(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 193789, 4817]*) (*NotebookOutlinePosition[ 194851, 4850]*) (* CellTagsIndexPosition[ 194807, 4846]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Hoop Algebra Supplement.", "Title", PageWidth->WindowWidth], Cell["R.H.Beresford. 21/2/07.", "Subsubtitle", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["1. Introduction.", "Section", PageWidth->WindowWidth], Cell[TextData[{ "This document is supplementary to ", StyleBox["\"HoopAlg&Physics.nb\"", FontSlant->"Italic"], "[in 1]", StyleBox[",", FontSlant->"Italic"], " which is a brief account of Hoop Algebras (also available as ", StyleBox["Hoops&Physics.doc", FontSlant->"Italic"], "[2])." }], "Text"], Cell[TextData[{ "\tHoops are ", "defined as ", "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 \ with 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}], " [1] as a Mathematica package ", StyleBox["Hoops.m", FontSlant->"Italic"], ". This contains over 80 named Hoops (with a few 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 inverse symbolic Cayley table). Multiplication, division, and (in many \ cases) inter-conversion between Cartesian vector {a,b,...} and polar {r,\ \[Theta]...} form are provided.\n\tThe figures in this document are \ self-explanatory output from a ", StyleBox["Mathematica", FontSlant->"Italic"], " session (with results of calculations appearing in ", StyleBox["bold-face", FontWeight->"Bold"], "), comprehensible to readers with no knowledge of ", StyleBox["Mathematica", FontSlant->"Italic"], ". They use the ", StyleBox["Hoops.m", FontSlant->"Italic"], " package. The input data are usually integer vectors, for clarity. ", 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 instructions:-." }], "Text"], Cell[BoxData[ \(Quit[]\)], "Input"], Cell[BoxData[ \(<< "\"\)], "Input", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["2. Demonstrations.", "Section", PageWidth->WindowWidth], Cell[TextData[{ "Summary.\n\tExamples demonstrate multiplication and division of vectors 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 ", StyleBox["multi-phase sinusoidal ", PageWidth->WindowWidth], " \"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[{ "\tWhen ", StyleBox["hoops.m", FontSlant->"Italic"], " has been activated a 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["hoopTbl, sh, topol, tovec, mm, nn, 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 information allows the \ procedures ", StyleBox["hoopTimes, hoopInverse,", FontSlant->"Italic"], " &", StyleBox[" hoopPower", FontSlant->"Italic"], " to calculate vector products, inverses and (where appropriate) 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, the symbolic form, \ the shape, the polar form and the vector reversion for the C3, C4 & C2 \ algebras", StyleBox[". C2 is unusual, having a \[OpenCurlyDoubleQuote]dual\ \[CloseCurlyDoubleQuote] form that does not involve an angle. ", PageWidth->WindowWidth], "(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 /. 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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 \ Ai=", StyleBox["hoopInverse[A]", FontSlant->"Italic"], "; division of B by A is ", StyleBox["hoopTimes[Ai,B].", FontSlant->"Italic"], "\n\tHoops are \"conservative\" - the real factors of the symbolic \ determinant are \"size\" functions that are conserved on multiplication, ", StyleBox["size[AB] = \[PlusMinus] size[A]size[B]", FontSlant->"Italic"], ". (Sizes are actually conjugates, but conjugation has no effect in most \ cases.) The sizes are listed 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 loops in general.)" }], "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. (", StyleBox["hoopInverse ", FontSlant->"Italic"], "calculates left-inverses; as C3 is Abelian the distinction is irrelevant \ here.)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\<\nAinverse = \>", Ai = hoopInverse[A], "\<\nShape of Ainverse = \>", sh /. \ as[Ai], "\<\nBinverse = \>", Bi = hoopInverse[B], "\<\nShape of Binverse = \>", sh /. \ as[Bi], "\<\nAinverse.AB recovers B\>", hoopTimes[Ai, AB], "\<\nAB.Binverse recovers A\>", hoopTimes[AB, Bi]}\)], "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.Binverse 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. This 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"], "). One 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"], ".