\\(\\* StyleBox[\\\"hoopPower\ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"[\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"A3\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\",\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"2\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"/\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"3\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"]\\\",\\nFontSlant->\\\"Italic\\\"]\\) gives the 2/3 power of A3\ \"\>", ",", \({1.5`, 2.4999999999999996`, \(-1.5`\), \(-1.4999999999999996`\)}\), ",", "\<\"\\nwhich folds to 3+4i and matches A^2\"\>", ",", \({1.5000000000000002`, 2.5`, \(-1.5000000000000002`\), \(-1.4999999999999998`\)}\), ",", "\<\"\\nwith shape\"\>", ",", \({1.0000000000000002`, \(-1.0000000000000002`\), 25.000000000000004`}\), ",", "\<\"\\nwhilst the repeated product AA\"\>", ",", \({2.`, 2.`, \(-1.`\), \(-2.`\)}\), ",", "\<\"\\nhas the second size positive\"\>", ",", \({1.`, 1.`, 25.`}\), ",", "\<\"\\nA^3 is the same as the repeated product AAA\"\>", ",", \({1.`, 6.`, \(-1.`\), \(-5.`\)}\), ",", "\<\"\\nThe second size of A^2 is negative, of AA is \ positive:-\\nPolar A =\"\>", ",", \({1.`, \(-1.`\), 5.`, 0.4636476090008061`}\), ",", "\<\"\\nPolar A^2 =\"\>", ",", \({1.0000000000000002`, \(-1.0000000000000002`\), 25.000000000000004`, 0.9272952180016122`}\), ",", "\<\"\\nPolar AA =\"\>", ",", \({1.`, 1.`, 25.`, 0.9272952180016122`}\), ",", "\<\"\\nPolar AAA =\"\>", ",", \({1.`, \(-1.`\), 125.`, 1.3909428270024182`}\), ",", "\<\"\\n\\n A^1.99\"\>", ",", \({1.4971602576129839`, 2.4770499321073896`, \(-1.4971602576129839`\), \ \(-1.4770499321073898`\)}\), ",", "\<\"\\n& A^2.01\"\>", ",", \({1.5027553889648462`, 2.5231485205881197`, \(-1.5027553889648462`\), \ \(-1.5231485205881197`\)}\), ",", "\<\"\\n bracket A^2\\nwhilst AA =\"\>", ",", \({2.`, 2.`, \(-1.`\), \(-2.`\)}\)}], "}"}]], "Output"] }, Open ]], Cell["\<\ \tIf a polar-form angle parameter is large, raising to a power (which \ multiplies the angle by the exponent) may give an angle greater than 2\[Pi]. \ This will become a negative angle and the roots will be rotated. Their sizes \ will be correct. (Taking roots makes the angle parameter smaller, and so \ cannot cause wrap-round.)\ \>", "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Use["\"]\), ";", RowBox[{"{", RowBox[{ "\"\\"", ",", \(A = {2. , 2. , 1. , 0. }\), ",", "\"\\"Italic\"]\)\!\(\* StyleBox[\"[\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"A\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"]\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"gives\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"the\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"cube\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\) A3= \>\"", ",", \(A3 = hoopPower[A, 3]\), ",", "\"\<\\nwhich folds to -11-2i\\n\!\(\* StyleBox[\"hoopPower\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"[\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"A3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"1\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"/\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"3\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"]\",\nFontSlant->\"Italic\"]\) gives rA3, the 1/3 power of A3\>\"", ",", \(rA3 = hoopPower[A3, 1/3]\), ",", "\"\<\\nwhich folds to 1.232-1.866i (which cubes correctly)\\nrA3 \ differs from A, but has the same sizes\>\"", ",", "\"\<\\nThe angle of A^3 exceeds 2Pi and so is negated:-\\nPolar A \ =\>\"", ",", \(topol /. \ as[A]\), ",", "\"\<\\nPolar AAA =\>\"", ",", \(topol /. \ as[A3]\), ",", "\"\<\\nPolar rA3 =\>\"", ",", \(topol /. \ as[rA3]\)}], "}"}]}]], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 6. C4 with large angle. (A^3)^(1/3) differs from \ A.\\nIf A =\"\>", ",", \({2.`, 2.`, 1.`, 0.`}\), ",", "\<\"which folds to 1+2i\\n\\!\\(\\* StyleBox[\\\"hoopPower\\\",\ \\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"[\\\",\\nFontSlant->\\\ \"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"A\\\",\\nFontSlant->\\\"Italic\\\"]\\)\ \\!\\(\\* StyleBox[\\\",\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"3\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"]\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"gives\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"the\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* \ StyleBox[\\\"cube\\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\",\ \\\",\\nFontSlant->\\\"Plain\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Plain\\\"]\\) A3= \"\>", ",", \({26.`, 30.`, 37.`, 32.`}\), ",", "\<\"\\nwhich folds to -11-2i\\n\\!\\(\\* StyleBox[\\\"hoopPower\ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"[\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"A3\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\",\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"1\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"/\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\"3\\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"]\\\",\\nFontSlant->\\\"Italic\\\"]\\) gives rA3, the 1/3 power \ of A3\"\>", ",", \({2.1160254037844384`, 0.0669872981077807`, 0.8839745962155612`, 1.9330127018922187`}\), ",", "\<\"\\nwhich folds to 1.232-1.866i (which cubes \ correctly)\\nrA3 differs from A, but has the same sizes\"\>", ",", "\<\"\\nThe angle of A^3 exceeds 2Pi and so is negated:-\\nPolar \ A =\"\>", ",", \({5.`, 1.`, 5.`, 1.1071487177940904`}\), ",", "\<\"\\nPolar AAA =\"\>", ",", \({125.`, 1.`, 125.`, \(-2.961739153797315`\)}\), ",", "\<\"\\nPolar rA3 =\"\>", ",", \({4.999999999999999`, 1.`, 4.9999999999999964`, \(-0.9872463845991049`\)}\)}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.5 Signed Tables, Generalized Signs.", "Subsection", PageWidth->PaperWidth], Cell[TextData[{ "\tMany Hoops are \"signed tables\" with signed products in the body of the \ multiplication table, but the elements are unsigned (unlike the second table \ in Example 4, where two elements are \"labeled\" -1 & -i). As these products \ are not members of the defining set, such tables are not loops. Signed tables \ are equivalence relations, folded from unsigned Moufang loops with ", StyleBox["r", FontSlant->"Italic"], "-fold symmetry. The commonest cases involve negation, where ", StyleBox["r", FontSlant->"Italic"], " = 2 and ", "the signed table is obtained from the top left quarter of the Moufang \ loop. Complex algebra is the \"C4c\" hoop, folded from C4. Compare the \ following table with the 2\[Cross]2 quarter-tables in Example 4. Index 3 \ becomes -1, corresponding to index 2 becoming ", StyleBox["i", FontWeight->"Bold"], " with ", StyleBox["i.i =", FontWeight->"Bold"], "-1. Example 7 is equivalent to the complex multiplication (3+1 ", StyleBox["i", FontWeight->"Bold"], ")(x+y ", StyleBox["i", FontWeight->"Bold"], ") = (3x-y)+", StyleBox["i", FontWeight->"Bold"], "(x+3y)." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\<\nThe table is\>", hoopTbl // tf, "\<\nA = \>", A = {3, 1}, "\<\nB = \>", B = {x, y}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nAinv = \>", Ainv = hoopInverse[A], "\<\nAinv.AB = B \>", Simplify[hoopTimes[hoopInverse[A], AB]], "\<\nComplex AB (3+\[ImaginaryI])(x+\[ImaginaryI]y) = (3x-y) \ + \[ImaginaryI](x+3y)\>"}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 7. C4c multiplication & division is Complex \ Algebra\"\>", ",", "\<\"\\nThe table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2"}, {"2", \(-1\)} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nA = \"\>", ",", \({3, 1}\), ",", "\<\"\\nB = \"\>", ",", \({x, y}\), ",", "\<\"\\nAB = \"\>", ",", \({3\ x - y, x + 3\ y}\), ",", "\<\"\\nAinv = \"\>", ",", \({3\/10, \(-\(1\/10\)\)}\), ",", "\<\"\\nAinv.AB = B \"\>", ",", \({x, y}\), ",", "\<\"\\nComplex AB (3+\[ImaginaryI])(x+\[ImaginaryI]y) = (3x-y) \ + \[ImaginaryI](x+3y)\"\>"}], "}"}]], "Output"] }, Open ]], Cell["\<\ \tThe Davenport Algebra is another signed table. It is folded from the \ following C4C2 isomorph, with 5\[RightArrow]-1, 6\[RightArrow]-2, 7\ \[RightArrow]-3:-\ \>", "Text", PageWidth->WindowWidth], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6", "7", "8"}, {"2", "5", "4", "7", "6", "1", "8", "3"}, {"3", "4", "5", "6", "7", "8", "1", "2"}, {"4", "7", "6", "1", "8", "3", "2", "5"}, {"5", "6", "7", "8", "1", "2", "3", "4"}, {"6", "1", "8", "3", "2", "5", "4", "7"}, {"7", "8", "1", "2", "3", "4", "5", "6"}, {"8", "3", "2", "5", "4", "7", "6", "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\<\nThe table is \>", hoopTbl // tf, "\<\nA= \>", A = {3, 4, 6, 1}, "\<\nB= \>", B = {w, x, y, z}, "\<\nAB =\>", AB = hoopTimes[A, B], "\<\nAinv = \>", \[IndentingNewLine]hoopInverse[ A], "\<\nAinv.AB = B \>", \n Simplify[hoopTimes[hoopInverse[A], AB]]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 8. Dav division\"\>", ",", "\<\"\\nThe table is \"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4"}, {"2", \(-1\), "4", \(-3\)}, {"3", "4", \(-1\), \(-2\)}, {"4", \(-3\), \(-2\), "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nA= \"\>", ",", \({3, 4, 6, 1}\), ",", "\<\"\\nB= \"\>", ",", \({w, x, y, z}\), ",", "\<\"\\nAB =\"\>", ",", \({3\ w - 4\ x - 6\ y + z, 4\ w + 3\ x - y - 6\ z, 6\ w - x + 3\ y - 4\ z, w + 6\ x + 4\ y + 3\ z}\), ",", "\<\"\\nAinv = \"\>", ",", \({57\/520, 1\/520, \(-\(51\/520\)\), 47\/520}\), ",", "\<\"\\nAinv.AB = B \"\>", ",", \({w, x, y, z}\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "\tSome hoops have complex products, with ", StyleBox["i", FontWeight->"Bold"], " occurring in the body of the multiplication table. The g2401c algebra has \ 6 elements; it is \"4-folded\" from the group g2401, i.e. the first group \ with 24 elements in the GAP Atlas (not shown):-" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\", hoopTbl // tf, "\<\nA= \>", A = {3, 4, 6, 1, 2, 7}, "\<\nB= \>", B = {u, v, w, x, y, z}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nAinv = \>", \[IndentingNewLine]hoopInverse[ A], "\<\nAinv.AB = B \>", \n Simplify[hoopTimes[hoopInverse[A], AB]]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 9. g2401c division\\n\"\>", ",", "\<\"The table is \"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6"}, {"2", "3", "1", "6", "4", "5"}, {"3", "1", "2", "5", "6", "4"}, {"4", "5", "6", "\[ImaginaryI]", \(2\ \[ImaginaryI]\), \(3\ \[ImaginaryI]\ \)}, {"5", "6", "4", \(3\ \[ImaginaryI]\), "\[ImaginaryI]", \(2\ \[ImaginaryI]\)}, {"6", "4", "5", \(2\ \[ImaginaryI]\), \(3\ \[ImaginaryI]\), "\[ImaginaryI]"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nA= \"\>", ",", \({3, 4, 6, 1, 2, 7}\), ",", "\<\"\\nB= \"\>", ",", \({u, v, w, x, y, z}\), ",", "\<\"\\nAB = \"\>", ",", \({3\ u + 6\ v + 4\ w + \[ImaginaryI]\ x + 2\ \[ImaginaryI]\ y + 7\ \[ImaginaryI]\ z, 4\ u + 3\ v + 6\ w + 7\ \[ImaginaryI]\ x + \[ImaginaryI]\ y + 2\ \[ImaginaryI]\ z, 6\ u + 4\ v + 3\ w + 2\ \[ImaginaryI]\ x + 7\ \[ImaginaryI]\ y + \[ImaginaryI]\ z, u + 7\ v + 2\ w + 3\ x + 4\ y + 6\ z, 2\ u + v + 7\ w + 6\ x + 3\ y + 4\ z, 7\ u + 2\ v + w + 4\ x + 6\ y + 3\ z}\), ",", "\<\"\\nAinv = \"\>", ",", \({189877\/19473305 - \(578094\ \[ImaginaryI]\)\/19473305, 237907\/7789322 + \(485997\ \[ImaginaryI]\)\/7789322, 649681\/38946610 + \(39203\ \[ImaginaryI]\)\/38946610, 60863\/38946610 + \(2452579\ \[ImaginaryI]\)\/38946610, \(-\(104532\ \/19473305\)\) + \(628594\ \[ImaginaryI]\)\/19473305, \ \(-\(1558699\/38946610\)\) - \(4719767\ \[ImaginaryI]\)\/38946610}\), ",", "\<\"\\nAinv.AB = B \"\>", ",", \({u, v, w, x, y, z}\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "A few hoops have ", StyleBox["J ", FontSlant->"Italic"], "& ", Cell[BoxData[ \(TraditionalForm\`J\^2\)]], " (primitive cube roots of 1) as signs. \"C9J\" is 3-folded from the \ following C9 isomorph by 9\[RightArrow]", Cell[BoxData[ \(TraditionalForm\`J\^2\)]], "3, 5\[RightArrow] ", StyleBox["J", FontSlant->"Italic"], " 2:-" }], "Text", PageWidth->WindowWidth], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6", "7", "8", "9"}, {"2", "9", "1", "5", "3", "4", "8", "6", "7"}, {"3", "1", "5", "6", "4", "8", "9", "7", "2"}, {"4", "5", "6", "7", "8", "9", "1", "2", "3"}, {"5", "3", "4", "8", "6", "7", "2", "9", "1"}, {"6", "4", "8", "9", "7", "2", "3", "1", "5"}, {"7", "8", "9", "1", "2", "3", "4", "5", "6"}, {"8", "6", "7", "2", "9", "1", "5", "3", "4"}, {"9", "7", "2", "3", "1", "5", "6", "4", "8"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Input", Editable->False, PageWidth->PaperWidth, Evaluatable->False, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ "Results need extensive simplification, using the rule ", StyleBox["js", FontSlant->"Italic"], ", i.e. ", StyleBox["J", FontSlant->"Italic"], "^any_\[RightArrow]", StyleBox["J", FontSlant->"Italic"], "^Mod[any,3]:-" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\", hoopTbl // tf, "\<\n A= \>", A = {3, 4, 6}, "\<\n B= \>", B = {x, y, z}, "\<\nAB =\>", AB = hoopTimes[A, B], "\<\nAinv = \>", \[IndentingNewLine]hoopInverse[ A], "\<\nAinv.AB unsimplified \>", hoopTimes[hoopInverse[A], AB], "\<\nSimplified Ainv.AB = B \>", \n Simplify[Simplify[hoopTimes[hoopInverse[A], AB]] //. js]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 10. C9j division.