(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 601202, 13786] NotebookOptionsPosition[ 576376, 13012] NotebookOutlinePosition[ 578823, 13082] CellTagsIndexPosition[ 578647, 13075] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Synergetics Coordinates Applications", "Title"], Cell["\<\ Written by Clifford J. Nelson, 1996 revised April 1997, and June 1997 and \ September 1997 and May 1, 1998 and May 9, 1998 and Jan 3, 2002 and October 1 \ 2002 and Feb 2003, May 2003 and October 14 2003, 29 March 2008 and October \ 4, 6, 10, 18, 19 2009, October 9 and 12 2014, April 23 2016, May 2 2016.\ \>", "Commentary", CellChangeTimes->{{3.463714154325485*^9, 3.463714184204135*^9}, { 3.463726825756113*^9, 3.463726952143014*^9}, {3.463727289837957*^9, 3.463727303224372*^9}, {3.4637276676545258`*^9, 3.4637276690195723`*^9}, { 3.463856408746874*^9, 3.463856448778351*^9}, {3.464194310592246*^9, 3.464194319109267*^9}, {3.464901884465357*^9, 3.464901887982286*^9}, { 3.464992107957437*^9, 3.464992148122756*^9}, {3.6218476303975983`*^9, 3.621847661514492*^9}, {3.622103449347795*^9, 3.622103452987483*^9}, { 3.670458203487466*^9, 3.670458205222821*^9}, {3.67045831249732*^9, 3.670458343476348*^9}, {3.67120446865528*^9, 3.6712044814231787`*^9}}], Cell["Copyright Clifford J. Nelson.", "Commentary"], Cell[CellGroupData[{ Cell["Reference", "Section"], Cell["Based on books :", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.464199129291295*^9, 3.464199129297714*^9}, { 3.46419917098542*^9, 3.464199171475408*^9}}, FontSize->14], Cell["\<\ Synergetics : explorations in the geometry of thinking.1975 by R.Buckminster \ Fuller. ISBN 0 - 02 - 065320 - 4 Macmillan Publishing Company 866 Third Avenue, New York, N.Y.10022 Collier Macmillan Canada, Inc.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.464199129291295*^9, 3.464199129297714*^9}, { 3.46419917098542*^9, 3.464199171472701*^9}, {3.464325665798353*^9, 3.4643256825894814`*^9}}, FontSize->14], Cell["\<\ Synergetics 2, 1979 by R.Buckminster Fuller. ISBN 0 - 02 - 541870 - X (v.1) ISBN 0 - 02 - 541880 - 7 (v.2) Macmillan Publishing Co., Inc.866 Third Avenue, New York, N.Y.10022 Collier Macmillan Canada, Ltd.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.464199129291295*^9, {3.464325688302617*^9, 3.464325698469859*^9}}, FontSize->14], Cell[BoxData["\<\"http://library.wolfram.com/infocenter/MathSource/600/\"\>"],\ "Output", CellChangeTimes->{3.670464917603948*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Foreword", "Section", CellChangeTimes->{{3.4637140327532997`*^9, 3.463714043560717*^9}}], Cell["\<\ I'll post some things from time to time to try to get some help understanding \ Bucky Fuller's books Synergetics 1 and 2 and my interpretation of the \ Synergetics Coordinate System. I've been working on this stuff all alone for \ years. Bucky advised not to work alone and he wrote that this coordinate \ system, or something like it, is very important. 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Bisect the edges and connect them by removing \ pool balls to make an octahedron and bisect the edges of the octahedron to \ make a cuboctahedron of thirteen balls. The four planes that defined the \ tetrahedron could move inward one layer of balls and meet at the origin of \ the coordinate system (4 dimensions) which is at the center ball of the \ cuboctahedron. 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Buckminster Fuller wrote the books, Synergetics, published in 1975, and \ Synergetics 2, published in 1979, which I read and thought about off and on \ until I discovered what he meant by the term \"Synergetics 60 degree \ coordinate system\" in 1994. Bucky wrote that he discovered the coordinate \ system of nature in 1940(?), but his books don't give concrete examples of \ them.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.463728946946948*^9, 3.463728948441123*^9}}, FontSize->14], Cell["\<\ The idea can be seen by starting with a paper ruled with equilateral \ triangles, and using three numbers representing the perpendicular \ displacement of the three defining lines of an equiangular triangle that is \ positioned at the center of the paper {0,0,0}. That triangle has an edge \ length of zero and an area of zero. All three of the vertices of that \ triangle are at the same place on the plane.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.463728175629279*^9, 3.463728196669446*^9}, { 3.463728953378438*^9, 3.463728954833345*^9}}, FontSize->14], Cell[TextData[{ StyleBox[" ", CellFrame->True], "If all three defining lines of the reference triangle move one unit away \ from the center of the origin triangle you get the triangle {1,1,1}. 