\n\tFollowing 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\". ", StyleBox["The multiplication tables also fold; an ", PageWidth->WindowWidth], StyleBox["m\[Times]m", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox[" table with ", PageWidth->WindowWidth], StyleBox["r", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox["-fold symmetry can undergo ", PageWidth->WindowWidth], StyleBox["r", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox["-folding to an ", PageWidth->WindowWidth], StyleBox["(m/r)\[Times](m/r)", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox[" table. ", PageWidth->WindowWidth], "\n\tExample 4 shows that \"C4\" folds to complex algebra. It demonstrates \ multiplication, division, raising to a power, and extracting a root. In each \ case, the result is shown to fold to the corresponding complex number:-" }], "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\>\"", ",", \(AAA = hoopTimes[A, hoopTimes[A, A]]\), ",", "\"\<\\nhoopTimes[A,hoopTimes[A,A]] (=\!\(A\^3\))\>\"", ",", \(hoopTimes[A, hoopTimes[A, A]]\), ",", \(hoopTimes[A, hoopTimes[A, A]]\), StyleBox[",", FontWeight->"Plain"], StyleBox["\[IndentingNewLine]", FontWeight->"Plain"], "\!\(\* StyleBox[\"\\\"\",\nFontWeight->\"Plain\"]\)\\nPolar Form\>\"", ",", \(topol /. as[A]\), ",", "\"\\"Plain\"]\)\!\(r\^2\)=5, \[CurlyPhi]=-.4638 \ OK\>\""}], "}"}]}]], "Input", FormatType->TraditionalForm], 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\"\>", ",", \({87.`, 80.`, 85.`, 91.`}\), ",", "\<\"\\nhoopTimes[A,hoopTimes[A,A]] (=\\!\\(A\\^3\\))\"\>", ",", \({87.`, 80.`, 85.`, 91.`}\), ",", \({87.`, 80.`, 85.`, 91.`}\), ",", "\<\"\\nPolar Form\"\>", ",", \({7.`, 1.`, 5.`, \(-0.4636476090008061`\)}\), ",", "\<\"has\\!\\(\\* StyleBox[\\\" \ \\\",\\nFontWeight->\\\"Plain\\\"]\\)\\!\\(r\\^2\\)=5, \[CurlyPhi]=-.4638 \ OK\"\>"}], "}"}]], "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 three reasons, negative \ sizes (example 5), \"angle wrap-round\" (example 6), ", StyleBox["and non-commutativity (section 2.6)", PageWidth->WindowWidth], ".\n\tIf a size is negative for A it alternates in sign for repeated \ products but the power is 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["\<\ \tExample 6 shows that if 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). ", StyleBox["(C4c is demonstrated in more detail later, in Figure S1.)", PageWidth->WindowWidth] }], "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[TextData[{ "\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", StyleBox[", and gives two copies of the complex field, with only the origin \ in common. It has application in binocular vision.", PageWidth->WindowWidth] }], "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 Group 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' ", 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 [4], 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. ", StyleBox["Example 11 demonstrates the non-Abelian (AB differs from BA) D3 \ algebra and Clifford(2,1) algebra. ", PageWidth->WindowWidth], "\n\t", StyleBox["hoopInverse", FontSlant->"Italic"], " calculates a left inverse, so Ai.AB divides AB by A, recovering B. \ Similarly Bi.BA recovers A, whilst BA.Ai recovers B because A is the left \ inverse of Ai. ", StyleBox["(Examples 16 & 18 also use D3.)", PageWidth->WindowWidth] }], "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 \"basis vectors\", \ \"vectors\" or \"scalars\" for single element generators. Their products are \ bivectors, trivectors, etc.) The symbols are chosen to match those in [3, \ p11] for quaternions (Qr) and CL2, and to match [3, 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[\(a\_1\ b\_1 + a\_2\ b\_2 + a\_3\ b\_3\), FontSlant->"Italic"]]], " 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 direction, having coefficient 0 in the multiplicands). " }], "Text", PageWidth->WindowWidth], Cell[TextData[{ "\tAll Clifford algebras are hoops. Their tables are created by using the \ ", StyleBox["GroupLoopHoop ", FontSlant->"Italic"], "procedure ", StyleBox["cl", FontSlant->"Italic"], " or by folding a specific group isomorph. Wedge (exterior) algebras are \ the bivector part of the Clifford product [3 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[TextData[{ "\tFinite Lie algebras appear to be Clifford hoops for which the real \ (scalar) elements have been zeroed. This has yet to be explored.