\\n\"\>", ",", "\<\"The table is \"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3"}, {"2", \(3\ J\^2\), "1"}, {"3", "1", \(2\ J\)} }], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\n A= \"\>", ",", \({3, 4, 6}\), ",", "\<\"\\n B= \"\>", ",", \({x, y, z}\), ",", "\<\"\\nAB =\"\>", ",", \({3\ x + 6\ y + 4\ z, 4\ x + 3\ y + 6\ J\ z, 6\ x + 4\ J\^2\ y + 3\ z}\), ",", "\<\"\\nAinv = \"\>", ",", \({\(-\(15\/\(\(-189\) + 216\ J + 64\ J\^2\)\)\), \(12\ \((\(-1\) + 3\ J)\)\)\/\(\(-189\) + \ 216\ J + 64\ J\^2\), \(2\ \((\(-9\) + 8\ J\^2)\)\)\/\(\(-189\) + 216\ J + 64\ \ J\^2\)}\), ",", "\<\"\\nAinv.AB unsimplified \"\>", ",", \({\(12\ \((\(-1\) + 3\ J)\)\ \((6\ x + 4\ J\^2\ y + 3\ z)\)\)\/\ \(\(-189\) + 216\ J + 64\ J\^2\) - \(15\ \((3\ x + 6\ y + 4\ z)\)\)\/\(\(-189\ \) + 216\ J + 64\ J\^2\) + \(2\ \((\(-9\) + 8\ J\^2)\)\ \((4\ x + 3\ y + 6\ J\ \ z)\)\)\/\(\(-189\) + 216\ J + 64\ J\^2\), \(2\ J\ \((\(-9\) + 8\ J\^2)\)\ \ \((6\ x + 4\ J\^2\ y + 3\ z)\)\)\/\(\(-189\) + 216\ J + 64\ J\^2\) + \(12\ \ \((\(-1\) + 3\ J)\)\ \((3\ x + 6\ y + 4\ z)\)\)\/\(\(-189\) + 216\ J + 64\ \ J\^2\) - \(15\ \((4\ x + 3\ y + 6\ J\ z)\)\)\/\(\(-189\) + 216\ J + 64\ \ J\^2\), \(-\(\(15\ \((6\ x + 4\ J\^2\ y + 3\ z)\)\)\/\(\(-189\) + 216\ J + 64\ J\^2\)\)\) + \(2\ \((\(-9\) + 8\ J\^2)\)\ \((3\ x + 6\ \ y + 4\ z)\)\)\/\(\(-189\) + 216\ J + 64\ J\^2\) + \(12\ J\^2\ \((\(-1\) + 3\ \ J)\)\ \((4\ x + 3\ y + 6\ J\ z)\)\)\/\(\(-189\) + 216\ J + 64\ J\^2\)}\), ",", "\<\"\\nSimplified Ainv.AB = B \"\>", ",", \({x, y, z}\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["J", FontSlant->"Italic"], " and ", Cell[BoxData[ \(TraditionalForm\`J\^2\)]], " are the only \"generalized signs\" used in this document; the ", StyleBox["Hoops.m ", FontSlant->"Italic"], "package includes instructions that make ", StyleBox["J", FontSlant->"Italic"], " behave as a sign. Other generalized signs are developed in a similar way \ in [2], as powers of primitive roots of unity." }], "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["2.6 Non-Abelian Hoops, Left & Right Inverses.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tSo far all the examples (except g2401c) have used Abelian (commutative) \ hoops. The D3 algebra and the Clifford(2,1) algebra \"CL21\" are non-Abelian, \ and so AB differs from BA.\n\t", StyleBox["hoopInverse", FontSlant->"Italic"], " calculates a left inverse, so Ainv.AB divides AB by A, recovering B. \ Similarly Binv.BA recovers A, whilst BA.Ainv recovers B because A is the left \ inverse of Ainv." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \({"\", Use["\"]; "\", hoopTbl // tf, "\<\nA= \>", A = {3, 4, 6, 1, 2, 7}, "\<\nB= \>", B = {4, \(-1\), 2, 2, 5, 3}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nAinv = \>", \[IndentingNewLine]Ainv = hoopInverse[A], "\<\nAinv.AB = B \>", \n Simplify[hoopTimes[Ainv, AB]], \[IndentingNewLine]"\<\nBA = \>", BA = hoopTimes[B, A], "\<\nBinv.BA = A \>", \n Simplify[hoopTimes[hoopInverse[B], BA]], "\<\nBA.Ainv = B \>", \n Simplify[hoopTimes[BA, Ainv, AB]], \[IndentingNewLine]Use["\"]; "\", hoopTbl // tf, "\<\nA= \>", A = {3, 4, 6, 1, 2, 7, 2, 1}, "\<\nB= \>", B = {4, \(-1\), 2, 2, 5, 3, 2, \(-3\)}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nAinv = \>", \[IndentingNewLine]Ainv = hoopInverse[A], "\<\nAinv.AB = B \>", \n Simplify[hoopTimes[Ainv, AB]], \[IndentingNewLine]"\<\nBA = \>", BA = hoopTimes[B, A], "\<\nBinv.BA = A \>", \n Simplify[hoopTimes[hoopInverse[B], BA]], "\<\nBA.Ainv = B \>", \n Simplify[hoopTimes[BA, Ainv, AB]]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 11. D3 & CL21 Left and Right division\\n\"\>", ",", "\<\"The D3 table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6"}, {"2", "1", "6", "5", "4", "3"}, {"3", "4", "5", "6", "1", "2"}, {"4", "3", "2", "1", "6", "5"}, {"5", "6", "1", "2", "3", "4"}, {"6", "5", "4", "3", "2", "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nA= \"\>", ",", \({3, 4, 6, 1, 2, 7}\), ",", "\<\"\\nB= \"\>", ",", \({4, \(-1\), 2, 2, 5, 3}\), ",", "\<\"\\nAB = \"\>", ",", \({65, 72, 65, 44, 39, 60}\), ",", "\<\"\\nAinv = \"\>", ",", \({\(-\(18\/161\)\), 4\/23, \(-\(13\/322\)\), \(-\(13\/322\)\), \(-\(15\/46\)\), 125\/322}\), ",", "\<\"\\nAinv.AB = B \"\>", ",", \({4, \(-1\), 2, 2, 5, 3}\), ",", "\<\"\\nBA = \"\>", ",", \({65, 50, 44, 69, 60, 57}\), ",", "\<\"\\nBinv.BA = A \"\>", ",", \({3, 4, 6, 1, 2, 7}\), ",", "\<\"\\nBA.Ainv = B \"\>", ",", \({4, \(-1\), 2, 2, 5, 3}\), ",", "\<\"The CL21 table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6", "7", "8"}, {"2", \(-1\), "4", \(-3\), "6", \(-5\), "8", \(-7\)}, {"3", \(-4\), "1", \(-2\), "7", \(-8\), "5", \(-6\)}, {"4", "3", "2", "1", "8", "7", "6", "5"}, {"5", \(-6\), \(-7\), "8", "1", \(-2\), \(-3\), "4"}, {"6", "5", \(-8\), \(-7\), "2", "1", \(-4\), \(-3\)}, {"7", "8", \(-5\), \(-6\), "3", "4", \(-1\), \(-2\)}, {"8", \(-7\), \(-6\), "5", "4", \(-3\), \(-2\), "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nA= \"\>", ",", \({3, 4, 6, 1, 2, 7, 2, 1}\), ",", "\<\"\\nB= \"\>", ",", \({4, \(-1\), 2, 2, 5, 3, 2, \(-3\)}\), ",", "\<\"\\nAB = \"\>", ",", \({54, 36, 45, 15, 11, 73, 42, \(-22\)}\), ",", "\<\"\\nAinv = \"\>", ",", \({5\/26, \(-\(1\/13\)\), 1\/10, \(-\(2\/13\)\), \(-\(19\/130\)\), 0, 8\/65, 27\/130}\), ",", "\<\"\\nAinv.AB = B \"\>", ",", \({4, \(-1\), 2, 2, 5, 3, 2, \(-3\)}\), ",", "\<\"\\nBA = \"\>", ",", \({54, \(-2\), 51, 3, 33, 33, 12, \(-22\)}\), ",", "\<\"\\nBinv.BA = A \"\>", ",", \({3, 4, 6, 1, 2, 7, 2, 1}\), ",", "\<\"\\nBA.Ainv = B \"\>", ",", \({4, \(-1\), 2, 2, 5, 3, 2, \(-3\)}\)}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.7 Many Algebras are Hoops.", "Subsection", PageWidth->WindowWidth], Cell["\<\ \tExample 12 shows that traditional vector dot & cross products are hoop \ multiplications where only the \"univectors\" are multiplied i.e. the scalar \ and multivector elements of the multiplicands are kept at zero. (I use \ \"univectors\" where many authors confusingly use \"vectors\" or \"scalars\" \ for single element generators. Their products are bivectors, trivectors, \ etc.) The symbols are chosen to match those in [5, p11] for quaternions (Qr) \ and CL2, and to match [5, p37] for octonions (Octr) and CL3.\ \>", "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`{Use["\"]; \*"\"\\"", hoopTimes[{0, e\_1, e\_2, 0}, {0, u\_1, u\_2, 0}], \[IndentingNewLine]Use["\"]; \*"\"\<\\n\\nOctr \ {0,\!\(a\_1\),\!\(a\_2\),0,\!\(a\_3\),0,0,0}\[Times]{0,\!\(b\_1\),\!\(b\_2\),\ 0,\!\(b\_3\),0,0,0}=\>\"", \ hoopTimes[{0, a\_1, a\_2, 0, a\_3, 0, 0, 0}, {0, b\_1, b\_2, 0, b\_3, 0, 0, 0}]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ \({"Example 12. Dot & Cross products with Qr & Octr.\n{0,\!\(e\_1\),\!\(e\ \_2\),0}\[Times]{0,\!\(u\_1\),\!\(u\_2\),0} gives\n Qr Scalar & Dot \ products\n", {\(-e\_1\)\ u\_1 - e\_2\ u\_2, 0, 0, \(-e\_2\)\ u\_1 + e\_1\ u\_2}, "\n\nOctr {0,\!\(a\_1\),\!\(a\_2\),0,\!\(a\_3\),0,0,0}\[Times]{0,\!\(b\ \_1\),\!\(b\_2\),0,\!\(b\_3\),0,0,0}=", {\(-a\_1\)\ b\_1 - a\_2\ b\_2 - a\_3\ b\_3, 0, 0, \(-a\_2\)\ b\_1 + a\_1\ b\_2, 0, \(-a\_3\)\ b\_1 + a\_1\ b\_3, \(-a\_3\)\ b\_2 + a\_2\ b\_3, 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "\tIn both cases the first element of the result is the scalar product (up \ to a sign), whilst the univectors (and octonion trivector) in the product are \ zero. The Qr trivector element ", Cell[BoxData[ \(TraditionalForm\`\(e\_2\) u\_1 - \(e\_1\) u\_2\)]], " is the 2D cross product, and Octc gives the (negated) 3D scalar product \ ", Cell[BoxData[ StyleBox[\(a\_1\ b\_1 + a\_2\ b\_2 + a\_3\ b\_3\), FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}]]], " together with elements that add to give the vector cross product {", Cell[BoxData[ \(TraditionalForm\`\((\(-a\_2\)\ b\_1 + a\_1\ b\_2)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(e\_1\) e\_2\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\((\(-a\_3\)\ b\_1 + a\_1\ b\_3)\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(e\_1\) e\_3\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\((\(-a\_3\)\ b\_2 + a\_2\ b\_3)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(e\_2\) e\_3\)]], "}. \n(The bivector ", Cell[BoxData[ \(TraditionalForm\`\(e\_1\) e\_2\)]], " is the 4th element basis direction, etc., having coefficient 0 in the \ multiplicands). " }], "Text", PageWidth->WindowWidth], Cell[TextData[{ "\tAll Clifford algebras are hoops. Their tables are created in [2] by \ using the procedure ", StyleBox["cl", FontSlant->"Italic"], " or by folding a specific group isomorph. Wedge (exterior) algebras are \ the bivector part of the Clifford product ([5] p.10). They multiply \ univectors using Clifford algebra rules; the scalars and multivectors are \ kept at zero in the multiplicands so various products are zero (i.e. these \ are Grassmann algebras). Multiplying univectors ", Cell[BoxData[ \(TraditionalForm\`u\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`u\_2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`v\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`v\_2\)]], " in CL2 gives a scalar ", Cell[BoxData[ \(TraditionalForm\`\(u\_1\) v\_1 - \(u\_2\) v\_2\)]], " and a bivector ", Cell[BoxData[ \(TraditionalForm\`\(u\_1\) v\_2 - \(u\_2\) v\_1\)]], "; the univectors in the product are zero. CL3 gives the wedge product of \ three univectors when the scalar and multivector terms are zero in the \ multiplicands. CL2 and CL3 univector products only differ from the quaternion \ and octonion univector products in the sign of the first term.\n\tDifferent \ 16-element Clifford algebras multiply 4 univectors to give scalars and three \ bivectors with assorted signs:- " }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{\(Use["\"]; "\"\), ",", \(hoopTimes[{0, u\_1, u\_2, 0}, {0, v\_1, v\_2, 0}]\), ",", "\[IndentingNewLine]", \(Use["\"]; "\<\nCL3\>"\), ",", \(hoopTimes[{0, u\_1, u\_2, 0, u\_3, 0, 0, 0}, {0, v\_1, v\_2, 0, v\_3, 0, 0, 0}]\), ",", "\[IndentingNewLine]", \(Use["\"]; "\<\nCL4\>"\), ",", FormBox[\(hoopTimes[{0, e\_1, e\_2, 0, e\_3, 0, 0, e\_4}, {0, u\_1, u\_2, 0, u\_3, 0, 0, u\_4}]\), "TraditionalForm"], ",", "\n", \(Use["\"]; "\<\nCL04\>"\), ",", FormBox[\(hoopTimes[{0, e\_1, e\_2, 0, e\_3, 0, 0, e\_4}, {0, u\_1, u\_2, 0, u\_3, 0, 0, u\_4}]\), "TraditionalForm"], ",", "\[IndentingNewLine]", \(Use["\"]; "\<\nCL31\>"\), ",", FormBox[\(hoopTimes[{0, e\_1, e\_2, 0, e\_3, 0, 0, e\_4}, {0, u\_1, u\_2, 0, u\_3, 0, 0, u\_4}]\), "TraditionalForm"]}], "}"}], TraditionalForm]], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ \({"Example 13. Wedge Products, Exterior Algebras. Clifford Algebras.\n\ CL2", {u\_1\ v\_1 - u\_2\ v\_2, 0, 0, \(-u\_2\)\ v\_1 + u\_1\ v\_2}, "\nCL3", {u\_1\ v\_1 + u\_2\ v\_2 + u\_3\ v\_3, 0, 0, \(-u\_2\)\ v\_1 + u\_1\ v\_2, 0, \(-u\_3\)\ v\_1 + u\_1\ v\_3, \(-u\_3\)\ v\_2 + u\_2\ v\_3, 0}, "\nCL4", {e\_1\ u\_1 + e\_2\ u\_2 + e\_3\ u\_3 - e\_4\ u\_4, 0, 0, \(-e\_2\)\ u\_1 + e\_1\ u\_2 + e\_4\ u\_3 + e\_3\ u\_4, 0, \(-e\_3\)\ u\_1 - e\_4\ u\_2 + e\_1\ u\_3 - e\_2\ u\_4, e\_4\ u\_1 - e\_3\ u\_2 + e\_2\ u\_3 + e\_1\ u\_4, 0, 0, 0, 0, 0, 0, 0, 0, 0}, "\nCL04", {\(-e\_1\)\ u\_1 - e\_2\ u\_2 - e\_3\ u\_3 + e\_4\ u\_4, 0, 0, \(-e\_2\)\ u\_1 + e\_1\ u\_2 - e\_4\ u\_3 - e\_3\ u\_4, 0, \(-e\_3\)\ u\_1 + e\_4\ u\_2 + e\_1\ u\_3 + e\_2\ u\_4, \(-e\_4\)\ u\_1 - e\_3\ u\_2 + e\_2\ u\_3 - e\_1\ u\_4, 0, 0, 0, 0, 0, 0, 0, 0, 0}, "\nCL31", {\(-e\_1\)\ u\_1 + e\_2\ u\_2 + e\_3\ u\_3 + e\_4\ u\_4, 0, 0, \(-e\_2\)\ u\_1 + e\_1\ u\_2 + e\_4\ u\_3 + e\_3\ u\_4, 0, \(-e\_3\)\ u\_1 - e\_4\ u\_2 + e\_1\ u\_3 - e\_2\ u\_4, \(-e\_4\)\ u\_1 - e\_3\ u\_2 + e\_2\ u\_3 - e\_1\ u\_4, 0, 0, 0, 0, 0, 0, 0, 0, 0}}\)], "Output"] }, Open ]], Cell["\<\ \tFinite Lie algebras appear to be Clifford hoops for which the real (scalar) \ elements have been zeroed. This has yet to be explored. \tOther hoops define many algebras, some of which appear to be new.\ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["2.8 Most loops are not conservative.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tThe majority of loops (quasigroups used as multiplication tables) are \ not conservative. The data contains a few non-conservative loops as examples, \ identified by names ending in ", StyleBox["n", FontSlant->"Italic"], ". The determinants of \"Q4n\" and \"C6n\" factorize nicely, but their \ quadratic factors are not conserved:-" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\", hoopTbl // tf, "\<\nShape\>", sh, \[IndentingNewLine]"\<\nA = \>", A = {4, 2, 3, 5}, \[IndentingNewLine]"\<\nB = \>", B = {1, 6, 2, 2}, \[IndentingNewLine]"\<\nAB= \>", AB = hoopTimes[A, B], "\<\nShape of A sa =\>", sa = sh /. \ as[A], "\<\nShape of B sb =\>", sb = sh /. \ as[B], "\<\nShape of AB sab=\>", \ sh /. \ as[AB], "\<\nThis is not sa*sb =\>", sa\ sb}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 14. Q4n is not conservative.