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agglomeration of spheres above keeps the same shape when more and more \ layers of closest packed spheres are added. The shape is called a \ cuboctahedron. It is the Vector Equilibrium. It is an equilibrium of vectors \ from equal diameter objects when the centers of the objects are connected to \ the centers of their nearest neighbors by vectors, and the spheres removed. \ It has six square faces and eight triangular faces. If you know the formulas \ for the square numbers (", Cell[BoxData[ FormBox[ SuperscriptBox["n", "2"], TraditionalForm]]], ") and the triangular numbers (", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"n", "(", " ", RowBox[{"1", "+", "n"}], ")"}], "2"], TraditionalForm]]], ") you can compute the shell growth rate of closest packed spheres. Bucky \ Fuller published the formula in 1941(?) and it can be used to find the number \ of protein nodes on the outer shell of a virus." }], "Text", CellDingbat->"\[FilledUpTriangle]", Evaluatable->False, CellChangeTimes->{{3.463729089037468*^9, 3.463729090743947*^9}}, AspectRatioFixed->True, FontSize->14], Cell[TextData[{ StyleBox[" ", CellFrame->True], "Objects under consideration can be assumed to act in equal amounts in all \ directions sometimes, i.e. like spheres. If you know the Avogadro constant \ that 22.4 liters of ANY gas at one atmosphere at zero degree Celsius contains \ 6.02252 times ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "23"], TraditionalForm]]], "molecules, then you can compute the size of a tetrahedron with a volume of \ 22.4 liters and think of the molecules as being like stacked cannon balls in \ a court yard. ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"n", "(", RowBox[{"1", "+", "n"}], ")"}], RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}]}], "6"], TraditionalForm]]], " is the formula for the number of balls in a tetrahedron of edge length \ n-1, because it is the sum of the triangular numbers, ", Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{"k", "(", " ", RowBox[{"1", "+", "k"}], ")"}], "2"]}], TraditionalForm]]], ". So, you can figure out the spherical influence of any molecule of gas." }], "Text", CellDingbat->"\[FilledUpTriangle]", Evaluatable->False, CellChangeTimes->{{3.463728343340371*^9, 3.463728345052417*^9}, { 3.463729095317758*^9, 3.4637290967995787`*^9}}, AspectRatioFixed->True, FontSize->14], Cell[TextData[{ StyleBox[" ", CellFrame->True], "{1,1,1} is a triangle with an edge length of three and {1,1,1,1} is a \ tetrahedron with an edge length of four, shown below. 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Their \ intersection after they move defines a point in three dimensional space.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.463729133998946*^9, 3.463729135638995*^9}}, FontSize->14], Cell[TextData[{ StyleBox[" ", CellFrame->True], "Four Synergetics coordinates represent the independent perpendicular \ movements of the four planes of a regular tetrahedron. Their intersections \ after the planes move define a point in four dimensional space, represented \ by a regular tetrahedron in three dimensional space. Whether you move the \ planes toward the center (a negative direction) of the reference tetrahedron, \ or away from the center (positive), you always get another regular \ tetrahedron. The edge length of the tetrahedron defined by the intersections \ of the planes, in terms of unit vectors or Euclid distance, is the sum of the \ coordinates if the coordinates are in terms of the height of a tetrahedron of \ edge length one. If the sum is negative, the tetrahedron is upside down and \ inside out. If the sum is zero it represents a \"fix\", location, or three \ dimensional point, and the zero edge length tetrahedron corresponds to three \ Cartesian coordinates." }], "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.463729140599057*^9, 3.463729141982883*^9}}, FontSize->14], Cell["\<\ The function StoP transforms Synergetics coordinates which add up to zero, to \ perpendicular 90 degree coordination. All movements of the four planes are \ kept with reference to a volume zero tetrahedron at {0,0,0,0}. The four \ planes of the reference tetrahedron coincide with the four hexagonal planes \ of the cuboctahedron. The edge length of the resulting tetrahedron found by \ adding the four coordinates is both the StraightLineDistance (Euclidean \ distance) and the (unit) VectorDistance from vertex to vertex.