\n\tMany \ algebras, some of which appear to be new", StyleBox[", are defined by other hoops", PageWidth->WindowWidth] }], "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\" factorise nicely, but their \ quadratic factors are not conserved:-" }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(Use["\"]; {"\", "\", hoopTbl // tf, "\<\nDet factors\>", sh, \[IndentingNewLine]"\<\nA = \>", A = {4, 2, 3, 5}, \[IndentingNewLine]"\<\nB = \>", B = {1, 6, 2, 2}, \[IndentingNewLine]"\<\nAB= \>", AB = hoopTimes[A, B], "\<\nDet factors of A sa =\>", sa = sh /. \ as[A], "\<\nDet factors of B sb =\>", sb = sh /. \ as[B], "\<\nDet factors 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], ",", "\<\"\\nDet factors\"\>", ",", \({a\^2 - b\^2 - c\^2 - d\^2, a\^2 + b\^2 + c\^2 + d\^2}\), ",", "\<\"\\nA = \"\>", ",", \({4, 2, 3, 5}\), ",", "\<\"\\nB = \"\>", ",", \({1, 6, 2, 2}\), ",", "\<\"\\nAB= \"\>", ",", \({32, 22, 37, \(-1\)}\), ",", "\<\"\\nDet factors of A sa =\"\>", ",", \({\(-22\), 54}\), ",", "\<\"\\nDet factors of B sb =\"\>", ",", \({\(-43\), 45}\), ",", "\<\"\\nDet factors of AB sab=\"\>", ",", \({\(-830\), 2878}\), ",", "\<\"\\nThis is not sa*sb =\"\>", ",", \({946, 2430}\)}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Use["\"];\)\), "\[IndentingNewLine]", \({"\", "\", hoopTbl // tf, "\<\nDet factors\>", sh, "\<\n A =\>", A = {4, 2, 3, 5, 1, 6}, "\<\n B =\>", B = {0, 6, 2, 4, 3, 2}, "\<\nAB = \>", AB = hoopTimes[A, B], "\<\nDet factors of A sa = \>", sa = sh /. \ as[A], "\<\nDet factors of B sb = \>", sb = sh /. \ as[B], "\<\nDet factors of AB sab= \>", sab = sh /. \ as[AB], "\<\nThis is not sa*sb \>", sa\ sb}\)}], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ RowBox[{"{", RowBox[{"\<\"Example 14a. 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], ",", "\<\"\\nDet factors\"\>", ",", \({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 = \"\>", ",", \({53, 70, 57, 66, 70, 41}\), ",", "\<\"\\nDet factors of A sa = \"\>", ",", \({\(-3\), 21, \(-18\), 24}\), ",", "\<\"\\nDet factors of B sb = \"\>", ",", \({\(-1\), 17, 25, 31}\), ",", "\<\"\\nDet factors of AB sab= \"\>", ",", \({3, 357, \(-504\), 978}\), ",", "\<\"\\nThis is not sa*sb \"\>", ",", \({3, 357, \(-450\), 744}\)}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "\tMany non-conservative loops, with various properties, have been ", StyleBox["tested by multiplying random vectors and comparing the \ determinants. (No necessary and sufficient loop property has been found for \ conservation.) ", PageWidth->WindowWidth], " The only conservative tables appear to be groups, a few non-commutative \ Moufang Loops [5] (including octonions and split-octonions), their direct \ composition 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", StyleBox[", in sizes, ", PageWidth->WindowWidth], "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 \ size 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. ([4] 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[TextData[{ StyleBox["Example 16", FontWeight->"Bold"], StyleBox[". Using D3, Division and Polar-Vector interconversions work, but \ repeated products AA differ from pseudo-powers ", FontFamily->"Courier New"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["A", FontSlant->"Plain"], "2"], TraditionalForm]], FontFamily->"Courier New"] }], "Text", FontSize->9], Cell[CellGroupData[{ Cell[BoxData[{ \(Use["\"]; {"\", \[IndentingNewLine]hoopTbl // tf}\), "\[IndentingNewLine]", \(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[ RowBox[{"{", RowBox[{"\<\"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]}], "}"}]], "Output"], 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\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"and 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 ]], Cell[TextData[{ "The repeated quadratic factor of D3, (", Cell[BoxData[ \(TraditionalForm\`a\^2 - b\^2 + c\^2 - d\^2 + e\^2 - f\^2 - c\ a - e\ a\)]], Cell[BoxData[ \(TraditionalForm\`+\)], PageWidth->PaperWidth, FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`b\ d\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(-c\)\ \ e + b\ f + d\ f\)]], ")/2, has 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\", \ i.e. \[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 left and right remainders A/B = C +Rl \ +Rr; sizes that are zero in C but not in A go into the left remainder; sizes \ that are zero in C but not in B go into the right remainder. This is a \ generalization of integer division. As ", StyleBox["hoopTimes", FontSlant->"Italic"], " implements both division and multiplication, remainders also extend to \ multiplication A*B=P+Rl+Rr so that P/A+Rr=B and P/B+Rl=A. Example 17 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 17. 