\\n\"\>", ",", "\<\"Q4n Table\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4"}, {"2", "1", "4", \(-3\)}, {"3", \(-4\), "1", "2"}, {"4", "3", \(-2\), "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nShape\"\>", ",", \({a\^2 - b\^2 - c\^2 - d\^2, a\^2 + b\^2 + c\^2 + d\^2}\), ",", "\<\"\\nA = \"\>", ",", \({4, 2, 3, 5}\), ",", "\<\"\\nB = \"\>", ",", \({1, 6, 2, 2}\), ",", "\<\"\\nAB= \"\>", ",", \({32, 22, 37, \(-1\)}\), ",", "\<\"\\nShape of A sa =\"\>", ",", \({\(-22\), 54}\), ",", "\<\"\\nShape of B sb =\"\>", ",", \({\(-43\), 45}\), ",", "\<\"\\nShape of AB sab=\"\>", ",", \({\(-830\), 2878}\), ",", "\<\"\\nThis is not sa*sb =\"\>", ",", \({946, 2430}\)}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Use["\"];\)\), "\[IndentingNewLine]", \({"\", "\", hoopTbl // tf, "\<\nShape\>", sh, "\<\n A =\>", A = {4, 2, 3, 5, 1, 6}, "\<\n B =\>", B = {0, 6, 2, 4, 3, 2}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nshape of A sa = \>", sa = sh /. \ as[A], "\<\nshape of B sb = \>", sb = sh /. \ as[B], "\<\nshape of AB sab= \>", sab = sh /. \ as[AB], "\<\nThis is not sa*sb \>", sa\ sb}\)}], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 15. C6n only conserves the linear factors.\\n\"\>", ",", "\<\"C6n Table\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6"}, {"2", "3", "1", "6", "4", "5"}, {"3", "1", "2", "5", "6", "4"}, {"4", "5", "6", "2", "1", "3"}, {"5", "6", "4", "1", "3", "2"}, {"6", "4", "5", "3", "2", "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nShape\"\>", ",", \({a + b + c - d - e - f, a + b + c + d + e + f, 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 - \ \((d - e)\)\^2 - \((e - f)\)\^2 - \((\(-d\) + f)\)\^2)\), 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 + \ \((d - e)\)\^2 + \((e - f)\)\^2 + \((\(-d\) + f)\)\^2)\)}\), ",", "\<\"\\n A =\"\>", ",", \({4, 2, 3, 5, 1, 6}\), ",", "\<\"\\n B =\"\>", ",", \({0, 6, 2, 4, 3, 2}\), ",", "\<\"\\nAB = \"\>", ",", \({53, 70, 57, 66, 70, 41}\), ",", "\<\"\\nshape of A sa = \"\>", ",", \({\(-3\), 21, \(-18\), 24}\), ",", "\<\"\\nshape of B sb = \"\>", ",", \({\(-1\), 17, 25, 31}\), ",", "\<\"\\nshape of AB sab= \"\>", ",", \({3, 357, \(-504\), 978}\), ",", "\<\"\\nThis is not sa*sb \"\>", ",", \({3, 357, \(-450\), 744}\)}], "}"}]], "Output"] }, Open ]], Cell["\<\ \tMany non-conservative loops, with various properties, have been \ investigated. The only conservative tables appear to be groups, a few \ non-commutative Moufang Loops (including octonions and split-octonions), the \ direct composition of these loops with abelian groups, and tables folded from \ conservative loops; all known conservative tables have the Moufang \ property.\ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2.9 Quadratic sizes, Abelian Polar-Vector Duals, Roots & Powers.\ \>", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tReal quadratic hoop determinant factors are conserved functions (though \ their complex-conjugate-pair factors are not conserved). They are symmetric \ real polynomials that can often be \"fragmented\", i.e. split into signed \ sums of smaller squared terms or \"fragments\". A generalisation (the \ \"PolyHelix Identity\") of ", Cell[BoxData[ \(TraditionalForm\`Cos\^2\)]], "[\[Theta]]+", Cell[BoxData[ \(TraditionalForm\`Sin\^2\)]], "[\[Theta]]=1 then leads to \"polar-dual\" formulations. E.g. C3 conserves \ ", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], " =", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`b\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`c\^2\)]], StyleBox["- a b - b c - c a", FontSlant->"Italic"], " which fragments into", StyleBox[" ", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`\((a - b)\)\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`\((b - c)\)\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`\((c - a)\)\^2\)]], ")/2. The associated angle is ", StyleBox["ArcTan[2a-b-c,-", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\@3\)], FontSlant->"Italic"], StyleBox["(b-c)]", FontSlant->"Italic"], ". (see Example 1.) Note that the \"-\" sign always occurs in squared terms \ so it corresponds to a \"symmetric difference\" that does not imply negation \ or subtraction, which only arise in hoops after 2-folding.\n\tIn effect, each \ conserved quadratic \"releases\" a degree of freedom, which is taken up by an \ angle. Angles are \"hidden variables\" for cartesian vectors. Polar-duals are \ the generalization of the {r,\[Theta]} formulation of the complex plane (or \ Argand - Wessel diagram) to other algebras. Linear sizes provide \"offsets\" \ for polar forms; these are extra degrees of freedom that are missing from \ complex algebra.\n\tAs angles add on abelian hoop multiplication, powers and \ roots can be obtained by raising sizes to the exponent and multiplying angles \ by the exponent. This is implemented by ", StyleBox["hoopPower", FontSlant->"Italic"], ", which converts a cartesian vector to the polar form, applies the \ exponent, and then reverts to cartesian form. It has been demonstrated in \ previous examples. Example 16 demonstrates a \"Hexal\" algebra, C3C4c, which \ is \"supersymmetric\" (see section 3) to one of the \"Dozal\" algebras, C3C4. \ These two algebras are unusual - they have sizes that involve ", Cell[BoxData[ \(TraditionalForm\`\(\(\@3\)\(.\)\)\)]], " Note the \"If\" statements that handle specific cases in the angle \ calculations. These are essential when dealing with orbits (section 2.13); \ they are omitted (for simplicity) in most ", StyleBox["topol", FontSlant->"Italic"], " definitions in the data. ([2] supplies the complete versions.)" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", hoopTbl // tf, "\<\nThe polar form is\>", topol, "\<\nA simplified cartesian form, in terms of radii \[Epsilon],\ \[Eta],\[Kappa] and angles \[Sigma],\[Chi],\[Psi] is\n\>", \(tovec /. \ as[{\[Epsilon]\^2, \[Sigma], \[Eta]\^2, \[Chi] - \[Pi]/ 6, \[Kappa]\^2, \[Psi] - \[Pi]/6}]\) /. \@any_\^2 \[Rule] 6 any, "\<\nWhen A = \>", A = {6. , 2. , 4. , 3. , 1. , 1. }, "\<\nthe shape of A is \>", sh /. \ as[A], "\<\nthe polar form is \>", pA = topol /. \ as[A], \[IndentingNewLine]"\", tovec /. \ as[pA], \[IndentingNewLine]"\<\nThe square root is\>", rootA = hoopPower[A], "\<\nwith polar form \>", topol /. \ as[rootA], "\<\nrootA.rootA = A \>", hoopTimes[rootA, rootA]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 16. C3C4c Polar form and square root\\nThe table \ is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6"}, {"2", "3", "1", "5", "6", "4"}, {"3", "1", "2", "6", "4", "5"}, {"4", "5", "6", \(-1\), \(-2\), \(-3\)}, {"5", "6", "4", \(-2\), \(-3\), \(-1\)}, {"6", "4", "5", \(-3\), \(-1\), \(-2\)} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nThe polar form is\"\>", ",", \({\((a + b + c)\)\^2 + \((d + e + f)\)\^2, If[d + e + f \[Equal] 0, If[a + b + c < 0, \[Pi], 0], ArcTan[a + b + c, d + e + f]], 3\/4\ \((\((\(a + b - 2\ c\)\/\@3 - d + e)\)\^2 + \((a - b + \(d + \ e - 2\ f\)\/\@3)\)\^2)\), \(-\(\[Pi]\/6\)\) + If[\(a + b - 2\ c\)\/\@3 - d + e \[Equal] 0, If[a - b + \(d + e - 2\ f\)\/\@3 < 0, \[Pi], 0], ArcTan[a - b + \(d + e - 2\ f\)\/\@3, \(a + b - 2\ c\)\/\@3 - d + e]], 3\/4\ \((\((\(a + b - 2\ c\)\/\@3 + d - e)\)\^2 + \((a - b - \(d + \ e - 2\ f\)\/\@3)\)\^2)\), \(-\(\[Pi]\/6\)\) + If[\(a + b - 2\ c\)\/\@3 + d - e \[Equal] 0, If[a - b - \(d + e - 2\ f\)\/\@3 < 0, \[Pi], 0], ArcTan[a - b - \(d + e - 2\ f\)\/\@3, \(a + b - 2\ c\)\/\@3 + d - e]]}\), ",", "\<\"\\nA simplified cartesian form, in terms of radii \ \[Epsilon],\[Eta],\[Kappa] and angles \[Sigma],\[Chi],\[Psi] is\\n\"\>", ",", \({2\ \[Epsilon]\ Cos[\[Sigma]] + \@3\ \[Eta]\ Cos[\[Chi]] + \@3\ \ \[Kappa]\ Cos[\[Psi]] + \[Eta]\ Sin[\[Chi]] + \[Kappa]\ Sin[\[Psi]], 2\ \[Epsilon]\ Cos[\[Sigma]] - \@3\ \[Eta]\ Cos[\[Chi]] - \@3\ \ \[Kappa]\ Cos[\[Psi]] + \[Eta]\ Sin[\[Chi]] + \[Kappa]\ Sin[\[Psi]], 2\ \[Epsilon]\ Cos[\[Sigma]] - 2\ \[Eta]\ Sin[\[Chi]] - 2\ \[Kappa]\ Sin[\[Psi]], \[Eta]\ Cos[\[Chi]] - \[Kappa]\ Cos[\ \[Psi]] + 2\ \[Epsilon]\ Sin[\[Sigma]] - \@3\ \[Eta]\ Sin[\[Chi]] + \@3\ \ \[Kappa]\ Sin[\[Psi]], \[Eta]\ Cos[\[Chi]] - \[Kappa]\ Cos[\[Psi]] + 2\ \[Epsilon]\ Sin[\[Sigma]] + \@3\ \[Eta]\ Sin[\[Chi]] - \@3\ \ \[Kappa]\ Sin[\[Psi]], \(-2\)\ \[Eta]\ Cos[\[Chi]] + 2\ \[Kappa]\ Cos[\[Psi]] + 2\ \[Epsilon]\ Sin[\[Sigma]]}\), ",", "\<\"\\nWhen A = \"\>", ",", \({6.`, 2.`, 4.`, 3.`, 1.`, 1.`}\), ",", "\<\"\\nthe shape of A is \"\>", ",", \({169.`, 22.928203230275507`, 9.071796769724491`}\), ",", "\<\"\\nthe polar form is \"\>", ",", \({169.`, 0.3947911196997615`, 22.928203230275507`, \(-0.8937137001742207`\), 9.071796769724491`, 0.08908002307310836`}\), ",", "\<\"tovec recovers A \"\>", ",", \({6.`, 2.`, 4.`, 2.9999999999999996`, 0.9999999999999998`, 1.`}\), ",", "\<\"\\nThe square root is\"\>", ",", \({2.414224259011073`, 0.3099888766376571`, 0.8113207702840068`, 0.576661967608032`, 0.13438651755665915`, \(-0.0039417039781438346`\)}\), ",", "\<\"\\nwith polar form \"\>", ",", \({12.999999999999995`, 0.19739555984988075`, 4.788340341942652`, \(-0.4468568500871104`\), 3.01194235830045`, 0.04454001153655429`}\), ",", "\<\"\\nrootA.rootA = A \"\>", ",", \({5.9999999999999964`, 1.9999999999999991`, 3.9999999999999982`, 2.999999999999999`, 0.9999999999999994`, 0.9999999999999996`}\)}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2.10 Non-Abelian Polar-Vector Duals, Rotated-roots, Uncertainty?\ \>", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tNon-abelian hoops have repeated determinant factors. They do not have \ ordinary polar forms because the repeated sizes release extra degrees of \ freedom. (As the tables are non-abelian, the determinant should possibly be \ found and factorised by noncommutative procedures. No progress has been made \ with this topic.) Non-abelian quadratic sizes often have more than three \ fragments, and do not fit the patterns that give abelian polar forms. \ Splitting them into sets of two or three fragments allows the formulation of \ invertible pseudo-polar-powers with ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[\(A\^\(1/p\)\), "TraditionalForm"], ")"}], "p"], TraditionalForm]]], "=A, but as the angles are not additive this gives \"pseudo-roots\" that do \ not multiply to recover the original vector i.e. ", Cell[BoxData[ \(TraditionalForm\`A\^\(1/2\)\)]], ".", Cell[BoxData[ \(TraditionalForm\`A\^\(1/2\)\)]], "\[NotEqual]A; the product has the correct sizes but the other elements of \ the polar form are modified in a non-obvious way. The angles are rotated. \ Conjecture - powers and roots may be associated with the introduction of \ uncertainty on making an observation of a quantum system.\n\tPseudo-roots are \ demonstrated for D3:-" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", hoopTbl // tf, "\<\nThe symbolic pseudo-polar-vector forms are\>", topol, tovec, "\<\nA = \>", A = {3. \ , 1. , 4. , 2. , 6. , 9. }, "\<\nThe pseudo-polar form is \>", pA = topol /. \ as[A], \[IndentingNewLine]"\<\ntovec recovers A \>", tovec /. \ as[pA], \[IndentingNewLine]"\<\nThe half-power is\>", r2A = hoopPower[A], "\<\nwith pseudo-polar form \>", topol /. \ as[r2A], "\<\nr2A.r2A \[NotEqual] A \>", r2a2 = hoopTimes[r2A, r2A], "\<\nwith pseudo-polar form \>", topol /. \ as[r2a2], "\<\nr2A^2=A reversion OK\>", hoopPower[r2A, 2], \[IndentingNewLine]"\<\nThe one-third power is\>", r3A = hoopPower[A, 1/3], "\<\nwith pseudo-polar form \>", topol /. \ as[r3A], "\<\nr3A.r3A.r3A \[NotEqual] A \>", r3a3 = hoopTimes[r3A, hoopTimes[r3A, r3A]], "\<\nbut has three correct sizes\>", topol /. \ as[r3a3], "\<\nr3A^3=A reversion OK\>", hoopPower[r3A, 3]}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 17. D3 Pseudo-polar form, rotated-roots.\\nThe \ table is\"\>", ",", TagBox[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6"}, {"2", "1", "6", "5", "4", "3"}, {"3", "4", "5", "6", "1", "2"}, {"4", "3", "2", "1", "6", "5"}, {"5", "6", "1", "2", "3", "4"}, {"6", "5", "4", "3", "2", "1"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], "TraditionalForm"], TraditionalForm, Editable->True], ",", "\<\"\\nThe symbolic pseudo-polar-vector forms are\"\>", ",", \({a + b + c + d + e + f, a - b + c - d + e - f, 1\/2\ \((\((a - c)\)\^2 - \((b - d)\)\^2 + \((c - e)\)\^2 + \((\(-a\ \) + e)\)\^2 - \((d - f)\)\^2 - \((\(-b\) + f)\)\^2)\), ArcTan[2\ a - c - e, \@3\ \((\(-c\) + e)\)], 1\/2\ \((\((a - c)\)\^2 + \((c - e)\)\^2 + \((\(-a\) + e)\)\^2)\), ArcTan[2\ b - d - f, \@3\ \((\(-d\) + f)\)]}\), ",", \({1\/6\ \((a + b + 4\ \@e\ Cos[d])\), 1\/6\ \((a - b + 4\ \@\(\(-c\) + e\)\ Cos[f])\), 1\/6\ \((a + b + 4\ \@e\ Cos[d + \(2\ \[Pi]\)\/3])\), 1\/6\ \((a - b + 4\ \@\(\(-c\) + e\)\ Cos[f + \(2\ \[Pi]\)\/3])\), 1\/6\ \((a + b + 4\ \@e\ Cos[d - \(2\ \[Pi]\)\/3])\), 1\/6\ \((a - b + 4\ \@\(\(-c\) + e\)\ Cos[f - \(2\ \[Pi]\)\/3])\)}\), ",", "\<\"\\nA = \"\>", ",", \({3.`, 1.`, 4.`, 2.`, 6.`, 9.`}\), ",", "\<\"\\nThe pseudo-polar form is \"\>", ",", \({25.`, 1.`, \(-50.`\), 2.4278682746450277`, 7.`, 2.209356022893902`}\), ",", "\<\"\\ntovec recovers A \"\>", ",", \({3.`, 0.9999999999999993`, 4.`, 1.9999999999999991`, 6.`, 9.`}\), ",", "\<\"\\nThe half-power is\"\>", ",", \({1.378814323049811`, 1.6006200942110909`, \(-0.06934573050425892`\), \ \(-1.4080220717003953`\), 1.6905314074544475`, 1.8074019774893046`}\), ",", "\<\"\\nwith pseudo-polar form \"\>", ",", \({5.