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.4637283525871553`*^9, {3.463729148631288*^9, 3.463729150383091*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"StraightLineDistance", "[", "x_List", "]"}], ":=", SqrtBox[ FractionBox[ RowBox[{"Plus", "@@", SuperscriptBox["x", "2"]}], "2"]]}]], "Input", CellChangeTimes->{3.621847137925391*^9}], Cell[BoxData[ RowBox[{ RowBox[{"VectorDistance", "[", "x_List", "]"}], ":=", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"Plus", "@@", RowBox[{"Sqrt", "/@", SuperscriptBox["x", "2"]}]}]}]}]], "Input", CellChangeTimes->{3.621847137941063*^9}], Cell["\<\ The VectorDistance function is the distance going by way of adjacent \ neighbors similar to the \"Manhattan distance\" with Cartesian coordinates. \ The synergetics coordinates add up to zero. There are six adjacent neighbors \ in two dimensions, twelve neighbors in three dimensions, d(d+1) neighbors in \ d dimensions with d+1 synergetics coordinates. When the coordinates are \ integers they are the positions of the centers of unit diameter closest \ packed circles in two dimensions and closest packed spheres in three \ dimensions. Notice the difference between VectorDistance and \ StraightLineDistance. VectorDistance gives the results I need for complex \ coordinates (that is why the Abs function is not used).\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.463729159239628*^9, 3.4637291607994633`*^9}}, FontSize->14], Cell["\<\ The edge length of a tetrahedron is just the sum of the four synergetics \ coordinates, like the tetrahedron {1,2,3,4} as shown below.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", Evaluatable->False, CellChangeTimes->{{3.463729164895934*^9, 3.463729166583642*^9}}, AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tet1234", "=", RowBox[{"VertexesOf", "[", RowBox[{"{", RowBox[{"1", ",", "2", ",", "3", ",", "4"}], "}"}], "]"}]}]], "Input", CellChangeTimes->{3.621847153454815*^9}, AspectRatioFixed->True], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3", ",", RowBox[{"-", "6"}]}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "2", ",", RowBox[{"-", "7"}], ",", "4"}], "}"}], 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dimensional space. If the triangle has an edge length of zero \ the point is in the plane. If the triangle is positive the point is above the \ plane. If the triangle is negative the point is below the plane.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", Evaluatable->False, CellChangeTimes->{3.463729185560878*^9}, AspectRatioFixed->True, FontSize->14], Cell[TextData[{ StyleBox[" ", CellFrame->True], "Five synergetics coordinates that add up to zero are points in four \ dimensional space. Just as the first three synergetics coordinates of a point \ in three dimensional space can represent the point as a triangle in the \ plane, the first four synergetics coordinates of a point in four dimensional \ space can represent the point as a tetrahedron in three dimensional space. It \ is implied that to transform the point represented by the tetrahedron in \ synergetics coordinates, to find the perpendicular fourth coordinate, find \ the midpoint of the tetrahedron and make a line perpendicular to all three \ mutually perpendicular XYZ Cartesian coordinate axes until the end point of \ the line is the same distance from all four vertices of the tetrahedron as \ they are from each other. The four perpendicular Cartesian coordinates are \ usually called XYZT for traditional reasons." }], "Text", CellDingbat->"\[FilledUpTriangle]", Evaluatable->False, CellChangeTimes->{3.463729192642531*^9}, AspectRatioFixed->True, FontSize->14], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"a", ",", "b", ",", "c", ",", "d"}], "]"}]], "Input", CellChangeTimes->{3.621847182793682*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"StoP", "[", RowBox[{"{", RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", RowBox[{ RowBox[{"-", "a"}], "-", "b", "-", "c", "-", "d"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{3.6218471828100157`*^9}, AspectRatioFixed->True], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "b"}], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{"a", "+", "b", "+", "c", "+", "d"}], ")"}]}], "+", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "d"}], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{"a", "+", "b", "+", "c", "+", "d"}], ")"}]}]}], ")"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "c"}], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{"a", "+", "b", "+", "c", "+", "d"}], ")"}]}], "+", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "d"}], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{"a", "+", "b", "+", "c", "+", "d"}], ")"}]}]}], ")"}]}]}], ")"}]}]}], ",", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], " ", SqrtBox["3"], " ", RowBox[{"(", RowBox[{"c", "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "a"}], "-", "b", "-", "c", "-", "d"}], ")"}]}], "+", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{ RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "a"}], "-", "b", "-", "c", "-", "d"}], ")"}]}], "+", "d"}], ")"}]}]}], ")"}]}], ",", RowBox[{ RowBox[{"-", SqrtBox[ FractionBox["2", "3"]]}], " ", RowBox[{"(", RowBox[{ RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "a"}], "-", "b", "-", "c", "-", "d"}], ")"}]}], "+", "d"}], ")"}]}], ",", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], " ", SqrtBox[ FractionBox["5", "2"]], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "a"}], "-", "b", "-", "c", "-", "d"}], ")"}]}]}], "}"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQA+JdVedKGs69dGTT3JfZCKS3MV+dDaIv7RC6CaLb1hqI NQHpDyK7nUH0rmfq70C02ccj8u1A+sDf2+Eguv61wyoQfY8xXrYTSD8J2K3Q BaT/HI7QAtEf2leFgWjr3hkJPUA6YWfKzH4gHVFWrToZSD9o35x/FEhn/Okp OQakDQq6OE8AaTlWKY+c8y8dPZbOYsoD0sduBQuC6Ameb6TygfSkw5/DQPSG jDS7AiB99NPyrBIg3SQdFrIWSB8IuKPxFEjL5fiagOiW/18efwDSgom+1scu vHTU6z3kDqKTFgR7Sv155ahYdktQAUhHaOxMBdF5F+ub5wBplr/SEXOBtIAD w6ybEa8dc4ys3W8B6W2n/mvcBtInme8bgWgjq9837gBpMaHKi42Rrx3j886e lIh67bhu3mlJFyANALvdzAM= "]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", "%", "]"}]], "Input", CellChangeTimes->{3.621847182859036*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"a", "-", "b"}], "2"], ",", FractionBox[ RowBox[{"a", "+", "b", "-", RowBox[{"2", " ", "c"}]}], RowBox[{"2", " ", SqrtBox["3"]}]], ",", FractionBox[ RowBox[{"a", "+", "b", "+", "c", "-", RowBox[{"3", " ", "d"}]}], RowBox[{"2", " ", SqrtBox["6"]}]], ",", RowBox[{ FractionBox["1", "2"], " ", SqrtBox[ FractionBox["5", "2"]], " ", RowBox[{"(", RowBox[{"a", "+", "b", "+", "c", "+", "d"}], ")"}]}]}], "}"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQBeJZyrKlDedeOnac3p/ZCKSPyt6YDaItrERugugFvkZi TUA6bvseZxDdFqn5DkTnbDkm3w6kv5beDQfRIaucVoHo2MMJsp1AOvXjHoUu IL3JLEoLRDe4rQkD0WFPZib0gGi1tJn9QLpfslZ1MpBec21L/lEgvdSrr+QY kJ5/o5vzBJC2nCntkXP+peMbhzlMeUB6ilqoIIg2EngnlQ+kAxK+hoHoL3/T 7QqA9LuJK7NKgPTRueEha4G0ENs9jadA+sNrPxMQ/WL+t8cfgHSaob/1sQtA +6KOuIPoNWtCPKX+vHK8Y70rVQFIq11uaJ4DpN9wy0bMBdKPpRhn3Yx47VjD Y+N+C0gH7WDQvA2kA7/fNwLRNWJ/btwB0q+1qy42Rr52jGo9d1Ii6rWjXdMZ SRcgDQANKsah "]] }, Open ]], Cell[TextData[{ "The triangle {1,1,1} can be looked at by finding the three vertices and \ plotting them in two dimensional space. 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Weisstein's MathWorld entry about Synergetics Coordinates \ is on line at:", CellFrame->True], "\n", "Weisstein, Eric W. \"Synergetics Coordinates.\" From MathWorld--A Wolfram \ Web Resource. ", StyleBox["\n", CellFrame->True], ButtonBox["mathworld.wolfram.com/SynergeticsCoordinates.html", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://mathworld.wolfram.com/SynergeticsCoordinates.html"], None}] }], "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463729269522987*^9, {3.464440732696431*^9, 3.464440732698654*^9}, 3.464440790928156*^9}, FontSize->14], Cell[TextData[{ StyleBox[" ", CellFrame->True], "Both volumes of the books Synergetics were on line the last time I looked \ at:\n", ButtonBox["www.rwgrayprojects.com/synergetics/synergetics.html", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.rwgrayprojects.com/synergetics/synergetics.html"], None}] }], "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{3.463729275227046*^9}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Area", " ", "and", " ", "Volume", " ", "With", " ", "Synergetics", " ", "Coordinates"}]], "Section", CellChangeTimes->{3.6218474007839613`*^9}], Cell["\<\ Area is measured by how many edge length one equilateral triangles instead of \ how many squares it takes to cover a region on a surface. Volume is measured \ by how many unit edge length tetrahedrons instead of cubes it takes to fill a \ region of space. Unit edge length regular tetrahedrons can't fill all-space \ by joining them face to face, but volume can still be measured with them. The \ tetra-volume of a polyhedron is always a rational number if the synergetics \ coordinates of its vertices are rational numbers.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{{3.463714080499013*^9, 3.463714091187274*^9}, 3.463728404405136*^9, 3.463735280289453*^9}, FontSize->14], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{ "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "x1", ",", "x2", ",", "x3"}], "]"}]], "Input", CellChangeTimes->{3.6218474008642797`*^9, 3.6704609716217318`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"generalTriangle", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"a", ",", "b", ",", RowBox[{ RowBox[{"-", "a"}], "-", "b"}]}], "}"}], ",", RowBox[{"{", RowBox[{"c", ",", "d", ",", RowBox[{ RowBox[{"-", "c"}], "-", "d"}]}], "}"}], ",", RowBox[{"{", RowBox[{"e", ",", "f", ",", RowBox[{ RowBox[{"-", "e"}], "-", "f"}]}], "}"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{3.6218474008802032`*^9, 3.670460971626894*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"tetrahedronWithGeneralTriangleBase", "=", RowBox[{"Join", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3", ",", RowBox[{ RowBox[{"-", "x1"}], "-", "x2", "-", "x3"}]}], "}"}], "}"}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Append", "[", RowBox[{"#1", ",", "0"}], "]"}], "&"}], ")"}], "/@", "generalTriangle"}]}], "]"}]}]], "Input", CellChangeTimes->{3.621847400914201*^9, 3.670460971632029*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3", ",", RowBox[{ RowBox[{"-", "x1"}], "-", "x2", "-", "x3"}]}], "}"}], ",", RowBox[{"{", RowBox[{"a", ",", "b", ",", RowBox[{ RowBox[{"-", "a"}], "-", "b"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"c", ",", "d", ",", RowBox[{ RowBox[{"-", "c"}], "-", "d"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"e", ",", "f", ",", RowBox[{ RowBox[{"-", "e"}], "-", "f"}], ",", "0"}], "}"}]}], "}"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQAWJNZd2WhnMvHU3mHFvQCKR9loa+awLSKS8+ybcDafdw rggQ7fM0fxWIrmDsk+0E0icFPyl0AelzT7u1QPTJ0pthIHq57/GEHiA9h2v2 zH4gfcdnpepkIM0V/iz/KJB+l7ib8wSQthEM8sg5/9JxddU5pjwg3bGtXRBE X41Rls4H0o/2aoaD6E1Rc+wKgPSi/odZJUBaILIzZC2QlssW0XwKpJ8pN5qA 6EfKuk8+AGmPnRXWxy68dLQK+OIOou8tqPaU+vPKcUWynK4CkNb2vZ0Kotu2 NbDPBdJld80jQLTYpMMMNyNeOz5KKXEF0WfX/Q0A0W6G4rNA9I41Ue63gHRe jYzmbSAdd/2XEYhOyZW8eQdIT6mceLEx8rXjzPpXJyWiXjvqsbyQdAHSAESE vug= "]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", FractionBox[ RowBox[{"Det", "[", RowBox[{"tetrahedronWithGeneralTriangleBase", "-", FractionBox["1", "4"]}], "]"}], RowBox[{"Det", "[", RowBox[{"generalTriangle", "-", FractionBox["1", "3"]}], "]"}]], "]"}]], "Input", CellChangeTimes->{3.62184740099999*^9, 3.670460971642271*^9}], Cell[BoxData[ RowBox[{"x1", "+", "x2", "+", "x3"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQAeJHHI9bGs69dNTJOLGgEUhPiAh/1wSkdyz4It8OpBc5 8kSAaIXHhatA9M4//bKdQPph9WeFLiAdw9irBaLdTt8KA9H/4k8m9ABpzU1z ZvYD6bIrq1UnA+ko3xf5R4F0Be9ezhNAOigk2CPn/EvH5K3nmfKA9LNPHYIg WmKSinQ+kHYo0Q4H0VlM8+wKgPSq4MdZJUB6ya+ukLVAWktPTPMpkP7wtckE RNtkGj75AKQtWqusj1146bhA+ps7iD6xtcZT6s8rxztfFHQVgHRT0p1UEP1p ZSP7XCD965BFBIhmsz/CcDPiteO5rSWuIFpO5l8AiBZZID4LRO/viXa/BaQ7 vspo3gbSL/x/G4FocQapm3eAdJf+pIuNka8dE9+/OikR9dqx5NALSRcgDQA8 psJQ "]] }, Open ]], Cell["\<\ That (above) proves that the triangular area of the base of a tetrahedron \ times the height of the tetrahedron equals the tetra-volume of the \ tetrahedron, just like the square area of a square times the height of a cube \ equals the cubical volume of the cube, except that the tetrahedron can be a \ skew tetrahedron. 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His website is: http://www.secamlocal.ex.ac.uk/people/staff/rjchapma/rjc.html\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726681744603*^9}, { 3.463726778121602*^9, 3.463726779882474*^9}, 3.463728427531055*^9, { 3.463735413517891*^9, 3.46373541805315*^9}, {3.464200868384327*^9, 3.464200909080183*^9}}, FontSize->14], Cell["\<\ Rowland, Todd. \"Cyclotomic Field.\" From MathWorld--A Wolfram Web Resource, \ created by Eric W. Weisstein. \ http://mathworld.wolfram.com/CyclotomicField.html\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726681744603*^9}, { 3.463726778121602*^9, 3.463726779882474*^9}, 3.463728427531055*^9, { 3.463735413517891*^9, 3.46373541805315*^9}, {3.464200868384327*^9, 3.464200920320344*^9}}, FontSize->14], Cell["\<\ The number of dimensions the nth cyclotomic field has is given by the Euler \ Totient function. Weisstein, Eric W. \"Totient Function.\" From MathWorld--A Wolfram Web \ Resource. http://mathworld.wolfram.com/TotientFunction.html\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726681744603*^9}, { 3.463726778121602*^9, 3.463726779882474*^9}, 3.463728427531055*^9, { 3.463735413517891*^9, 3.46373541805315*^9}, {3.464200868384327*^9, 3.464200932120264*^9}}, FontSize->14], Cell["\<\ The proof that the nth cyclotomic field has n - 1 dimensions when n is a \ prime number is not trivial according to Robin Chapman.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726681744603*^9}, { 3.463726778121602*^9, 3.463726779882474*^9}, 3.463728427531055*^9, { 3.463735413517891*^9, 3.46373541805315*^9}, {3.464200868384327*^9, 3.464200940639277*^9}}, FontSize->14], Cell["\<\ Multiplication The n coordinates of a B_n number add up to zero. You make the outer product \ to make an n by n table of the multiplication products of the coordinates in \ two B_n numbers when you want to multiply B_n numbers. Then you have to \ decide how to add up the results and where to put them into the n coordinates \ of the result. The coordinates of each B_n number are labeled 1 to n and the \ rule is to add up the products in the table that have the same sum of labels \ (j-residue n). (j-residue n) is almost like congruence modulo n but gives \ results 1 to n instead of 0 to n-1. So 5 (mod 5) is 0 but 5 (j-residue 5) is \ 5. You could just work from the 5 by 5 table for B_5 multiplication, but, \ labels (1,2,3,4,5) + (5,4,3,2,1) (j-residue 5) = (1,1,1,1,1) for B_5 numbers, \ so the inner product of a B_5 number with the reverse of the other B_5 number \ goes in the first coordinate of the result. Then you rotate the coordinates \ of one of the B_5 numbers so the labels add up to (2,2,2,2,2), then \ (3,3,3,3,3), (4,4,4,4,4), and (5,5,5,5,5) (j-residue 5) and put the inner \ products into coordinates 2, 3, 4 and 5 respectively. You divide each inner \ product by (-n) for B_n numbers. Then some B_4 numbers, B[1,-1,1,-1] for \ instance, do not have a multiplicative inverse because B_4 numbers are \ isomorphic to the 4th cyclotomic field which has two dimensions and B_4 \ numbers have three dimensions, not two. B_n numbers work right when n is a \ prime number and the coordinates are exact rational numbers or B_m numbers, n \ not equal to m (m is prime too). B_n numbers with modulo p arithmetc is good \ if p is a prime number and is a quadratic nonresidue of n. Weisstein, Eric W. \[OpenCurlyDoubleQuote]Quadratic Residue.