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", FontSize->9] }, Open ]], Cell[TextData[{ "\t\[DoubleStruckCapitalC], \[DoubleStruckCapitalH], and \ \[DoubleStruckCapitalO] can have divisors of zero if their coefficients are \ complex.The hoop 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 18 shows this for left and right multiplication by a 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 18. 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 [5] requires over a \ hundred pages of \"Post-Modern Algebra\" before subtraction can be tackled. \ In hoops, negation is the result of a fold with ", StyleBox["r", FontSlant->"Italic"], "=2. 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"], "j"], 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 vector 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 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[{ \(Use["\"]; {"\", "\<\nOC301 is {0,1,a} with \ vector form\>", OC301 = tovec /. \ as[{0, 1, a}], "\<\nOC311 is {1,1,b} with vector form\>", OC311 = tovec /. \ as[{1, 1, b}], "\<\nProducts are orbits with added angles; the polar form \ of the product of OC301 & OC311 is\>", Simplify[topol /. \ as[hoopTimes[OC301, OC311]]], "\<\nand the remainders are\>", Rl, Rr}\), "\[IndentingNewLine]", \(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 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"], Cell[BoxData[ \({"Example 20b. 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", FontSize->9] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["3. Analogies with Particles.", "Section", PageWidth->WindowWidth], Cell[TextData[{ StyleBox["Summary\n\tThe \"Dozal\" algebras based on 12-element groups have \ elements that act as half-spin (fermionic) and unit-spin (bosonic) quantum \ operators. Their orbits 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. Four hoops ", FontSlant->"Italic"], StyleBox["{C3K, C4C4, D3C2, Q12} possibly ", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox["correspond to four forces - they conserve different but \ overlapping sets of properties. The non-Abelian hoops may introduce \ uncertainty. Linear sizes are additive properties.\n\tOrbits resemble \ multi-phase deBroglie waves, and quadratic sizes resemble Planck areas. Wave \ packets in stable orbits", FontSlant->"Italic"], StyleBox[" with 3, 6 or 12 phases ", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox["may relate to quarks whilst 4-phase orbits may relate to leptons. \ Resonant orbits, at integer multiples of the fundamental frequency, may \ relate to particles resonances.\n\t", FontSlant->"Italic"], StyleBox["Bosonic \"hexal\" ", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox["elements are supersymmetric to (but fewer than) ", FontSlant->"Italic"], StyleBox["fermionic", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox[" dozal", FontSlant->"Italic"], StyleBox[" elements. Bosons", PageWidth->WindowWidth, FontSlant->"Italic"], StyleBox[" are not subject to the Pauli exlusion principle because they are \ equivalence relations. ", 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 Loop symmetries (together withLie group symmetries). Sporadic \ discoveries, spread over fifteen years of research, have repeatedly related \ different physical phenomena to newly discovered aspects of hoop algebra, \ encouraging the conjecture that these algebras are \"physical \ mathematics\".\ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.2. Dozal elements act as Quantum Operators.", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "\tJ. J. 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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 ", StyleBox["Example 23.", PageWidth->WindowWidth, CellMargins->{{Inherited, 0}, {Inherited, Inherited}}, FontSize->10, FontWeight->"Bold"], StyleBox[" Banded Differential Equations and Multi-phase Sinusoids", PageWidth->WindowWidth, CellMargins->{{Inherited, 0}, {Inherited, Inherited}}, FontSize->10], "\n", StyleBox[" ", FontSize->10], 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", FontSize->10], StyleBox["\nSubstituting the trial function ", FontSize->10], Cell[BoxData[ \(x\_j = c[t] + \(c\_1\) Sin[c\_2\ t + \[Phi] + 2 j\ k\ \[Pi]/m]\)], FontFamily->"Times New Roman", FontSize->10], StyleBox[" leads to ", FontSize->10], Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \ \)\(x\_j = c[t] + \[Sum]\_p\( c\_p\) Sin[2 a\ Sin[2 k\ \[Pi]\ f\_p/m]\ t + \[Phi]\_p + 2 j\ k\ \[Pi]/m]/f\_p\)\)\)], "Input", FontFamily->"Times New Roman", FontSize->10, FontWeight->"Plain"], StyleBox[".