`, 0.9999999999999996`, \(-7.071067811865475`\), 1.2139341373225139`, 2.6457513110645907`, 1.104678011446951`}\), ",", "\<\"\\nr2A.r2A \[NotEqual] A \"\>", ",", \({9.47787941528671`, 5.061384806245875`, 0.7610602923566447`, 1.6422345701092436`, 2.761060292356644`, 5.296380623644881`}\), ",", "\<\"\\nwith pseudo-polar form \"\>", ",", \({24.999999999999993`, 1.`, 50.00000000000001`, 0.22079220972595706`, 62.54929737601914`, 1.1046780114469512`}\), ",", "\<\"\\nr2A^2=A reversion OK\"\>", ",", \({3.`, 1.`, 4.`, 1.9999999999999993`, 6.`, 9.`}\), ",", "\<\"\\nThe one-third power is\"\>", ",", \({1.2902351436656412`, 1.4891432367494186`, \(-0.24208388089593505`\), \ \(-1.180985456845474`\), 0.913857606336727`, 0.6538510892024889`}\), ",", "\<\"\\nwith pseudo-polar form \"\>", ",", \({2.9240177382128665`, 0.9999999999999996`, \(-3.684031498640386`\), 0.8092894248816759`, 1.9129311827723887`, 0.7364520076313007`}\), ",", "\<\"\\nr3A.r3A.r3A \[NotEqual] A \"\>", ",", \({13.682903427041946`, 12.56158618741602`, \(-4.576332396489963`\), \ \(-7.002858121295428`\), 3.893428969448026`, 6.441271933879416`}\), ",", "\<\"\\nbut has three correct sizes\"\>", ",", \({25.000000000000018`, 1.`, \(-50.00000000000002`\), 0.48189413225415095`, 250.4851803055853`, 0.7364520076313006`}\), ",", "\<\"\\nr3A^3=A reversion OK\"\>", ",", \({3.000000000000002`, 1.0000000000000022`, 4.000000000000002`, 2.0000000000000018`, 6.000000000000001`, 9.`}\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "The repeated quadratic factor (", Cell[BoxData[ \(TraditionalForm\`a\^2 - c\ a - e\ a\)]], Cell[BoxData[ \(TraditionalForm\`\(-b\^2\) + c\^2 - d\^2 + \)], PageWidth->PaperWidth, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`e\^2 - f\^2 + b\ d\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(-c\)\ \ e + b\ f + d\ f\)]], ")/2\nhas been handled as two sets of sums of three squares ", Cell[BoxData[ \(TraditionalForm\`\((\((e - a)\)\^2 + \((a - c)\)\^2 + \((c - e)\)\^2)\ \)/2\)], PageWidth->PaperWidth, FontFamily->"Times New Roman"], " and ", Cell[BoxData[ \(TraditionalForm\`\((\((f - b)\)\^2 + \((b - d)\)\^2 + \((d - f)\)\^2)\ \)/2\)]], " to allow the formulation of a pseudo-polar dual. The angles do not add, \ nor are the squared radii conserved. (The conserved length is their \ difference.) I have sought (without success) a reformulation with additive \ angles, using the distinction between left and right multiplication." }], "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["2.11 Vectors with some zero sizes, Remainders.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tSizes provide the denominators of the partial-fraction formulation of \ the multiplicative inverse (because it is found by Cramer's method). Vectors \ can have coefficients that make one or more size zero in a particular \ algebra; these algebras have \"divisors of zero\". Most mathematicians choose \ to restrict their work to the \"algebras without real divisors of zero\", \ \[DoubleStruckCapitalR], \[DoubleStruckCapitalC], \[DoubleStruckCapitalH] \ (Quaternions), and \[DoubleStruckCapitalO] (Octonions). A determinant is zero \ if it has one or more zero factor, but it may still have non-zero factors. \ Hoop algebras overcome division-by-zero via two innovations, ", StyleBox["projection", FontSlant->"Italic"], " into sub-algebras and ", StyleBox["ejection", FontSlant->"Italic"], " of remainders. Zeroed sizes correspond to operations in sub-algebras with \ these sizes constrained to be zero; inverses are in the same sub-algebras, \ and also have these zeroes. In effect, zeroes are \"factored out\". \ Conservation is maintained by ejecting remainders A/B = C +Rl +Rr; sizes that \ are zero in C but not in A go into the left remainder Rl; sizes that are zero \ in C but not in B go into the right remainder. This is a generalization of \ integer division. Remainders also extend to multiplication A*B=P+Rl+Rr so \ that P/A+Rr=B and P/B+Rl=A. Example 18 sets up two vectors with C4 shapes \ {6,2,0} & {0,10,10}; their product has shape {0,20,0}; Rl conserves the 6 \ from A0; Rr conserves the second 10 from B0. Dividing AB by A does not \ recover B until Rr is added; dividing AB by B does not recover A until Rl is \ added. " }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Use["\"]\), ";", RowBox[{"Chop", "[", RowBox[{"{", RowBox[{ "\"\\"", ",", "\"\<\\n A0 = \>\"", ",", \(A0 = {2, 1, 2, 1}\), ",", "\[IndentingNewLine]", "\"\<\\n\!\(\* StyleBox[\"shape\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"of\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)A0 = \>\"", ",", \(sA0 = sh /. \ as[A0]\), ",", "\[IndentingNewLine]", "\"\<\\n B0 = \>\"", ",", \(B0 = {3, \(-1\), 2, \(-4\)}\), ",", "\[IndentingNewLine]", "\"\<\\n\!\(\* StyleBox[\"shape\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"of\",\nFontSlant->\"Italic\"]\) B0 = \>\"", ",", \(sh /. \ as[B0]\), ",", "\[IndentingNewLine]", "\"\<\\n AB = \>\"", ",", \(AB = hoopTimes[A0, B0]\), ",", "\[IndentingNewLine]", "\"\<\\n\!\(\* StyleBox[\"shape\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"of\",\nFontSlant->\"Italic\"]\) AB = \>\"", ",", \(sh /. \ as[AB]\), ",", "\[IndentingNewLine]", "\"\<\\n\!\(\* StyleBox[\"Note\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"that\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"sizes\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\[UpArrow] \[UpArrow]\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"have\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"become\",\nFontSlant->\"Italic\"]\) 0\>\"", ",", "\[IndentingNewLine]", "\"\<\\n\!\(\* StyleBox[\"Remainder\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"Rl\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"=\",\nFontSlant->\"Italic\"]\)\>\"", ",", \(RA = Rl\), ",", "\"\<\\n\!\(\* StyleBox[\"has\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"shape\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\) \>\"", ",", \(sh /. \ as[Rl]\), ",", "\"\<\\n\!\(\* StyleBox[\"Remainder\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"Rr\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"=\",\nFontSlant->\"Italic\"]\)\>\"", ",", \(RB = Rr\), ",", "\"\<\\n\!\(\* StyleBox[\"has\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"a\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\"shape\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)\>\"", ",", \(sh /. \ as[Rr]\), ",", "\"\<\\n\!\(\* StyleBox[\"Ainverse\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)= \>\"", ",", \(Ainv = hoopInverse[A0]\), ",", "\"\<\\nwith shape \>\"", ",", \(sh /. \ as[Ainv]\), ",", "\"\<\\nAB/A0 = \>\"", ",", \(ABoA = hoopTimes[Ainv, AB]\), ",", "\"\<\\nwith shape \>\"", ",", \(sh /. \ as[ABoA]\), ",", "\"\<\\nAdding Rr recovers B0\>\"", ",", \(ABoA + RB\), ",", "\[IndentingNewLine]", "\"\<\\n\!\(\* StyleBox[\"Binverse\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Italic\"]\)= \>\"", ",", \(Binv = hoopInverse[B0]\), ",", "\"\<\\nwith shape \>\"", ",", \(sh /. \ as[Binv]\), ",", "\"\<\\nAB/B0 = \>\"", ",", \(ABoB = hoopTimes[AB, Binv]\), ",", "\"\<\\nwith shape \>\"", ",", \(sh /. \ as[ABoB]\), ",", "\"\<\\nAdding Rl recovers A0 \>\"", ",", \(ABoB + RA\)}], StyleBox["}", FontSlant->"Italic"]}], "]"}]}]], "Input", PageWidth->WindowWidth, CellOpen->False, FontFamily->"Courier New", FontSize->10], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 18. C4 multiplication with remainders\"\>", ",", "\<\"\\n A0 = \"\>", ",", \({2, 1, 2, 1}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"shape\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"of\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)A0 = \"\>", ",", \({6, 2, 0}\), ",", "\<\"\\n B0 = \"\>", ",", \({3, \(-1\), 2, \(-4\)}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"shape\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"of\\\",\\nFontSlant->\\\"Italic\\\"]\\) B0 = \"\>", ",", \({0, 10, 10}\), ",", "\<\"\\n AB = \"\>", ",", \({5, \(-5\), 5, \(-5\)}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"shape\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"of\\\",\\nFontSlant->\\\"Italic\\\"]\\) AB = \"\>", ",", \({0, 20, 0}\), ",", "\<\"\\n\\!\\(\\* StyleBox[\\\"Note\\\",\\nFontSlant->\\\"Italic\ \\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"that\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"sizes\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\[UpArrow] \[UpArrow]\ \\!\\(\\* StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"have\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"become\\\",\\nFontSlant->\\\"Italic\\\"]\\) 0\"\>", ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"Remainder\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"Rl\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"=\\\",\\nFontSlant->\\\"Italic\\\"]\\)\"\>", ",", \({3\/2, 3\/2, 3\/2, 3\/2}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"has\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"a\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"shape\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\) \"\>", ",", \({6, 0, 0}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"Remainder\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"Rr\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"=\\\",\\nFontSlant->\\\"Italic\\\"]\\)\"\>", ",", \({1\/2, 3\/2, \(-\(1\/2\)\), \(-\(3\/2\)\)}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"has\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"a\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\\\" \\\ \",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\"shape\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* \ StyleBox[\\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)\"\>", ",", \({0, 0, 10}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"Ainverse\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\ \\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)= \"\>", ",", \({1\/6, \(-\(1\/12\)\), 1\/6, \(-\(1\/12\)\)}\), ",", "\<\"\\nwith shape \"\>", ",", \({1\/6, 1\/2, 0}\), ",", "\<\"\\nAB/A0 = \"\>", ",", \({5\/2, \(-\(5\/2\)\), 5\/2, \(-\(5\/2\)\)}\), ",", "\<\"\\nwith shape \"\>", ",", \({0, 10, 0}\), ",", "\<\"\\nAdding Rr recovers B0\"\>", ",", \({3, \(-1\), 2, \(-4\)}\), ",", "\<\"\\n\\!\\(\\* \ StyleBox[\\\"Binverse\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\* StyleBox[\ \\\" \\\",\\nFontSlant->\\\"Italic\\\"]\\)= \"\>", ",", \({3\/40, \(-\(7\/40\)\), \(-\(1\/40\)\), 1\/8}\), ",", "\<\"\\nwith shape \"\>", ",", \({0, 1\/10, 1\/10}\), ",", "\<\"\\nAB/B0 = \"\>", ",", \({1\/2, \(-\(1\/2\)\), 1\/2, \(-\(1\/2\)\)}\), ",", "\<\"\\nwith shape \"\>", ",", \({0, 2, 0}\), ",", "\<\"\\nAdding Rl recovers A0 \"\>", ",", \({2, 1, 2, 1}\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "\t\[DoubleStruckCapitalC], \[DoubleStruckCapitalH], and \ \[DoubleStruckCapitalO] can have divisors of zero if their coefficients are \ complex.The algebras defined by the Pauli-\[Sigma] matrices (P4, P8, P16) \ conserve variations on the Minkowski metric ", Cell[BoxData[ \(TraditionalForm\`t\^2\)]], "-", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "-", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "-", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], " and can have real divisors of zero. The \"light cone\" is a zero \ sub-space.\n\tSome algebras can have \"annihilators\", real vectors with all \ sizes zero. Operations with them give results with all sizes zero (but with \ non-zero coefficients), so one remainder will be the other operand, \ unchanged. Example 18a shows this for left and right multiplication by the D3 \ annihilator A0." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; Chop[{"\", "\<\nD3 has shape\>", sh, "\<\nA0 = \>", A0 = {2, 0, \(-2\), \(-2\), 0, 2}, \[IndentingNewLine]"\<\nA0 shape\>", sh /. \ as[A0], "\<\nB = \>", B = {2, 3, \(-4\), \(-1\), 5, 2}, \[IndentingNewLine]"\<\nB shape \>", sh /. \ as[B], "\<\nAB = \>", AB = hoopTimes[A0, B], "\<\nAB shape\>", sh /. \ as[AB], "\<\nRl = \>", Rl, "\<\nRr = B\>", Rr, "\<\nBA = \>", BA = hoopTimes[B, A0], "\<\nBA shape\>", sh /. \ as[BA], "\<\nRl = B\>", Rl, "\<\nRr = \>", Rr}]\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ \({"Example 18a. D3 annihilator", "\nD3 has shape", {a + b + c + d + e + f, a - b + c - d + e - f, 1\/2\ \((\((a - c)\)\^2 - \((b - d)\)\^2 + \((c - e)\)\^2 + \((\(-a\) \ + e)\)\^2 - \((d - f)\)\^2 - \((\(-b\) + f)\)\^2)\)}, "\nA0 = ", {2, 0, \(-2\), \(-2\), 0, 2}, "\nA0 shape", {0, 0, 0}, "\nB = ", {2, 3, \(-4\), \(-1\), 5, 2}, "\nB shape ", {7, \(-1\), 50}, "\nAB = ", {0, 20, \(-20\), \(-20\), 20, 0}, "\nAB shape", {0, 0, 0}, "\nRl = ", {0, 0, 0, 0, 0, 0}, "\nRr = B", {2, 3, \(-4\), \(-1\), 5, 2}, "\nBA = ", {0, \(-10\), \(-10\), 0, 10, 10}, "\nBA shape", {0, 0, 0}, "\nRl = B", {2, 3, \(-4\), \(-1\), 5, 2}, "\nRr = ", {0, 0, 0, 0, 0, 0}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.12 Subtraction is Additive Elimination.