\ \[CloseCurlyDoubleQuote] From MathWorld--A Wolfram Web Resource. \ http://mathworld.wolfram.com/QuadraticResidue.html\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726681744603*^9}, { 3.463726778121602*^9, 3.463726779882474*^9}, 3.463728427531055*^9, { 3.463735413517891*^9, 3.46373541805315*^9}, {3.464200868384327*^9, 3.4642008905425653`*^9}, {3.670459522848157*^9, 3.67045954572967*^9}, { 3.670459577212882*^9, 3.6704597236735907`*^9}, {3.6704598154760838`*^9, 3.670459859720175*^9}, {3.670460002206889*^9, 3.6704600098840847`*^9}, { 3.670460142700245*^9, 3.67046016465212*^9}, {3.670808356564785*^9, 3.670808360648983*^9}}, FontSize->14], Cell["\<\ Division You can divide B_n numbers c/d by solving the matrix equation Mx = c for the \ unknown vector x where M is an n by n matrix for the B number d. It is only \ necessary to know the first row of the matrix M, because each subsequent row \ is a rotation to the right one position of the row above it. \ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726685830834*^9}, { 3.464209682684671*^9, 3.4642096886188602`*^9}, {3.464990072742599*^9, 3.464990074695141*^9}}, FontSize->14], Cell["\<\ FirstRowofM[x_List] := -Reverse[x]/Length[x] + 1/Length[x] FirstRowofM[{a, b, c}] == {1/3 - c/3, 1/3 - b/3, 1/3 - a/3} \ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726685830834*^9}, { 3.464209682684671*^9, 3.464209688616806*^9}, 3.464990063736661*^9}, FontSize->14], Cell["\<\ But, the multiplicative inverse of a B_n number, when n is a prime number, \ can be found without creating the matrix M and storing it in computer memory, \ and without having to solve the linear algebra problem. \ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726691683813*^9}, 3.464990080326346*^9}, FontSize->14], Cell["\<\ The matrix M is called a circulant matrix. Its eigen values can be computed \ quickly with a formula that uses powers of the nth roots of unity. The \ product of the eigen values is the determinant of the matrix M, and it is a \ rational number.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726700070771*^9}, 3.464990085446611*^9}, FontSize->14], Cell["\<\ Weisstein, Eric W. \"Circulant Determinant From MathWorld--A Wolfram Web \ Resource. http://mathworld.wolfram.com/CirculantDeterminant.html\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726707471335*^9}, 3.464990090198369*^9, {3.670457368230768*^9, 3.67045739348855*^9}}, FontSize->14], Cell["\<\ The nth roots of unity are usually defined as complex numbers e^(2*Pi*i*k/n), \ for k = 0 .. (n - 1), and i = Sqrt[-1], or equivalently Cos[2*Pi*k/n] + \ i*Sin[2*Pi*k/n]. But the nth roots of unity are rotations of the coordinates \ of unity for B_n numbers (nth roots of unity = RotateLeft[unity, k] for k = 0 \ .. (n - 1)). Unity for B_ 3 numbers is B[1, 1, -2]; for B_ 5 numbers B[1, 1, \ 1, 1, -4] etc .. So, the eigen values for the matrix M, for a B_n number b, \ are themselves B_n numbers . The first eigen value e[1] is always unity, and \ the last e[n], is always b itself. The partial product of the eigen values, \ e[1]*e[2]* ... e[n - 1], divided by the full product e[1]*e[2]* ... e[n], is \ equal to 1/e[n] = 1/b. You only need the first coordinate of the full product \ of the eigen values, so, you divide the partial product by a scalar. (and \ that' s the point : you don' t know how to divide by a B_n number unless the \ first n - 1 coordinates are all the same; then it' s the same as dividing by \ a scalar). The Initialization section has the Mathematica code for the BuckyNumber \ algorithms.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{ 3.463726648014382*^9, {3.4637266817256937`*^9, 3.463726735536882*^9}, { 3.464990095552843*^9, 3.464990111782752*^9}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell["Solving Matrix Problems Using Bucky Numbers", "Section", CellChangeTimes->{3.463724896636549*^9}], Cell[TextData[{ Cell[BoxData[ FormBox[ SubscriptBox["B", "n"], TraditionalForm]]], " Numbers have n numbers that add up to zero and are geometrically \ interpreted with the synergetics coordinate system as regular simplexes with \ an edge length of zero and n vertices at the same place." }], "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell[TextData[{ "Each coordinate in a B number fixes a geometric object. The objects fixed \ by