\n", FontSize->10], "\n\t", 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", FontSlant->"Italic"], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(f\_p\)], "Input", FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic"], " and ", Cell[BoxData[ StyleBox[\(\[Phi]\_p\), FontFamily->"Times New Roman"]], FontSlant->"Italic"], " are phase amplitude, resonant multiplier and phase offset constants which \ have to be subscripted because independent sets can arise; ", StyleBox["k", FontSlant->"Italic"], " is a phase step. ", StyleBox["Sin[2k \[Pi]/m]", FontSlant->"Italic"], " is a constant that determines the frequency & period. When ", StyleBox["m", FontSlant->"Italic"], "=4, ", StyleBox["k", FontSlant->"Italic"], " must be 1; folding then removes the c[t] term and reproduces the ordinary \ sinusoid equation-pair with doubled amplitude. Other values of ", StyleBox["m", FontSlant->"Italic"], " and ", StyleBox["k", FontSlant->"Italic"], " give a frequency reduction.\n\tThe period is ", StyleBox["2\[Pi]", FontSlant->"Italic"], Cell[BoxData[ \(f\_p\)], FontFamily->"Times New Roman", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["/Sin[2k \[Pi]/m]", FontSlant->"Italic"], ", and the function is non-sinusoidal (constant) if 2/m is integral.\n\t", StyleBox["The Dozal C3C4 loop gives half-spin ", PageWidth->WindowWidth, CellMargins->{{Inherited, 0}, {Inherited, Inherited}}], "with a period of 4\[Pi] when ", StyleBox["m", FontSlant->"Italic"], "=12, k=1, ", Cell[BoxData[ \(f\_p\)], FontFamily->"Times New Roman", FontWeight->"Plain"], "=1. The twelve components are elements of the C12 hoop.However, C3C4 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 multiples of the fundamental frequency) \ need study; what is the relationship between hoops and the theory of strings \ and branes, etc.? ", StyleBox["Before folding, the sinusoids are \[OpenCurlyDoubleQuote]free\ \[CloseCurlyDoubleQuote]; they do not have a zero mean because they include \ constants of integration c[t]. This may relate to position in space.", PageWidth->WindowWidth, CellMargins->{{Inherited, 0}, {Inherited, Inherited}}] }], "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[ FormBox[ RowBox[{ StyleBox["Example", FormatType->TraditionalForm], StyleBox[" ", FormatType->TraditionalForm], StyleBox["24.", FormatType->TraditionalForm], StyleBox[" ", FormatType->TraditionalForm], StyleBox["C3K", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox["Polar", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox["form", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox["and", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox["reversion", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox["to", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox["Cartesian", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontWeight->"Plain"], StyleBox[\(\(form\)\(.\)\), FormatType->TraditionalForm, FontWeight->"Plain"]}], TraditionalForm]], "Input"], 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"], 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"] }, 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]\)]], " have been made, to give a compact expression. Note 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 \ - how do these relate to space-time?" }], "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]} in common with \ C3K) and reverted as follows:-\ \>", "Text", PageWidth->WindowWidth], Cell[TextData[{ StyleBox["Example 25.", FontWeight->"Bold"], " C3C4 Full Polar form and Orbits." }], "Text"], Cell[BoxData[ \(Use["\"]; sh\)], "Input", CellOpen->False], 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[ \(TraditionalForm\`\@3\)]], StyleBox["to give a compact expression.", PageWidth->WindowWidth], "\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. Each set of angles involving \[Omega]=2\[Pi]/3 gives a \ chiral orbit; the chirality is reversed if the angle is negated. \n\tSimilar, \ but