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tNegation \"-a\" and subtraction \"b-a\" are conflated and treated as a \ single fundamental process by most mathematicians, but [3] requires over a \ hundred pages of \"Post-Modern Algebra\" before subtraction can be tackled. \ In hoops, negation is the result of an ", StyleBox["r", FontSlant->"Italic"], "=2 fold. This equivalences {", Cell[BoxData[ \(TraditionalForm\`a\_i\)]], ",", Cell[BoxData[ \(TraditionalForm\`a\_\(i + m/2\)\)]], "} to a real number ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["a", FontWeight->"Bold"], "i"], TraditionalForm]]], ", with {", Cell[BoxData[ \(TraditionalForm\`a\_\(i + m/2\)\)]], ",", Cell[BoxData[ \(TraditionalForm\`a\_i\)]], "}~-", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["a", FontWeight->"Bold"], "i"], TraditionalForm]]], " and {", Cell[BoxData[ \(TraditionalForm\`a\_i\)]], ",", Cell[BoxData[ \(TraditionalForm\`a\_i\)]], "}~", StyleBox["0", FontWeight->"Bold"], ". The first size, \[Sum]", Cell[BoxData[ \(TraditionalForm\`a\_i\)]], ", is destroyed, as it equivalences to zero. Subtraction can now be seen as \ \"additive elimination\", adding a set of coefficients that make all the \ terms of an un-folded vector identical, so that they equivalence to zero: \ {a,b}+{b,a} ={a+b,a+b} ~ ", StyleBox["0", FontWeight->"Bold"], ". (The commutative property of addition is inherited from the natural \ numbers.)\n\tWhen ", StyleBox["r", FontSlant->"Italic"], " is 3, the equivalence relationship is ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontWeight->"Bold"], "i"], TraditionalForm]]], "~{", Cell[BoxData[ \(TraditionalForm\`a\_j\)]], ",", Cell[BoxData[ \(TraditionalForm\`a\_\(j + m/3\)\)]], ",", Cell[BoxData[ \(TraditionalForm\`a\_\(j + 2 m/3\)\)]], "} ~{", Cell[BoxData[ \(TraditionalForm\`b\_j\)]], ",", Cell[BoxData[ \(TraditionalForm\`b\_\(j + m/3\)\)]], ",", Cell[BoxData[ \(TraditionalForm\`b\_\(j + 2 m/3\)\)]], "} iff (", Cell[BoxData[ \(TraditionalForm\`a\_j\)]], "+", Cell[BoxData[ \(TraditionalForm\`b\_\(j + m/3\)\)]], ") = (", Cell[BoxData[ \(TraditionalForm\`a\_\(j + m/3\)\)]], "+", Cell[BoxData[ \(TraditionalForm\`b\_\(j + 2 m/3\)\)]], ") = (", Cell[BoxData[ \(TraditionalForm\`a\_\(j + 2 m/3\)\)]], "+", Cell[BoxData[ \(TraditionalForm\`b\_j\)]], "). The \"terplex number\" {a,b,c} has a left negation (or rotation), \ {b,c,a} and a right negation {c,a,b}, with {a,b,c} +{b,c,a} +{c,a,b} ={a+b+c, \ a+b+c, a+b+c} ~ ", StyleBox["0", FontWeight->"Bold"], ". A number is eliminated by adding the two rotations of its primal form. \ This extends to other ", StyleBox["r", FontSlant->"Italic"], " values; a number is eliminated by adding the ", StyleBox["r", FontSlant->"Italic"], "-1 rotations of a primal form. Simultaneous equations can be solved in \ this way in hoop algebras, but this is not demonstrated here." }], "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["2.13. Orbits.", "Subsection", PageWidth->WindowWidth], Cell[TextData[{ "\tI define a \"unital\" sub-algebra by the restriction that \"the product \ of non-zero sizes is unity\". The name \"orbit\" is appropriate to polar \ forms of unital sub-algebras because the coefficients are restricted to \ circles in ", StyleBox["m", FontSlant->"Italic"], "-dimensional space, and the group orbits are points on these circles. \ Multiplying one orbit by another (using the same hoop) gives another orbit, \ and any remainders will also be orbits. If some sizes are zero, operations \ are restricted to a sub-algebra in which these sizes remain at zero, by \ conservation. This generates sub-orbits that have analogies with particles - \ discussed in section 3.\n \tI use the convention that orbit polar forms have \ names like OHijk, with vector forms VHijk, where H is the hoop name and \ ijk... gives the sizes. Linear sizes provide \"offsets\", displacing the \ orbit origin from zero. This may be a new concept; it introduces additive \ properties to orbits. The squared radius is a multiplicative property for \ abelian radius/angle formulations." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\<\nOC301 is {0,1,a} with \ vector form\>", OC301 = tovec /. \ as[{0, 1, a}], "\<\nOC311 is {1,1,b} with vector form\>", OC311 = tovec /. \ as[{1, 1, b}], "\<\nProducts are orbits with added angles; the polar form \ of the product of OC301 & OC311 is\>", Simplify[topol /. \ as[hoopTimes[OC301, OC311]]], "\<\nand the remainders are\>", Rl, Rr}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ \({"Example 19. C3 Orbits", "\nOC301 is {0,1,a} with vector form", {\(2\ Cos[a]\)\/3, 2\/3\ Cos[a + \(2\ \[Pi]\)\/3], 2\/3\ Cos[a - \(2\ \[Pi]\)\/3]}, "\nOC311 is {1,1,b} with vector form", {1\/3\ \((1 + 2\ Cos[b])\), 1\/3\ \((1 + 2\ Cos[b + \(2\ \[Pi]\)\/3])\), 1\/3\ \((1 + 2\ Cos[b - \(2\ \[Pi]\)\/3])\)}, "\nProducts are orbits with added angles; the polar form of the product \ of OC301 & OC311 is", {0, 1, ArcTan[Cos[a + b], Sin[a + b]]}, "\nand the remainders are", {0, 0, 0}, {1\/3, 1\/3, 1\/3}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\<\nOC4001 is \>", OC4001 = {0, 0, 1, a}, "\<\n with vector form\>", VC4001 = tovec /. \ as[OC4001], "\<\nOC4011 is\>", OC4011 = {0, 1, 1, b}, "\<\n with vector form\>"\ , VC4011 = tovec /. \ as[OC4011], "\<\nOC4101 =\>", OC4101\ = \ {1, 0, 1, c}, "\<\n with vector form\>", VC4101 = tovec /. \ as[OC4101], "\<\nOC4111 is\>", OC4111\ = \ {1, 1, 1, d}, "\<\nwith vector form\>", VC4111 = tovec /. \ as[OC4111], "\<\n\n\>", \[IndentingNewLine]Simplify[{"\", sh /. \ as[ hoopTimes[VC4001, VC4001]], "\<\nOC4101\[Times]OC4011 also has the 001 shape\n\ as the two offsets disappear \>", sh /. \ as[hoopTimes[VC4101, VC4011]], "\<\nwith remainders \>", Rl, Rr, "\<\nOC4101\[Times]OC4111 has the 101 shape\>", \ \[IndentingNewLine]sh /. \ as[hoopTimes[VC4101, VC4111]], "\<\nwith remainders \>", Rl, Rr, "\<\nOC4111\[Times]OC4111 has the 111 shape\>", sh /. \ as[hoopTimes[VC4111, VC4111]]}], "\<\nwith remainders \>", Rl, Rr}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ \({"Example 20. C4 Orbits", "\nOC4001 is ", {0, 0, 1, a}, "\n with vector form", {Cos[a]\/2, Sin[a]\/2, \(-\(Cos[a]\/2\)\), \(-\(Sin[a]\/2\)\)}, "\nOC4011 is", {0, 1, 1, b}, "\n with vector form", {1\/4\ \((1 + 2\ Cos[b])\), 1\/4\ \((\(-1\) + 2\ Sin[b])\), 1\/4\ \((1 - 2\ Cos[b])\), 1\/4\ \((\(-1\) - 2\ Sin[b])\)}, "\nOC4101 =", {1, 0, 1, c}, "\n with vector form", {1\/4\ \((1 + 2\ Cos[c])\), 1\/4\ \((1 + 2\ Sin[c])\), 1\/4\ \((1 - 2\ Cos[c])\), 1\/4\ \((1 - 2\ Sin[c])\)}, "\nOC4111 is", {1, 1, 1, d}, "\nwith vector form", {1\/4\ \((2 + 2\ Cos[d])\), Sin[d]\/2, 1\/4\ \((2 - 2\ Cos[d])\), \(-\(Sin[d]\/2\)\)}, "\n\n", {"OC4001\[Times]OC4001 has the 001 shape", {0, 0, 1}, "\nOC4101\[Times]OC4011 also has the 001 shape\nas the two offsets \ disappear ", {0, 0, 1}, "\nwith remainders ", {1\/4, 1\/4, 1\/4, 1\/4}, {1\/4, \(-\(1\/4\)\), 1\/4, \(-\(1\/4\)\)}, "\nOC4101\[Times]OC4111 has the 101 shape", {1, 0, 1}, "\nwith remainders ", {0, 0, 0, 0}, {1\/4, \(-\(1\/4\)\), 1\/4, \(-\(1\/4\)\)}, "\nOC4111\[Times]OC4111 has the 111 shape", {1, 1, 1}}, "\nwith remainders ", {0, 0, 0, 0}, {0, 0, 0, 0}}\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["3. Analogies with particles.", "Section", FontFamily->"Times New Roman"], Cell[TextData[StyleBox["Summary\n\tThe algebras based on 12-element groups, \ \"Dozal\", have elements that act as half-spin (fermionic) and unit-spin \ (bosonic) quantum operators, and have orbits that have analogies with \ fundamental particles. This leads to the possibility that hoop algebras may \ provide a new paradigm for physics. The following conjectures, which the \ author cannot develop further, are outlined:- \n\tDozal interactions, with \ remainders conserving different properties, resemble particle interactions \ and decay, with four hoops corresponding to four forces. \n\tHexal is \ supersymmetric to Dozal; it has fewer elements that are not subject to the \ Pauli exlusion principle because they are equivalence relations.\n\tStable \ orbits may relate to stable particles. \n\tOrbits resemble multi-phase \ deBroglie waves, and quadratic sizes resemble Planck areas. ", FontSlant->"Italic"]], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell["3.1. Dozal and Hexal.", "Subsubsection", PageWidth->WindowWidth], Cell["\<\ \tDozal and Hexal algebras, with 12 and 6 elements, have many analogies with \ fundamental particles. They may lead to a new paradigm for physics, based on \ finite Moufang Loops rather than Lie groups. Sporadic discoveries, spread \ over fifteen years of research, have repeatedly related different physical \ phenomena to new aspects of hoop algebra, encouraging the conjecture that \ these algebras are \"physical mathematics\".\ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.2. Elements act as Quantum Operators.", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "\tJ. J. Hamilton [4] developed a \"hypercomplex arithmetic\", with \ half-spin \"non-real, non-complex\" quantum operators {", StyleBox["a", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`\(a\^*\)\)], PageWidth->WindowWidth, Evaluatable->False], ",", StyleBox["b ", FontSlant->"Italic"], ",", Cell[BoxData[ \(TraditionalForm\`\(b\^*\)\)], PageWidth->WindowWidth, Evaluatable->False], "} as well as integer-spin quantum operators, having the following \ multiplication table:-" }], "Text", PageWidth->WindowWidth], Cell[BoxData[ RowBox[{" ", FormBox[GridBox[{ { " ", \(E\_0\), \(E\_1\), \(E\_2\), \(E\_3\), \(E\_4\), \ \(E\_5\), \(E\_6\), \(E\_7\)}, {\(E\_0\), "1", "i", "h", "k", "a", \(b\^*\), \(a\^*\), "b"}, {\(E\_1\), "i", \(-1\), "k", \(-h\), \(b\^*\), \(-a\), "b", \(-\(a\^*\)\)}, {\(E\_2\), "h", "k", "1", "i", \(a\^*\), "b", "a", \(b\^*\)}, {\(E\_3\), "k", \(-h\), "i", \(-1\), "b", \(-\(a\^*\)\), \(b\^*\), \(-a\)}, {\(E\_4\), "a", \(b\^*\), \(a\^*\), "b", "1", "i", "h", "k"}, {\(E\_5\), \(b\^*\), \(-a\), "b", \(-\(a\^*\)\), "i", \(-1\), "k", \(-h\)}, {\(E\_6\), \(a\^*\), "b", "a", \(b\^*\), "h", "k", "1", "i"}, {\(E\_7\), "b", \(-\(a\^*\)\), \(b\^*\), \(-a\), "k", \(-h\), "i", \(-1\)} }], "TraditionalForm"]}]], "Input", PageWidth->WindowWidth, Evaluatable->False], Cell[TextData[{ StyleBox["a", FontSlant->"Italic"], "* and ", StyleBox["b", FontSlant->"Italic"], "* are \"biconjugates\" (my term) with ", StyleBox["a", FontSlant->"Italic"], "**=-", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], "**=-", StyleBox["b", FontSlant->"Italic"], ". The table has four elements that square to -1; Hamilton also introduces \ several other elements and many aliases. Replacing the ambiguous \"-1\" by ii \ and including extra rows and columns that match his multiplication rules \ gives a 12\[Times]12 table with his table embedded (in boldface, with ii in \ place of -1):-" }], "Text", PageWidth->WindowWidth], Cell[BoxData[GridBox[{ {"1", "i", StyleBox["ii", FontWeight->"Plain"], "h", "k", StyleBox[\(-h\), FontWeight->"Plain"], "a", \(b\^*\), StyleBox[\(-a\), FontWeight->"Plain"], \(a\^*\), "b", StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"]}, {"i", "ii", StyleBox["1", FontWeight->"Plain"], "k", \(-h\), StyleBox["h", FontWeight->"Plain"], \(b\^*\), \(-a\), StyleBox["a", FontWeight->"Plain"], "b", \(-\(a\^*\)\), StyleBox[\(a\^*\), FontWeight->"Plain"]}, { StyleBox["ii", FontWeight->"Plain"], StyleBox["1", FontWeight->"Plain"], StyleBox["i", FontWeight->"Plain"], StyleBox[\(-h\), FontWeight->"Plain"], StyleBox["h", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain"], StyleBox[\(-a\), FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain"], StyleBox[\(b\^*\), FontWeight->"Plain"], StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"], StyleBox[\(a\^*\), FontWeight->"Plain"], StyleBox["b", FontWeight->"Plain"]}, {"h", "k", StyleBox[\(-h\), FontWeight->"Plain"], "1", "i", StyleBox["ii", FontWeight->"Plain"], \(a\^*\), "b", StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"], "a", \(b\^*\), StyleBox[\(-a\), FontWeight->"Plain"]}, {"k", \(-h\), StyleBox["h", FontWeight->"Plain"], "i", "ii", StyleBox["1", FontWeight->"Plain"], "b", \(-\(a\^*\)\), StyleBox[\(a\^*\), FontWeight->"Plain"], \(b\^*\), \(-a\), StyleBox["a", FontWeight->"Plain"]}, { StyleBox[\(-h\), FontWeight->"Plain"], StyleBox["h", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain"], StyleBox["ii", FontWeight->"Plain"], StyleBox["1", FontWeight->"Plain"], StyleBox["i", FontWeight->"Plain"], StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"], StyleBox[\(a\^*\), FontWeight->"Plain"], StyleBox["b", FontWeight->"Plain"], StyleBox[\(-a\), FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain"], StyleBox[\(b\^*\), FontWeight->"Plain"]}, {"a", \(b\^*\), StyleBox[\(-a\), FontWeight->"Plain"], \(a\^*\), "b", StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"], "1", "i", StyleBox["ii", FontWeight->"Plain"], "h", "k", StyleBox[\(-h\), FontWeight->"Plain"]}, {\(b\^*\), \(-a\), StyleBox["a", FontWeight->"Plain"], "b", \(-\(a\^*\)\), StyleBox[\(a\^*\), FontWeight->"Plain"], "i", "ii", StyleBox["1", FontWeight->"Plain"], "k", \(-h\), StyleBox["h", FontWeight->"Plain"]}, { StyleBox[\(-a\), FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain"], StyleBox[\(b\^*\), FontWeight->"Plain"], StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"], StyleBox[\(a\^*\), FontWeight->"Plain"], StyleBox["b", FontWeight->"Plain"], StyleBox["ii", FontWeight->"Plain"], StyleBox["1", FontWeight->"Plain"], StyleBox["i", FontWeight->"Plain"], StyleBox[\(-h\), FontWeight->"Plain"], StyleBox["h", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain"]}, {\(a\^*\), "b", StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"], "a", \(b\^*\), StyleBox[\(-a\), FontWeight->"Plain"], "h", "k", StyleBox[\(-h\), FontWeight->"Plain"], "1", "i", StyleBox["ii", FontWeight->"Plain"]}, {"b", \(-\(a\^*\)\), StyleBox[\(a\^*\), FontWeight->"Plain"], \(b\^*\), \(-a\), StyleBox["a", FontWeight->"Plain"], "k", \(-h\), StyleBox["h", FontWeight->"Plain"], "i", "ii", StyleBox["1", FontWeight->"Plain"]}, { StyleBox[\(-\(a\^*\)\), FontWeight->"Plain"], StyleBox[\(a\^*\), FontWeight->"Plain"], StyleBox["b", FontWeight->"Plain"], StyleBox[\(-a\), FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain"], StyleBox[\(b\^*\), FontWeight->"Plain"], StyleBox[\(-h\), FontWeight->"Plain"], StyleBox["h", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain"], StyleBox["ii", FontWeight->"Plain"], StyleBox["1", FontWeight->"Plain"], StyleBox["i", FontWeight->"Plain"]} }]], "Input", PageWidth->WindowWidth], Cell[TextData[{ "\tThis is isomorphic to C3K, so ", StyleBox["i", FontSlant->"Italic"], " and ", StyleBox["ii", FontSlant->"Italic"], " are interpreted as a cube roots of 1. His half-spin operators {", StyleBox["a, a*, -a\[Congruent] a**,b, b*, -b\[Congruent] b**\[Congruent] \ -a*", FontSlant->"Italic"], "} are {", StyleBox["a, a h, a i i, a h i, a i , a h i i", FontSlant->"Italic"], "}; conjugation ", StyleBox["a", FontSlant->"Italic"], "* is multiplication by ", StyleBox["h", FontSlant->"Italic"], " (with ", StyleBox["h h", FontSlant->"Italic"], " =1), but biconjugation ", StyleBox["a**", FontSlant->"Italic"], " becomes multiplication by ", StyleBox["i", FontSlant->"Italic"], " or ", StyleBox["ii", FontSlant->"Italic"], ". His half-spin operators are still valid, but they are now defined in \ terms of two distinct square roots of 1 (", StyleBox["h & a", FontSlant->"Italic"], ") and the cube roots (", StyleBox["i, ii", FontSlant->"Italic"], ") of 1. Complex numbers are not involved." }], "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.2. Half-spin and Dozal algebra.", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "\tBanded sets of differential equations generate multi-phase sinusoids \ (just as ", Cell[BoxData[ \(TraditionalForm\`\(x\_1\)\& . \)]], StyleBox[" = a ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\_\(\(2\)\(,\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(x\_2\)\& . \)\)\)]], StyleBox["= -a", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ", equivalent to ", Cell[BoxData[ \(TraditionalForm\`x\&\[DoubleDot]\)]], " = -", Cell[BoxData[ \(TraditionalForm\`a\^2\)]], "x, generates ordinary sinusoids). A relevant simple set is the ", StyleBox["m", FontSlant->"Italic"], " equations (with ", StyleBox["m", FontSlant->"Italic"], "-1 being independent):- \n ", Cell[BoxData[ StyleBox[\(D[x\_j, t] = \(c'\)[t] + a \((x\_\(Mod[j + k, m]\) - x\_\(Mod[j - k, m]\))\), j = \(0. .. \)\ m - 1, k < m/2. \), FontFamily->"Times New Roman", FontWeight->"Plain"]], "Input"], "\nSubstituting the trial function ", Cell[BoxData[ \(x\_j = c[t] + \(c\_1\) Sin[c\_2\ t + \[Phi] + 2 j\ k\ \[Pi]/m]\)], FontFamily->"Times New Roman"], " leads to ", Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \ \)\(x\_j = c[t] + \[Sum]\_p\( c\_p\) Sin[2 a\ Sin[2 k\ \[Pi]/m]\ t + \[Phi]\_p + 2 j\ k\ \[Pi]/m]\)\)\)], "Input", FontFamily->"Times New Roman", FontWeight->"Plain"], ".\n", StyleBox["j", FontSlant->"Italic"], " is one of the ", StyleBox["m", FontSlant->"Italic"], " phases; c[t] gives freedom from the origin;", StyleBox[" ", FontSlant->"Italic"], " ", Cell[BoxData[ \(c\_p\)], FontFamily->"Times New Roman"], " and ", Cell[BoxData[ StyleBox[\(\[Phi]\_p\), FontFamily->"Times New Roman"]]], " are phase amplitude and phase offset constants which hav to be \ subscripted because independent sets can arise; ", StyleBox["k", FontSlant->"Italic"], " is a phase step. Sin[2k \[Pi]/m] is a constant that determines the \ frequency & period. When ", StyleBox["m", FontSlant->"Italic"], "=4, ", StyleBox["k", FontSlant->"Italic"], " must be 1; deflating then removes the c[t] term and reproduces the \ ordinary sinusoid equation-pair with double the amplitude. Other values of ", StyleBox["m", FontSlant->"Italic"], " and ", StyleBox["k", FontSlant->"Italic"], " give a frequency reduction.\n\tThe period is 2\[Pi]/Sin[2k \[Pi]/m], and \ the function is non-sinusoidal (constant) if 2/m is integral.\n\tHalf-spin \ with a period of 4\[Pi] results when ", StyleBox["m", FontSlant->"Italic"], "=12, k=1. The twelve components are elements of the C12 hoop; its C3C4 \ isomorph is the preferred isomorph because it has sizes in common with the \ other Dozal hoops C3K, Q12, D3C2, and A4.\n\t A lot more research is needed \ here. The differential equations need to be related to hoops; the resonances \ (orbits with frequencies that are integer or half-integer multiples of the \ fundamental frequency) need study; what is the relationship between hoops and \ the theory of strings and branes, etc." }], "Text", PageWidth->WindowWidth, CellMargins->{{Inherited, 0}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3.3. Sizes and Orbits as Planck Areas & deBroglie Waves with polarization & \ chirality.\ \>", "Subsubsection", PageWidth->WindowWidth], Cell["\<\ \tC3K has a polar dual that can be expressed as {\[Alpha], \[Beta], \[Gamma], \ \[Delta], \[Zeta]\[Zeta], \[Tau], \[Lambda]\[Lambda], \[Phi], \[Eta]\[Eta], \ \[Chi], \[Kappa]\[Kappa], \[Psi]} and reverted to a vector as follows:-\ \>", "Text", PageWidth->WindowWidth], Cell[BoxData[ \(\(\(\ \ \ \ \)\(C3KO = {\[IndentingNewLine]\[Alpha] + \[Beta] + \ \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Lambda]\ \ Cos[\[Phi]]\ \ \ \ \ + \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ + \[Kappa]\ \ Cos[\[Psi]], \[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau] + \ \[Omega]] + \[Lambda]\ Cos[\[Phi] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \ \[Omega]] + \[Kappa]\ Cos[\[Psi] + \[Omega]], \ \[IndentingNewLine]\[Alpha] + \ \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ \ Cos[\[Phi] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \[Omega]] + \[Kappa]\ \ Cos[\[Psi] - \[Omega]], \[IndentingNewLine]\[Alpha] - \[Beta] + \[Gamma] - \ \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ + \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ - \[Kappa]\ Cos[\[Psi]], \ \[Alpha] - \ \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Lambda]\ \ Cos[\[Phi] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \[Omega]] - \[Kappa]\ \ Cos[\[Psi] + \[Omega]], \ \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ \ Cos[\[Tau] - \[Omega]] - \[Lambda]\ Cos[\[Phi] - \[Omega]] + \[Eta]\ Cos[\ \[Chi] - \[Omega]] - \[Kappa]\ Cos[\[Psi] - \[Omega]], \ \[Alpha] + \[Beta] - \ \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Lambda]\ \ Cos[\[Phi]]\ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ - \[Kappa]\ \ Cos[\[Psi]], \ \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] \ + \[Omega]] + \[Lambda]\ Cos[\[Phi] + \[Omega]] - \[Eta]\ Cos[\[Chi] + \ \[Omega]] - \[Kappa]\ Cos[\[Psi] + \[Omega]], \ \[Alpha] + \[Beta] - \[Gamma] \ - \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ Cos[\[Phi] - \ \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]] - \[Kappa]\ Cos[\[Psi] - \ \[Omega]], \ \[Alpha] - \[Beta] - \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ \ + \[Kappa]\ Cos[\[Psi]], \ \[Alpha] - \[Beta] - \[Gamma] + \[Delta] + \ \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Lambda]\ Cos[\[Phi] + \[Omega]] - \[Eta]\ \ Cos[\[Chi] + \[Omega]] + \[Kappa]\ Cos[\[Psi] + \[Omega]], \ \[Alpha] - \ \[Beta] - \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ \ Cos[\[Phi] - \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]] + \[Kappa]\ \ Cos[\[Psi] - \[Omega]]}/12;\)\)\)], "Input", FormatType->TraditionalForm, FontSize->9], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; topol\)], "Input", CellOpen->False], Cell[BoxData[ \({a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, a + b + c + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, 1\/2\ \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h \ - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\), ArcTan[2\ a - b - c + 2\ d - e - f + 2\ g - h - i + 2\ j - k - l, \(-\@3\)\ \((b - c + e - f + h - i + k - l)\)], 1\/2\ \((\((a - b - d + e + g - h - j + k)\)\^2 + \((\(-a\) + c + d - f \ - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\), ArcTan[2\ a - b - c - 2\ d + e + f + 2\ g - h - i - 2\ j + k + l, \(-\@3\)\ \((b - c - e + f + h - i - k + l)\)], 1\/2\ \((\((a - b + d - e - g + h - j + k)\)\^2 + \((\(-a\) + c - d + f \ + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\), ArcTan[2\ a - b - c + 2\ d - e - f - 2\ g + h + i - 2\ j + k + l, \(-\@3\)\ \((b - c + e - f - h + i - k + l)\)], 1\/2\ \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - c - e + f - h \ + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g - i - j + l)\)\^2)\), ArcTan[2\ a - b - c - 2\ d + e + f - 2\ g + h + i + 2\ j - k - l, \(-\@3\)\ \((b - c - e + f - h + i + k - l)\)]}\)], "Output", FontSize->9] }, Open ]], Cell[TextData[{ "\tHere the substitutions \[Zeta]=2", Cell[BoxData[ \(TraditionalForm\`\@\[Zeta]\[Zeta]\)]], ", \[Lambda]=2", Cell[BoxData[ \(TraditionalForm\`\@\[Lambda]\[Lambda]\)]], ", \[Eta]=2", Cell[BoxData[ \(TraditionalForm\`\@\[Eta]\[Eta]\)]], ", \[Kappa]=2", Cell[BoxData[ \(TraditionalForm\`\@\[Kappa]\[Kappa]\)]], " are made, to give a compact expression. \n\tNote that \[Zeta]\[Zeta], \ \[Lambda]\[Lambda], \[Eta]\[Eta], & \[Kappa]\[Kappa] represent quadratic \ sizes that are expressible as a sums of squares, so they are positive for \ real elements. I conjecture that they correspond to Planck areas, and provide \ a fundamental scale; their square roots provide radii (2", Cell[BoxData[ \(TraditionalForm\`\@\[Zeta]\[Zeta]\)]], "/12 etc) for multiphase deBroglie-like sinusoids. The angles \[Tau], \ \[Phi], \[Chi], & \[Psi] are \"hidden variables\" in the vector form. PROBLEM \ 1 - how do these relate to space-time? PROBLEM 2 - resonances and higher \ frequencies?" }], "Text", PageWidth->WindowWidth], Cell["\<\ \tC3C4 has a polar dual that can be expressed as {\[Alpha], \[Beta], \ \[Epsilon]\[Epsilon], \[Sigma], \[Zeta]\[Zeta], \[Tau], \[Lambda]\[Lambda], \ \[Phi], \[Eta]\[Eta], \[Chi], \[Kappa]\[Kappa], \[Psi]} (with {\[Alpha], \ \[Beta], \[Zeta]\[Zeta], \[Tau], \[Lambda]\[Lambda], \[Phi], \[Eta]\[Eta]} in \ common with C3K) :-\ \>", "Text"], Cell[BoxData[ \({\[Alpha] = a + b + c + d + e + f + g + h + i + j + k + l, \[Beta] = a + b + c - d - e - f + g + h + i - j - k - l, \[Epsilon]\[Epsilon] = \((a + b + c - g - h - i)\)\^2 + \((d + e \ + f - j - k - l)\)\^2, \[Sigma] = ArcTan[a + b + c - g - h - i, d + e + f - j - k - l], \[Zeta]\[Zeta] = \((\((a - b + d - e + g - h + j - k)\)\^2 + \ \((b - c + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + \ l)\)\^2)\)/2, \[Tau] = ArcTan[2\ a - b - c + 2\ d - e - f + 2\ g - h - i + 2\ j - k - l, \(-\@3\)\ \((b - c + e - f + h - i + k - l)\)], \[Lambda]\[Lambda] = \((\((a - b - d + e + g - h - j + \ k)\)\^2 + \((\(-a\) + c + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h \ - i - k + l)\)\^2)\)/2, \[Phi] = ArcTan[2\ a - b - c - 2\ d + e + f + 2\ g - h - i - 2\ j + k + l, \(-\@3\)\ \((b - c - e + f + h - i - k + l)\)], \[Eta]\[Eta] = \ \((\((a - b + d - e - g + h - j + \ k)\)\^2 + \((\(-a\) + c - d + f + g - i + j - l)\)\^2 + \((b - c + e - f - h \ + i - k + l)\)\^2)\)/2, \[Chi] = ArcTan[2\ a - b - c + 2\ d - e - f - 2\ g + h + i - 2\ j + k + l, \(-\@3\)\ \((b - c + e - f - h + i - k + l)\)], \[Kappa]\[Kappa] = \((\((a - b - d + e - g + h + j - \ k)\)\^2 + \((b - c - e + f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g \ - i - j + l)\)\^2)\)/2, \[Psi] = ArcTan[2\ a - b - c - 2\ d + e + f - 2\ g + h + i + 2\ j - k - l, \(-\@3\)\ \((b - c - e + f - h + i + k - l)\)]}\)], "Input", FontSize->9], Cell["\<\ \tIt can be reverted as follows:-\ \>", "Text", PageWidth->WindowWidth], Cell[BoxData[ \(\(\(\ \ \)\(C3C4O = {\[IndentingNewLine]\[Alpha] + \[Beta] + \ \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Lambda]\ \ Cos[\[Phi]]\ \ \ \ \ \ \ \ \ \ + \[Eta]\ Cos[x] + \[Eta]\ r3\ Sin[ x] + \[Kappa]\ Cos[y] + \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine]\[Alpha] + \[Beta] + \ \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Lambda]\ \ Cos[\[Phi] + \[Omega]]\ \ \ \ \ - \[Eta]\ Cos[x] + \[Eta]\ r3\ Sin[ x] - \[Kappa]\ Cos[y] + \[Kappa]\ r3\ Sin[ y], \ \[IndentingNewLine]\[Alpha] + \[Beta] + \ \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ \ Cos[\[Phi] - \[Omega]] - \[Eta]\ r6\ Sin[ x]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \[Kappa]\ r6\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] + \ \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ - \[Lambda]\ \ Cos[\[Phi]]\ \ \ \ \ \ + \[Eta]\ r3\ Cos[x]\ \ \ \ \ - \[Eta]\ Sin[ x] - \[Kappa]\ r3\ Cos[y] + \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] + \ \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Lambda]\ \ Cos[\[Phi] + \[Omega]] + \[Eta]\ r3\ Cos[x]\ \ \ \ \ + \[Eta]\ Sin[ x] - \[Kappa]\ r3\ Cos[y] - \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] + \ \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ \ Cos[\[Phi] - \[Omega]] - \[Eta]\ r6\ Cos[ x]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \[Kappa]\ r6\ Cos[ y], \[IndentingNewLine]\[Alpha] + \[Beta] - \ \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ + \[Lambda]\ \ Cos[\[Phi]]\ \ \ \ \ \ \ \ \ \ \ - \[Eta]\ Cos[x] - \[Eta]\ r3\ Sin[ x] - \[Kappa]\ Cos[y] - \[Kappa]\ r3\ Sin[ y]\ , \[IndentingNewLine]\[Alpha] + \[Beta] - \ \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Lambda]\ \ Cos[\[Phi] + \[Omega]]\ \ \ \ \ + \[Eta]\ Cos[x] - \[Eta]\ r3\ Sin[ x] + \[Kappa]\ Cos[y] - \[Kappa]\ r3\ Sin[ y], \ \[IndentingNewLine]\[Alpha] + \[Beta] - \ \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ \ Cos[\[Phi] - \[Omega]] + \[Eta]\ r6\ Sin[ x]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \[Kappa]\ r6\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] - \ \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ - \[Lambda]\ \ Cos[\[Phi]]\ \ \ \ \ \ - \[Eta]\ r3\ Cos[x]\ \ \ \ \ + \[Eta]\ Sin[ x] + \[Kappa]\ r3\ Cos[y] - \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] - \ \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Lambda]\ \ Cos[\[Phi] + \[Omega]] - \[Eta]\ r3\ Cos[x]\ \ \ \ \ - \[Eta]\ Sin[ x] + \[Kappa]\ r3\ Cos[y] + \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] - \ \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ \ Cos[\[Phi] - \[Omega]] + \[Eta]\ r6\ Cos[ x]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \[Kappa]\ \ r6\ Cos[y]}/12;\)\)\)], "Input", PageWidth->WindowWidth, CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, FontSize->9], Cell[TextData[{ "(Here \[Epsilon]=2", Cell[BoxData[ \(TraditionalForm\`\@\[Epsilon]\[Epsilon]\)]], ", \[Zeta]=2", Cell[BoxData[ \(TraditionalForm\`\@\[Zeta]\[Zeta]\)]], ", \[Lambda]=2", Cell[BoxData[ \(TraditionalForm\`\@\[Lambda]\[Lambda]\)]], ", \[Eta]=", Cell[BoxData[ \(TraditionalForm\`\@\(3 \[Eta]\[Eta]\)\)]], ", \[Kappa]=", Cell[BoxData[ \(TraditionalForm\`\@\(3 \[Kappa]\[Kappa]\)\)]], ", x=\[Chi]+\[Pi]/6, y=\[Psi]+\[Pi]/6, \[Omega]=2\[Pi]/3, r3=", Cell[BoxData[ \(TraditionalForm\`\@3\)]], ", r6=2", Cell[BoxData[ \(TraditionalForm\`\@3\)]], ")\n\tOrbits are obtained by setting some of the parameters {\[Alpha], \ \[Beta], \[Gamma], \[Delta], \[Epsilon]\[Epsilon], \[Zeta]\[Zeta] ,\[Lambda]\ \[Lambda] , ", Cell[BoxData[ \(TraditionalForm\`\[Eta]\[Eta]\)]], ", \[Kappa]\[Kappa]} to zero or 1 and constraining the product of the \ remainder to be 1.\n\tSimilar, but simpler, expressions give Q12 & D3C2 \ orbits.The parameters are related to those for C3K and C3C4." }], "Text", PageWidth->WindowWidth], Cell[BoxData[ \(\(Q12O = \[IndentingNewLine]{\ \ \((\[Alpha] + \[Beta] + \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Eta]\ Cos[\[Chi]])\), \ \[IndentingNewLine]\((\[Alpha] + \[Beta] + \[Epsilon]\ Cos[\[Sigma]] + \ \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] + \[Beta] + \[Epsilon]\ Cos[\[Sigma]] + \ \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] - \[Beta] + \[Epsilon]\ Sin[\[Sigma]]\ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ + \[CurlyKappa]\ Cos[\[Psi]])\), \ \[IndentingNewLine]\((\[Alpha] - \[Beta] + \[Epsilon]\ Sin[\[Sigma]]\ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi] + \[Omega]] + \[CurlyKappa]\ Cos[\[Psi] + \[Omega]])\ \), \[IndentingNewLine]\((\[Alpha] - \[Beta] + \[Epsilon]\ Sin[\[Sigma]]\ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi] - \[Omega]] + \[CurlyKappa]\ Cos[\[Psi] - \ \[Omega]])\), \[IndentingNewLine]\((\[Alpha] + \[Beta] - \[Epsilon]\ Cos[\ \[Sigma]] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ - \[Eta]\ Cos[\[Chi]])\), \ \[IndentingNewLine]\((\[Alpha] + \[Beta] - \[Epsilon]\ Cos[\[Sigma]] + \ \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Eta]\ Cos[\[Chi] + \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] + \[Beta] - \[Epsilon]\ Cos[\[Sigma]] + \ \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] - \[Beta] - \[Epsilon]\ Sin[\[Sigma]]\ \ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ - \[CurlyKappa]\ Cos[\[Psi]])\), \ \[IndentingNewLine]\((\[Alpha] - \[Beta] - \[Epsilon]\ Sin[\[Sigma]]\ \ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi] - \[Omega]] - \[CurlyKappa]\ Cos[\[Psi] + \ \[Omega]])\), \[IndentingNewLine]\((\[Alpha] - \[Beta] - \[Epsilon]\ Sin[\ \[Sigma]]\ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi] - \[Omega]] - \[CurlyKappa]\ \ Cos[\[Psi] - \[Omega]])\)}/12;\)\)], "Input", PageWidth->WindowWidth, FontSize->9], Cell["\<\ \tThis (and D3C2O, below) gives a 12-phase system if \ \[Psi]=\[Chi]\[PlusMinus]\[Pi]/6 & \[CurlyKappa]=\[Eta]. A polarised 3-phase \ system (with polarisation angle \[CurlyPhi]) is given by \[Phi]=\[Tau]+\ \[CurlyPhi] & \[Lambda]=\[PlusMinus]\[Zeta].\ \>", "Text", PageWidth->WindowWidth], Cell[BoxData[ \(\(D3C2O = \[IndentingNewLine]{\ \ \((\[Alpha] + \[Beta] + \[Gamma] + \ \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Eta]\ Cos[\[Chi]])\), \ \[IndentingNewLine]\((\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ \ Cos[\[Tau] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ \ Cos[\[Tau] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] - \[Beta] + \[Gamma] - \[Delta]\ \ \ \ \ \ - \ \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ - \[CurlyKappa]\ Cos[\[Psi]])\), \ \[IndentingNewLine]\((\[Alpha] - \[Beta] + \[Gamma] - \[Delta]\ \ \ \ \ \ - \ \[Lambda]\ Cos[\[Phi] + \[Omega]] - \[CurlyKappa]\ Cos[\[Psi] + \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] - \[Beta] + \[Gamma] - \[Delta]\ \ \ \ \ \ - \ \[Lambda]\ Cos[\[Phi] - \[Omega]] - \[CurlyKappa]\ Cos[\[Psi] - \[Omega]])\), \ \[IndentingNewLine]\((\[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ \ Cos[\[Tau]]\ \ \ \ \ \ - \[Eta]\ Cos[\[Chi]])\), \[IndentingNewLine]\((\ \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \ \[Eta]\ Cos[\[Chi] + \[Omega]])\), \[IndentingNewLine]\((\[Alpha] + \[Beta] - \ \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Eta]\ Cos[\[Chi] - \ \[Omega]])\), \[IndentingNewLine]\((\[Alpha] - \[Beta] - \[Gamma] + \[Delta]\ \ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ + \[CurlyKappa]\ Cos[\[Psi]])\ \), \[IndentingNewLine]\((\[Alpha] - \[Beta] - \[Gamma] + \[Delta]\ \ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi] + \[Omega]] + \[CurlyKappa]\ Cos[\[Psi] + \[Omega]])\ \), \[IndentingNewLine]\((\[Alpha] - \[Beta] - \[Gamma] + \[Delta]\ \ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi] - \[Omega]] + \[CurlyKappa]\ Cos[\[Psi] - \[Omega]])\ \)}/12;\)\)], "Input", PageWidth->WindowWidth, FontSize->9], Cell["\<\ \tEach set of angles involving \[Omega]=2\[Pi]/3 gives a chiral orbit; the \ chirality is reversed if the angle is negated.\ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.4. C5, C7 & C8 Orbits have unstable amplitudes.", "Subsubsection", PageWidth->WindowWidth, FontFamily->"Times New Roman"], Cell[TextData[{ "\tThe orbits of C3 have squared radius of magnitude 2/3, with offsets \ {0,0,0} or {1/3, 1/3, 1/3). Those of C4 have squared radii or ulnae of \ magnitude 1/2, and offsets {0,..} or {\[PlusMinus]1/4,..}. Orbits of C5, C7, \ Cp (p prime>3) etc can be developed, with p sinusoidal phases, but their \ radii are unconstrained, though their offsets are 0 or 1/p. Example 21 shows \ that OC5dr, a five-phase orbit with a displacement d and radius ", StyleBox["r", FontSlant->"Italic"], ", has a quartic size of zero and so is unconstrained. The angle, linear L1 \ and quartic parameters only use 3 of the 5 degrees of freedom. Similarly, a \ seven-phase orbit with radius ", StyleBox["r", FontSlant->"Italic"], " has a zero size in C7 (I have not found a polar expression for this \ group) and an eight-phase orbit with radius ", StyleBox["r", FontSlant->"Italic"], " has a zero size in C8. These orbits also appear to be unstable - \ numerical integration of the relevant differential-equation sets is \ unstable." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell["\<\ {Use[\"C5\"];\"Example 21. Zero-sized C5 orbit.\\nOC5dr is\",OC5dr = tovec/. as[{d, r, \[Sigma]}], \"\\nThe radius disappears from the shape\", Simplify[sh/. as[OC5dr]]}\ \>", "Input", PageWidth->WindowWidth, CellOpen->False, FontFamily->"Courier"], Cell[BoxData[ \({"Example 21. Zero-sized C5 orbit.\nOC5dr is", {1\/5\ \((d + r\ Sin[\[Sigma]])\), 1\/5\ \((d + r\ Sin[\(2\ \[Pi]\)\/5 + \[Sigma]])\), 1\/5\ \((d + r\ Sin[\(4\ \[Pi]\)\/5 + \[Sigma]])\), 1\/5\ \((d + r\ Sin[\(6\ \[Pi]\)\/5 + \[Sigma]])\), 1\/5\ \((d + r\ Sin[\(8\ \[Pi]\)\/5 + \[Sigma]])\)}, "\nThe radius disappears from the shape", {d, 0}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3.5. The limited number of stable orbits may limit the number of stable \ particles.\ \>", "Subsubsection", PageWidth->WindowWidth], Cell["\<\ \tI surmise that stable orbits represent stable particles, and that the \ number of stable particles is determined by the limited number of stable \ particles. \tStable, finite amplitude, orbits with 3, 4, & 6 phases have been found, \ some of which have polarization and chirality. Some of these require the \ angles to be related. The following examples are restrictions (with various \ radii and offsets set to zero) of the tables in section 3.3.\ \>", "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \({"\", \ 12 C3KO /. {\[Lambda] \[Rule] 0, \[Eta] \[Rule] 0, \[Kappa] \[Rule] 0, \[CurlyKappa] \[Rule] 0} // tf}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 22. 3-phase C3K Orbit (0mitting the 1/12 \ factor).\\n\"\>", ",", TagBox[ FormBox[\({\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ \ \(cos(\[Tau])\), \[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ \(cos(\ \[Tau] + \[Omega])\), \[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ \ \(cos(\[Tau] - \[Omega])\), \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \ \[Zeta]\ \(cos(\[Tau])\), \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ \ \(cos(\[Tau] + \[Omega])\), \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \ \[Zeta]\ \(cos(\[Tau] - \[Omega])\), \[Alpha] + \[Beta] - \[Gamma] - \[Delta] \ + \[Zeta]\ \(cos(\[Tau])\), \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \ \[Zeta]\ \(cos(\[Tau] + \[Omega])\), \[Alpha] + \[Beta] - \[Gamma] - \[Delta] \ + \[Zeta]\ \(cos(\[Tau] - \[Omega])\), \[Alpha] - \[Beta] - \[Gamma] + \ \[Delta] + \[Zeta]\ \(cos(\[Tau])\), \[Alpha] - \[Beta] - \[Gamma] + \[Delta] \ + \[Zeta]\ \(cos(\[Tau] + \[Omega])\), \[Alpha] - \[Beta] - \[Gamma] + \ \[Delta] + \[Zeta]\ \(cos(\[Tau] - \[Omega])\)}\), "TraditionalForm"], TraditionalForm, Editable->True]}], "}"}]], "Output"] }, Open ]], Cell["\<\ \tThe linear terms {\[Alpha], \[Beta], \[Gamma], \[Delta]} are offsets (\ \[Alpha]/12 etc) from zero. Square roots of the quadratic terms appear as \ radii with scaling factors. Each phase occurs four times in a 3-phase dozal \ orbit, with different linear offsets. \tThe offsets and scaled radii are omitted in later examples.\ \>", "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \({"\", \ 12 C3C4O/\[Zeta] /. {\[Alpha] \[Rule] 0, \[Beta] -> 0, \[Epsilon] \[Rule] 0, \[Lambda] \[Rule] 0, \[Eta] \[Rule] 0, \[Kappa] \[Rule] 0} // tf}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 23. 3-phase C3C4 Orbit (omitting \[Alpha], \ \[Beta], & the \[Zeta]/12 factor).\"\>", ",", TagBox[ FormBox[\({cos(\[Tau]), cos(\[Tau] + \[Omega]), cos(\[Tau] - \[Omega]), cos(\[Tau]), cos(\[Tau] + \[Omega]), cos(\[Tau] - \[Omega]), cos(\[Tau]), cos(\[Tau] + \[Omega]), cos(\[Tau] - \[Omega]), cos(\[Tau]), cos(\[Tau] + \[Omega]), cos(\[Tau] - \[Omega])}\), "TraditionalForm"], TraditionalForm, Editable->True]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["topol", FontSlant->"Italic"], " shows that the squared amplitude ", Cell[BoxData[ \(TraditionalForm\`\((\[Zeta]/12)\)\^2\)]], "is the sum of three squares:-\n", Cell[BoxData[ StyleBox[\(\((\((a - b - d + e + g - h - j + k)\)\^2 + \((\(-a\) + c + \ d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\)/72\), FontFamily->"Times New Roman", FontWeight->"Plain"]], "Input"] }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \({"\", \ 12 C3C4O/\[Epsilon] /. {\[Alpha] \[Rule] 0, \[Beta] -> 0, \[Zeta] \[Rule] 0, \[Lambda] \[Rule] 0, \[Eta] \[Rule] 0, \[Kappa] \[Rule] 0} // tf}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 24. 4-phase C3C4 Orbit \\n(omitting \[Epsilon]/12 \ factor).