the three coordinates of a ", Cell[BoxData[ FormBox[ SubscriptBox["B", "3"], TraditionalForm]]], " number are lines. The coordinate axes are perpendicular to the lines that \ define an equiangular triangle, and differ by 120 degrees in their \ directions. The movements of the lines are in terms of the height of an edge \ length one equilateral triangle, and the intersection of the lines after they \ move are the three vertices of an equiangular triangle whose edge length is \ the sum of the three numbers (zero)." }], "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{3.463728462159131*^9}, FontSize->14], Cell[TextData[{ "A regular ", Cell[BoxData[ FormBox[ SubscriptBox["T", "n"], TraditionalForm]]], " simplex has n vertices that are the same distance from each other. The \ first n-1 coordinates of a ", Cell[BoxData[ FormBox[ SubscriptBox["B", "n"], TraditionalForm]]], "number are a regular ", Cell[BoxData[ FormBox[ SubscriptBox["T", RowBox[{"n", "-", "1"}]], TraditionalForm]]], " simplex with and edge length of minus the ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "th"], TraditionalForm]]], " coordinate, in the synergetics coordinate system. " }], "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{3.463714788423243*^9}, FontSize->14], Cell["\<\ Here is a matrix m, and an example of the LinearSolve function. Then the \ problem is solved using B numbers. \ \>", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"m", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "4"}], "}"}]}], "}"}]}]], "Input", CellChangeTimes->{3.621847516388653*^9, 3.6704609717160892`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "4"}], "}"}]}], "}"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQA2KlxCvdDedeOqaaf17aCKTFG52tm4B0mmPWOxBd9Idd oR1I50yVjQDRO842rwLR6woWyXYC6anXOBS7gHRY0WItEP279W0YiDbQvp3Q A6TL6jfO7AfSa7z2qk4G0nNe/sk/CqT3RJ6uOAakl909x3kCSJ9vTvfIOf/S serNE6Y8IH1nx1xBEH0mwV46H0h7ibuGg+imns12BUBabc33rBIgzWQzP2Qt kJbp0dJ8CqTDvsw0AdEnlD2efADS8z0nWB+78NLxzTtuDxDt0NbpKfXnleO/ tz3WCkD6+O6XqSD6l9RTgblAetNVxwgQ/YH1NMPNiNeOEULlriB6GSdbIIiW 7JefBaJFJFPcbwHpAkNVzdtAmtec2RhETxBUvnkHSKvIz7zYGPnascv380mJ qNeOr7a8l3QB0gBsM8g+ "]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"LinearSolve", "[", RowBox[{"m", ",", RowBox[{"{", RowBox[{"5", ",", "6"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{3.621847516407398*^9, 3.670460971725812*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", FractionBox["9", "2"]}], "}"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQA+IcbumehnMvHWepui1rBNLvEt2sm4C0mF7OOxAtXM+p 0A6k/X3lI0B0BGvrKhAdp7VEthNICylwKXYB6Wu7lmiB6C1K78NAtNCzOwk9 QPrD0U0z+4H0kwP7VCcDaZlF//KPAunsesnKY0D6QNAFzhNA+s7eDI+c8y8d s0SeMeUB6ftv5wmC6EmvHKTzgbTeU7dwEH2EfatdAZAO8/yZVQJSx70wZC2Q 3sOio/kUSCtPnm0CoqttvZ58ANLHBSdZH7vw0nHDcl4PEL1mZpen1J9XjnIq vdYKQDru6KtUEL0n4IXAXCA9/6lTBIiOqzvNcDPiteOXxHJXEF1YyhYIoqN/ y88C0TbKqe63gPR2CTXN20B67mpmYxC9drryzTtA2rJ75sXGyNeOvns/n5SI eu2opvZB0gVIAwA+B8R0 "]] }, Open ]], Cell["\<\ LinearSolve[m,{5,6}] solves the matrix equation m times the solution vector \ of two scalars equals {5,6}. The solution vector is the scalars {x, y} such \ that x+2*y == 5 and 3*x+4*y == 6. The problem can be stated x(1,3)+y(2,4) = \ (5,6). In other words, x and y each scale a vector.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell[TextData[{ "The known vectors can be Complex numbers or ", Cell[BoxData[ FormBox[ SubscriptBox["B", "3"], TraditionalForm]]], " numbers." }], "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell["\<\ The functions to divide by a B number and the mz function are in the \ initialization cells section.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", " ", "mz"}]], "Input", CellChangeTimes->{ 3.621847516413991*^9, 3.670460971735642*^9, 3.6712033698665657`*^9, { 3.671203440489818*^9, 3.671203452620693*^9}}], Cell[CellGroupData[{ Cell["Global`mz", "Print", "PrintUsage", CellChangeTimes->{3.6712040475246477`*^9}, CellTags->"Info3671178847-9125241"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{"mz", "[", "x_List", "]"}], ":=", RowBox[{"Append", "[", RowBox[{"x", ",", RowBox[{"-", 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The three parts of a ", Cell[BoxData[ FormBox[ SubscriptBox["B", "3"], TraditionalForm]]], " number are synergetics coordinates. They add up to zero. The ", Cell[BoxData[ FormBox[ SubscriptBox["B", "3"], TraditionalForm]]], " numbers are points on the plane, but the last coordinate can be dropped or \ ignored and the first two coordinates fix the end points of a directed line \ segment. {1,1} is a line segment of length two. Minus the last coordinate of \ a ", Cell[BoxData[ FormBox[ SubscriptBox["B", "3"], TraditionalForm]]], " number is the width of the directed line segment fixed by the first two \ coordinates." }], "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell["First divide everything by the first