simpler, expressions give Q12 & D3C2 orbits. Their parameters are related \ to those for C3K and C3C4. A 12-phase system arises if \[Psi]=\[Chi]+\[Pi]/6. \ ", StyleBox["Problem - non-commutativity prevents angle additivity in these \ orbits", PageWidth->WindowWidth], StyleBox[". Does this introduce uncertainty?", PageWidth->WindowWidth, FontFamily->"Times New Roman"] }], "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["\<\ This (and D3C2O, below) gives a 12-phase system if \[Psi]=\[Chi]\[PlusMinus]\ \[Pi]/6 & \[Chi]=\[Eta]. A polarised 3-phase system (with polarisation angle \ \[CurlyPhi]) is given by \[Phi]=\[Tau] & \[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["\<\ \tFor these orbits to represent particles, the product of the non-zero \ quadratic sizes should define a fundamental size - a Planck Area. Section 3.7 \ explores this topic. The product of the non-zero linear sizes {\[Alpha], \ \[Beta], \[Gamma], \[Delta]} should be 1; some quantization process may \ constrain them to be \[PlusMinus]1. The orbits may represent multi-phase \ deBroglie waves. \ \>", "Text"] }, 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 a 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. \ Orbit instability ", StyleBox["is a vague concept. ", PageWidth->WindowWidth], "It may be related to the occurrence of determinant factors that are larger \ than quadratic." }], "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell["\<\ {Use[\"C5\"];\"Example 26. 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, FontFamily->"Courier"], Cell[BoxData[ \({"Example 26. 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 & 12 phases have been found, \ some with polarization and chirality. 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 27. 3-phase C3K Orbit (Omitting 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]} define offsets (\ \[Alpha]/12 etc) from zero. Scaled square roots of the quadratic terms appear \ as radii. Each phase occurs four times in a 3-phase dozal orbit, with \ different linear offsets.\ \>", "Text", PageWidth->WindowWidth], Cell[TextData[{ StyleBox["\ttopol", FontSlant->"Italic"], " shows that the squared amplitude ", Cell[BoxData[ \(TraditionalForm\`\((\[Zeta]/12)\)\^2\)]], "is the sum of three squares:- \n", StyleBox["((a-b-d+e+g-h-j+k)", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\^2\)], FontSlant->"Italic"], StyleBox["+(-a+c+d-f-g+i+j-l)", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\^2\)], FontSlant->"Italic"], StyleBox["+(b-c-e+f+h-i-k+l)", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\^2\)], FontSlant->"Italic"], StyleBox[")/72", FontSlant->"Italic"] }], "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 28. 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 ", Cell[BoxData[ StyleBox[\(\((\((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - \ l)\)\^2)\)/144\), FontSlant->"Italic"]], "Text", FontFamily->"Times New Roman"], "." }], "Text", PageWidth->WindowWidth], Cell[BoxData[ \({"\", \ Simplify[C3KO /. {\[Psi] \[Rule] 0, \[Zeta] \[Rule] 0, \[CurlyKappa] \[Rule] 0}] // tf}\)], "Input", PageWidth->WindowWidth, CellOpen->False], Cell[BoxData[ StyleBox[ RowBox[{\(Example\ 29. \ 6\), "-", RowBox[{"phase", " ", "C3K", " ", RowBox[{ RowBox[{"Orbit", ".", TagBox[ FormBox[\(\(\[IndentingNewLine]\)\({\[Alpha] + \[Beta] + \ \[Gamma] + \[Delta] + \[Kappa]\ \ \ \ \ \ + \[Lambda]\ \(cos(\[Phi])\)\ \ \ \ \ \ \ \ + \[Eta]\ \(cos(\[Chi])\), \[Alpha] + \[Beta] + \[Gamma] + \[Delta] \ - \[Kappa]/ 2 + \[Lambda]\ \(cos(\[Phi] + \[Omega])\) + \[Eta]\ \ \(cos(\[Chi] + \[Omega])\), \[Alpha] + \[Beta] + \[Gamma] + \[Delta] - \ \[Kappa]\ / 2 + \[Lambda]\ \(cos(\[Phi] - \[Omega])\) + \[Eta]\ \ \(cos(\[Chi] - \[Omega])\), \[Alpha] - \[Beta] + \[Gamma] - \[Delta] - \ \[Kappa]\ \ \ \ \ \ \ - \[Lambda]\ \(cos(\[Phi])\)\ \ \ \ \ \ + \[Eta]\ \ \(cos(\[Chi])\), \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[Kappa]\ / 2 - \[Lambda]\ \(cos(\[Phi] + \[Omega])\) + \[Eta]\ \ \(cos(\[Chi] + \[Omega])\), \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \ \[Kappa]\ / 2 - \[Lambda]\ \(cos(\[Phi] - \[Omega])\) + \[Eta]\ \ \(cos(\[Chi] - \[Omega])\), \[Alpha] + \[Beta] - \[Gamma] - \[Delta] - \ \[Kappa]\ \ \ \ \ \ \ + \[Lambda]\ \(cos(\[Phi])\)\ \ \ \ \ \ - \[Eta]\ \ \(cos(\[Chi])\), \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Kappa]\ / 2 + \[Lambda]\ \(cos(\[Phi] + \[Omega])\) - \[Eta]\ \ \(cos(\[Chi] + \[Omega])\), \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \ \[Kappa]\ / 2 + \[Lambda]\ \(cos(\[Phi] - \[Omega])\) - \[Eta]\ \ \(cos(\[Chi] - \[Omega])\), \[Alpha] - \[Beta] - \[Gamma] + \[Delta] + \ \[Kappa]\ \ \ \ \ \ \ - \[Lambda]\ \(cos(\[Phi])\)\ \ \ \ \ \ - \[Eta]\ \ \(cos(\[Chi])\), \[Alpha] - \[Beta] - \[Gamma] + \[Delta] - \[Kappa]/ 2 - \[Lambda]\ \(cos(\[Phi] + \[Omega])\) - \[Eta]\ \ \(cos(\[Chi] + \[Omega])\), \[Alpha] - \[Beta] - \[Gamma] + \[Delta] - \ \[Kappa]/2 - \[Lambda]\ \(cos(\[Phi] - \[Omega])\) - \[Eta]\ \(cos(\[Chi] - \ \[Omega])\)}\)\), "TraditionalForm"], TraditionalForm, Editable->True]}], "/", "12"}]}]}], FontWeight->"Plain"]], "Input"], Cell["\<\ \tThe linear terms 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[TextData[{ "\tAs Dozal elements resemble quantum operators, the addition/splitting and \ multiplication/division of Dozal vectors should resemble particle decays and \ interactions. C3K and D3C2 conserve the linear sizes {\[Alpha], \[Beta], \ \[Gamma], \[Delta]} and C3C4 and Q12 conserve {\[Alpha], \[Beta]} on \ addition and subtraction, corresponding to additive particle properies. \ 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?", StyleBox[" It appears to discriminate between different classes of orbit.", PageWidth->WindowWidth], ") Many other functions are conserved on multiplication of orbits, rather \ than of general vectors. Example 30 lists the known conserved functions, \ together with two {L4Q,L4\[Sigma]} conserved by two signed tables related to \ quaternions and Pauli-\[Sigma] hoops.\n\tI have not been able to correlate \ these functions 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[BoxData[ FormBox[ RowBox[{ StyleBox["Example", FormatType->TraditionalForm, FontFamily->"Times New Roman"], StyleBox[" ", FormatType->TraditionalForm, FontFamily->"Times New Roman"], StyleBox["30.", FormatType->TraditionalForm, FontFamily->"Times New Roman"], StyleBox[" ", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["Functions", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["conserved", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["in", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["Dozal", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox[" ", FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox[\(\(interactions\)\(.\)\), FormatType->TraditionalForm, FontFamily->"Times New Roman", FontWeight->"Plain"]}], TraditionalForm]], "Input"], 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[{ "\tThese ", StyleBox["sh12", FontSlant->"Italic"], " functions are conserved quantities in some 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? The \"Large numbers \ hypothesis\"?\ \>", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "\tThe Dozal ", StyleBox["quadratic", PageWidth->WindowWidth], " sizes,{\[Zeta], \[Lambda], \[Eta], \[Kappa]}, must multiply to 1 ", StyleBox["(excluding any zeroes) in any orbits. ", PageWidth->WindowWidth], "\[Zeta]", StyleBox[" has (leptonic?) 4-fold symmetry.", PageWidth->WindowWidth], " The three with ternary (hadronic?) symmetry could have one (\[Lambda]) \ corresponding to a Planck area ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["p", FontSlant->"Plain"], "2"], TraditionalForm]]], " and the other two (\[Eta], \[Kappa]) 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 then gives three size regimes. There is an analogy here with \ the \"T-duality\" of M-theory, where equivalent theories are created by \ exchanging a large dimension and a small dimension. ", StyleBox["The \"Dirac large numbers hypothesis\" ", PageWidth->WindowWidth], "could be a result of this possible reciprocal relationship between two \ sizes", StyleBox[", with size ratios being powers of ", PageWidth->WindowWidth], Cell[BoxData[ \(TraditionalForm\`10\^40\)]], StyleBox[" approximately. ", PageWidth->WindowWidth], " More work is needed." }], "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3.8. Mass and the Unit Velocity Equation.", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "\tThe basic wave equation in many directions, ", Cell[BoxData[ \(TraditionalForm\`d\^2\)], FontSlant->"Italic"], StyleBox["G/d", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`t\^2\)], FontSlant->"Italic"], "-\[Sum][", Cell[BoxData[ \(TraditionalForm\`d\^2\)], FontSlant->"Italic"], StyleBox["G/d", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^2\)], FontSlant->"Italic"], "]=0, ", StyleBox["has an infinite number of planar wave solutions, each with a unit \ velocity. These project onto ", PageWidth->WindowWidth], "the different directions as velocity components. ", StyleBox["This is proved in Example 31, which ", PageWidth->WindowWidth], "sums the second differentials of a general travelling wave G[-t+x lx+y \ ly+z lz] with three space directions." }], "Text", PageWidth->WindowWidth], Cell[TextData[{ StyleBox["Example 31.", FontWeight->"Bold"], " Unit Velocity Equation with 3 space directions." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\[PartialD]\_{t, 2}G[\(-t\) + x\ lx + y\ ly + z\ lz] - \[IndentingNewLine]\[PartialD]\_{x, 2}G[\(-t\) + x\ lx + y\ ly + z\ lz] - \[IndentingNewLine]\ \[PartialD]\_{y, 2}G[\(-t\) + x\ lx + y\ ly + z\ lz] - \[PartialD]\_{z, 2}G[\(-t\) + x\ lx + y\ ly + z\ lz]]\)], "Input", PageWidth->WindowWidth, FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[ RowBox[{\(-\((\(-1\) + lx\^2 + ly\^2 + lz\^2)\)\), " ", RowBox[{ SuperscriptBox["G", "\[Prime]\[Prime]", MultilineFunction->None], "[", \(\(-t\) + lx\ x + ly\ y + lz\ z\), "]"}]}]], "Output", PageWidth->WindowWidth] }, Open ]], Cell[TextData[{ "\tThe result is zero if the ", StyleBox["lx", FontSlant->"Italic"], " are direction sines so that their squares sum to 1. (Direction sines are \ used because the angles are measured towards the x direction and the lx \ contribution disappears in sub-algebras where this angle is zero.). This \ generalises to any number of space directions. Each additional space \ direction adds the square of its direction sine. The partial differential \ equation is therefore satisfied by any unit velocity function moving forward \ in any number of directions. This gives a light-like limiting velocity in two \ possible ways:-\n1. If pairs of directions fold to single dimensions, the \ disturbance travels at a speed that is less than unity unless there is a zero \ velocity in one of each of the folded pairs.\n2. If all but 3 space \ directions are curled-up (Kaluza-Klein) directions, KK velocities reduce the \ velocity in the spatial directions; light-like disturbances have no velocity \ in the KK directions. KK \"orbital\" velocities give mass to the \ disturbances. Massless particle waves would be restricted to space-like \ dimensions, whilst others would have velocities in Kaluza-Klein dimensions. \ This relates to the reciprocal radii conjecture. Constant orbital frequencies \ in small dimensions would contribute less mass than the same in larger \ dimensions." }], "Text", PageWidth->WindowWidth], Cell[TextData[{ "\tAn article [7] on Huygen's principle develops both the linear and the \ spherical wave versions for n spatial dimensions. n=3 gives sinusoidal waves \ with the square law of decay, amplitude=A/", Cell[BoxData[ \(TraditionalForm\`r\^\(n - 2\)\)]], ". n=2 gives Bessel function solutions. [8] applies the spherical wave \ versions to spaces with more than 3 space dimensions. Odd n's give expanding \ spherical waves that decay according to r^[n-2]; the square law decay is \ restricted to 3 spatial dimensions." }], "Text", PageWidth->WindowWidth], Cell["\<\ \tThe Schrodinger equation provides \"information about particles\" as wave \ packets which disperse with the passage of time. As dispersion arises when \ the velocity is frequency-dependent, the unit velocity equation describes \ non-dispersing wave packets. Generalizing the Schrodinger equation to 12 \ directions (instead of the 4 implied by using \[ImaginaryI]) might lead to a \ more general description of non-dispersing particles. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["3.9. The 6-vectors and Dependent Time.", "Subsubsection", PageWidth->WindowWidth], Cell[TextData[{ "\tThe 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"], " and can be expressed as ", StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["s26\.b2-abef =", PageWidth->WindowWidth, FontFamily->"Times New Roman", FontSlant->"Italic"], Cell[BoxData[ StyleBox[\(\((\((ab\^2 + bc\^2 + ca\^2 + de\^2 + ef\^2 + fd\^2)\)/ 2)\)\^2\), FontSlant->"Italic"]], FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{ RowBox[{"-", StyleBox["3", FontSlant->"Italic"]}], StyleBox[\(\((ab\ ef - bc\ de)\)\^2\), FontSlant->"Italic"]}]], "Input", FontFamily->"Times New Roman", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], ". 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