\"\>", ",", TagBox[ FormBox[\({cos(\[Sigma]), cos(\[Sigma]), cos(\[Sigma]), sin(\[Sigma]), sin(\[Sigma]), sin(\[Sigma]), \(-\(cos(\[Sigma])\)\), \(-\(cos(\[Sigma])\)\), \ \(-\(cos(\[Sigma])\)\), \(-\(sin(\[Sigma])\)\), \(-\(sin(\[Sigma])\)\), \ \(-\(sin(\[Sigma])\)\)}\), "TraditionalForm"], TraditionalForm, Editable->True]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "Each of the four phases occur three times. ", StyleBox["topol", FontSlant->"Italic"], " shows that the squared amplitude ", Cell[BoxData[ \(TraditionalForm\`\((\[Epsilon]/12)\)\^2\)]], " is the sum of two squares:-\n", Cell[BoxData[ \(\((\((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - l)\)\^2)\)/ 144\)], "Text", FontFamily->"Times New Roman"] }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \({"\", \ Simplify[12 C3KO/\[Eta] /. {\[Alpha] \[Rule] 0, \[Beta] \[Rule] 0, \[Gamma] \[Rule] 0, \[Delta] \[Rule] 0, \[Lambda] \[Rule] 0, \[Psi] \[Rule] 0, \[Zeta] \[Rule] 0, \[CurlyKappa] \[Rule] 0}] // tf}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 25. 6-phase C3K Orbit.\\n\"\>", ",", TagBox[ FormBox[\({cos(\[Chi]), cos(\[Chi] + \[Omega]), cos(\[Chi] - \[Omega]), cos(\[Chi]), cos(\[Chi] + \[Omega]), cos(\[Chi] - \[Omega]), \(-\(cos(\[Chi])\)\), \(-\(cos(\[Chi] + \ \[Omega])\)\), \(-\(cos(\[Chi] - \[Omega])\)\), \(-\(cos(\[Chi])\)\), \ \(-\(cos(\[Chi] + \[Omega])\)\), \(-\(cos(\[Chi] - \[Omega])\)\)}\), "TraditionalForm"], TraditionalForm, Editable->True]}], "}"}]], "Output"] }, Open ]], Cell["\<\ \tThe linear terms, and the \[Eta]/12 factor, have been omitted. Each phase \ occurs twice, with different offsets. \ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.6. Particle Interactions.", "Subsubsection", PageWidth->WindowWidth], Cell["\<\ \tAs Dozal elements resemble quantum operators, the addition/splitting and \ multiplication/division of Dozal vectors should resemble particle decays and \ interactions. All sizes are conserved on multiplication using one or more of \ the groups, so different groups could correspond to interactions according to \ specific force laws. C3K and D3C2 conserve the linear sizes {\[Alpha], \ \[Beta], \[Gamma], \[Delta]} and C3C4 and Q12 conserve {\[Alpha], \[Beta]} \ on addition and subtraction, corresponding to absorption or emission of \ particles. Multiplication, with remainders, conserves overlapping sets of \ sizes for each group. Conjecture - this may correspond to interactions \ subject to the four forces. (What is the role of A4 here?) Many other \ functions are conserved on multiplication of orbits, rather than general \ vecs. I have not been able to correlate them with conserved particle \ properties. Resonances are an outstanding problem; if hoops are related to \ differential equations (as the multi-phase sinusoidal orbits suggest) there \ should be solutions with multiple frequencies, corresponding to the plethora \ of short-lived particles. \ \>", "Text", PageWidth->WindowWidth], Cell["\<\ The sh12 functions (given in section 2) are conserved quantities in Dozal \ interactions; different hoops conserve different selections, and some are \ only conserved for orbit interactions. This is a rich topic that has yet to \ be explored, but has promise as an explanation of particle properties.\ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.7. Reciprocal Radii; Three size regimes?", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "The Dozal sizes with ternary symmetry could have three sizes that multiply \ to 1, one corresponding to a Planck area ", Cell[BoxData[ \(TraditionalForm\`P\^2\)]], " and the others in a reciprocal relationship, two squared radii ", Cell[BoxData[ \(TraditionalForm\`r\^2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], " corresponding to the dimensions of \"pseudo-point\" particles and the \ universe. p=r.R gives three size regimes. More work needed." }], "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.8. Mass and the Unit Velocity Equation.", "Subsubsection", PageWidth->WindowWidth], Cell["\<\ The basic wave equation in many dimensions gives a unit velocity, which is \ projected onto the different dimensions as velocity components. Massless \ particle waves would be restricted to space-like dimensions, whilst others \ would have velocities in Kaluza-Klein dimensions. Generalizing the \ Schrodinger equation to 12 directions (instead of the 4 implied by using \ \[ImaginaryI]) might lead to a more general description of \"information \ about particles\", and a non-dispersing form (including non-linear terms) \ could describe stable particles.More work needed.\ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.9. The 6-vector and Dependent Time.", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "The C3C4 hoop has a conserved quartic ", StyleBox["l4s", FontSlant->"Italic"], " which factorizes into two sums of squares ", StyleBox["l22a", FontSlant->"Italic"], " & ", StyleBox["l22b", FontSlant->"Italic"], ":- " }], "Text", PageWidth->WindowWidth], Cell[BoxData[ \({ab = a - b - g + h, ag = a - b + g - h, bc = b - c - h + i, bh = b - c + h - i, ca = a - c - g + i, ci = a - c + g - i, de = d - e - j + k, dj = d - e + j - k, ef = e - f - k + l, ek = e - f + k - l, fd = d - f - j + l, fl = d - f + j - l, f1 = a - b - g + h + \((d + e - 2\ f - j - k + 2 l)\)/\@3, \[IndentingNewLine]f2 = \((a + b - 2 c - g - h + 2 i)\)/\@3 - d + e + j - k, \[IndentingNewLine]f3 = a - b - g + h - \((d + e - 2\ f - j - k + 2 l)\)/\@3, \[IndentingNewLine]f4 = \((a + b - 2 c - g - h + 2 i)\)/\@3 + d - e - j + k, \[IndentingNewLine]l4s = \n\((ab\^2 + bc\^2 + ca\^2 + de\^2 + ef\^2 + fd\^2)\)/2 - 3 \((ab\ ef - bc\ de)\)\^2, \[IndentingNewLine]l4f = 9 \((f1\^2 + f2\^2)\) \((f3\^2 + f4\^2)\)/16}\)], "Input", PageWidth->WindowWidth], Cell[TextData[{ " This may correspond to a 6-vector ", Cell[BoxData[ \(\((ab\^2 + bc\^2 + ca\^2 + de\^2 + ef\^2 + fd\^2)\)/2\)], FontFamily->"Times New Roman"], " and a dependent \"time\" ", Cell[BoxData[ \(\(-3\) \((ab\ ef - bc\ de)\)\^2\)], "Input", FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], ". To be investigated." }], "Text", PageWidth->WindowWidth] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4. Outstanding problems.", "Section", FontFamily->"Times New Roman"], Cell["\<\ \tThe author, as an amateur experimental mathematician, who interprets \ Goedels theorem as \"Truth transcends peoof\", has only concerned himself \ with the demonstrability of hoop algebras and their properties. Rigorous \ analysis, based on primal numbers rather than real numbers, is required. \tThe only general \[OpenCurlyDoubleQuote]proof\[CloseCurlyDoubleQuote] that \ a table is conservative is by demonstration using random vectors. Direct \ factorization is only possible for small tables. Frobenius\[CloseCurlyQuote]s \ proof has not been extended to folded tables; it does not apply to general \ Moufang loops. \t \tOrbits are related to the interpretation of hoops as banded sets of \ differential equations with cyclic solutions; resonances at multiples of the \ fundamental frequency may account for the resonances of particle physics. \t \tConservation and orbits are \[OpenCurlyDoubleQuote]emergent properties\ \[CloseCurlyDoubleQuote]; others can be expected when techniques are \ developed to find the shapes of larger hoops. Higher-order sizes may lead to \ new emergent properties. They may be difficult to identify - C3 & C4 have \ duals with only a single angle each, but several years of effort were needed \ before \[OpenCurlyDoubleQuote]L4\[CloseCurlyDoubleQuote], a quartic size of \ C3C4, could be factorized and shown to define two orbits. A4 has one linear \ size, one quadratic size (giving an orbit) and a repeated cubic size that \ appears to discriminate between other types of Dozal orbit. Quartic sizes are \ common; if (like \[OpenCurlyDoubleQuote]L4\[CloseCurlyDoubleQuote]) they have \ a minimum of zero, they may be factorizable. \ \>", "Text"], Cell[TextData[{ "\tSome non-abelian hoops have \"pseudo-roots\" that may introduce \ uncertainty because their angles do not add on hoop multiplication. They have \ repeated quadratic roots that release extra degrees of freedom. D3 is the \ smallest non-commutative group. Figure 7 demonstrates that AB differs from BA \ in the D3 algebra; B & A are recovered by pre-mu;tiplication by the \ appropriate inverse. The \"pseudo-power\" A", Cell[BoxData[ \(TraditionalForm\`\^2\)]], " differs from the repeated product, but the \"pseudo-root\" recovers A. \ (There may be some new mathematics to discover here!)" }], "Text"], Cell[TextData[{ "Figure 7.", StyleBox[" Using D3, Division & Polar-Vector interconversions are OK, but \ repeated prodcts AA differ from pseudo-powers ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["A", FontWeight->"Plain", FontSlant->"Plain"], "2"], TraditionalForm]]], "." }], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; Chop[{Flatten[{"\", A = {2. , 3. , \(-3. \), 0. , \(-2. \), 1. }, "\", topol /. as[A]}], Flatten[{"\", tovec /. as[ topol /. as[ A]], "\", "\<\>", "\<\>", "\<\>", "\<\>", \ "\<\>", "\<\>"}], \[IndentingNewLine]Flatten[{"\", B = {1. , \(-5. \), 3. , \(-1. \), 2. , 2. }, "\", topol /. as[B]}], \[IndentingNewLine]Flatten[{"\", AB = hoopTimes[A, B], "\", topol /. as[AB]}], \[IndentingNewLine]Flatten[{"\", ABA = hoopTimes[hoopInverse[A], AB], "\", "\<\>", "\<\>", "\<\>", "\<\>", "\<\>", \ "\<\>"}], \[IndentingNewLine]Flatten[{"\", BA = hoopTimes[B, A], "\", topol /. as[BA]}], \[IndentingNewLine]Flatten[{"\", BAB = hoopTimes[hoopInverse[B], BA], "\", "\<\>", "\<\>", "\<\>", "\<\>", "\<\>", \ "\<\>"}], Flatten[{"\", AA = hoopTimes[A, A], "\", topol /. as[AA]}], Flatten[{\*"\"\<\!\(A\^2\)=\>\"", A2 = hoopPower[A, 2], \*"\"\<\!\(polarA\^2\)=\>\"", topol /. as[A2]}], Flatten[{\*"\"\<(\!\(A\^2\)\!\(\()\^\(1/2\)\)\)=\>\"", RA2 = hoopPower[A2, 1/2], "\", topol /. as[RA2]}]}] // tf\)], "Input", CellOpen->False], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"\<\"If A=\"\>", "2.`", "3.`", \(-3.`\), "0", \(-2.`\), "1.`", "\<\"polarA=\"\>", "1.`", \(-7.`\), "14.`", "0.19012560334646675`", "21.`", "0.3334731722518321`"}, {"\<\"Vec(pol(A))=\"\>", "2.`", "3.`", \(-2.9999999999999987`\), "0", \(-1.9999999999999993`\), "1.0000000000000002`", "\<\"OK, =A\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"If B=\"\>", "1.`", \(-5.`\), "3.`", \(-1.`\), "2.`", "2.`", "\<\"polarB=\"\>", "2.`", "10.`", \(-34.`\), \(-2.6179938779914944`\), "3.`", "2.700286221232442`"}, {"\<\"AB=\"\>", \(-23.`\), \(-9.`\), "4.`", "18.`", \(-15.`\), "27.`", "\<\"polarAB=\"\>", "2.`", \(-70.`\), \(-476.`\), \(-2.3869764961206146`\), "577.`", "2.899028779494308`"}, {"\<\"Ai.AB=\"\>", "1.`", \(-4.999999999999998`\), "2.9999999999999987`", \(-1.0000000000000002`\), "1.9999999999999993`", "2.0000000000000018`", "\<\"OK, =B\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"BA=\"\>", \(-23.`\), \(-5.`\), \(-9.`\), "13.`", \(-2.`\), "28.`", "\<\"polarBA=\"\>", "2.`", \(-70.`\), \(-476.`\), "2.808119481337961`", "343.`", "2.6704321487405127`"}, {"\<\"Bi.BA=\"\>", "2.`", "2.999999999999999`", \(-3.0000000000000004`\), "0", \(-1.9999999999999993`\), "0.9999999999999991`", "\<\"OK, =A\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"AA=\"\>", "26.`", "7.`", \(-5.`\), \(-20.`\), "4.`", \(-11.`\), "\<\"polarAA=\"\>", "1.`", "49.`", "196.`", "0.2860552964471535`", "763.`", "0.3334731722518321`"}, {"\<\"\\!\\(A\\^2\\)=\"\>", "5.`", "16.532249250832564`", \(-18.999999999999996`\), \ \(-1.3562945691657533`\), \(-9.999999999999996`\), "9.824045318333194`", "\<\"\\!\\(polarA\\^2\\)=\"\>", "1.0000000000000124`", \(-49.`\), "195.99999999999977`", "0.38025120669293355`", "440.99999999999983`", "0.6669463445036641`"}, {"\<\"(\\!\\(A\\^2\\)\\!\\(\\()\\^\\(1/2\\)\\)\\)=\"\>", "2.000000000000001`", "3.0000000000000018`", \(-2.9999999999999982`\), "0", \(-1.9999999999999984`\), "1.0000000000000013`", "\<\"polar=\"\>", "1.0000000000000082`", \(-7.`\), "13.99999999999999`", "0.19012560334646675`", "20.999999999999993`", "0.3334731722518321`"} }], "\[NoBreak]", ")"}], TraditionalForm]], "Output", FontSize->9] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["5. References.", "Section", FontFamily->"Times New Roman"], Cell[TextData[{ "[1] H.-D Ebbinghaus et al, Numbers, Springer-Verlag N.Y. 1991 p22.", "\n", "[2] ", " ", ButtonBox["http://library.wolfram.com/infocenter/MathSource/4894/", ButtonData:>{ URL[ "http://library.wolfram.com/infocenter/MathSource/4894/"], None}, ButtonStyle->"Hyperlink"], " R.H.Beresford 2003-6.", "\n", "[3] J.D.H. Smith, A.B. Romanowska, Post-Modern Algebra, Wiley Interscience \ 1999.", "\n", "[4] ", "J.J.Hamilton, Hypercomplex numbers and the description of spin states, \ J.Math.Phys. ", StyleBox["38", FontWeight->"Bold"], "(10) Oct. 1997. pp4914-4928.", "\n", "[5] P. Lounesto, Clifford Algebras & Spinors 2nd ed., Cambridge University \ Press 2001." }], "Text", PageWidth->WindowWidth] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 887}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{1050, 682}, WindowMargins->{{65, Automatic}, {Automatic, 39}}, PrintingCopies->1, PrintingPageRange->{1, 1}, PageHeaders->{{Inherited, Inherited, Inherited}, {None, Inherited, None}}, PageFooters->{{Inherited, Inherited, Inherited}, {Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Cell[ TextData[ "21/2/07"]], Cell[ TextData[ "Hoop Algebras."]]}}, Magnification->1.5 ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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