coefficient.", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ FractionBox["ee", RowBox[{ "ee", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]]], "Input", CellChangeTimes->{3.621847516620984*^9, 3.670460971795615*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"B", "[", RowBox[{"1", ",", "1", ",", RowBox[{"-", "2"}]}], "]"}], ",", RowBox[{"B", "[", RowBox[{ FractionBox["22", "13"], ",", FractionBox["16", "13"], ",", RowBox[{"-", FractionBox["38", "13"]}]}], "]"}], ",", RowBox[{"B", "[", RowBox[{ FractionBox["47", "13"], ",", FractionBox["20", "13"], ",", RowBox[{"-", FractionBox["67", "13"]}]}], "]"}]}], "}"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQB+Jt0ju2Npx76Rg279mKRiAtcrPXuglIZ0WsfQei7cLt FdqB9CGx5AgQ7b7uxCoQHffjpWwnkFZwsVPsAtI9Vte0QPSbBfLhILqJSyKx B0grib+c2Q+k33H9Up0MpB3UDQqOAunFaY+rjwFpIY7fnCeA9A+nhR455186 bquXZs4D0v09NwVBtMKMYul8IF2j0RIOotmKX9kVAGmmm0bZJUD6QMV55lIg HX/kSshaIP1GOVbzKZC+xXPaBERL8jc8+QCk300+aH3swkvHg43WHiD695RF nlJ/XjnuEppqrQCkg+2Z0kA01/5/AnOBdPH8sAgQfeTgRYabEa8dHRZUuoLo kjk8gSD6pILaLBB9yTPf/RaQjlmlo3kbSFu94jYG0e6ntG/eAdKXN86/2Bj5 2jFj8u+TElGvHZVcvkm6AGkAj63KVw== "]] }, Open ]], Cell["\<\ Each simplex represented by a B number has a width and a mid point. The mid \ point of {1,1} is {0,0}. Matrix problems seek scalar answers. The vector \ equation is true for both the midpoints and the widths of the simplexes \ represented by the B numbers, so all of the B numbers can be replaced by \ their mid points. Then at least one of the variables drops out (it is \ multiplied by zero, the mid point of unity B[1,1,-2]).\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mid", "/@", "%"}]], "Input", CellChangeTimes->{3.621847516707513*^9, 3.670460971805538*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"B", "[", RowBox[{"0", ",", "0", ",", "0"}], "]"}], ",", RowBox[{"B", "[", RowBox[{ FractionBox["3", "13"], ",", RowBox[{"-", FractionBox["3", "13"]}], ",", "0"}], "]"}], ",", RowBox[{"B", "[", RowBox[{ FractionBox["27", "26"], ",", RowBox[{"-", FractionBox["27", "26"]}], ",", "0"}], "]"}]}], "}"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQAeL1d/qsm869dDxhuP4diD6m76jQDqT/vE+JANH6HadW geiAi69lO4H0DiUHxS4gPUfmhhaIDs5SDAfR8jclE3uAtMCnVzP7gbTCvd+q k4H0vBzDgqNAWmn5lppjQPqJ3R/OE0D6HOdij5zzLx03ecsy5wHpXwm3BUH0 ttRS6XwgfS67IxxE5zi+sSsA0nuWmWSXAOlLPddC1gLpJ1/iNJ8C6ahTZ0xA 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FontSize->14], Cell["\<\ The problem of finding the minimum or maximum of a linear function \ constrained by linear inequalities (called Linear Programming) can be stated \ as a problem of finding unknown scalars of known vectors and solved using B \ numbers. The simplex method invented by Dantzig (I think) is usually used \ (and I think is much faster). You just make so called \"slack variables\" \ (unknowns), to turn an inequality, such as greater than or equal, into an \ equality. If you can find a solution to the system with all the slack \ variables non negative, it solves the inequality.\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell["Finding Roots Of Equations Numerically With Bucky Numbers", "Section"], Cell["\<\ Here is some inspiration from Bucky Fuller's books Synergetics and \ Synergetics 2, section 260.42:\ \>", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell["\<\ 260.42 The synergetic coordinate system of nature and its finite macro-micro \ turnaround-limited hierarchy of primitive ascending or descending \ timeless-sizeless, omnisymmetrically concentric, polyhedral components \ provides the human mind with a rational means of resolving problems by \ bringing nature into a corner__a convergent terminus center, a \ four-dimensional corner of the four-dimensional planes of the tetrahedron. \ Only with the four-dimensional convergence and divergence of synergetics can \ the human mind reduce problems to comprehension as minimum-limit systems. The \ minimum polygon is a triangle; the minimum polyhedron is a tetrahedron; both \ of their structural behaviors are unique (see Secs. 614.00 and 621.00). By \ their academic training humans think only in terms of parallel and \ rectilinear coordination, and so they tend to hold to the unresolvable \ parallel interpretations of their lives\[CloseCurlyQuote] experiences. They \ seek to maintain the status quo and__despite the organic and biologic \ manifests of birth and death__they fail to be able to take advantage of the \ cornerability of comprehension and the positional fixes provided by the \ four-dimensional, synergetic, convergent-divergent coordination. \ \>", "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"rootOfUnity", "[", "5", "]"}]], "Input", CellChangeTimes->{3.621847551034457*^9, 3.67046097185542*^9}], Cell[BoxData[ RowBox[{"B", "[", RowBox[{"1", ",", "1", ",", "1", ",", RowBox[{"-", "4"}], ",", "1"}], "]"}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQBWIvE273pnMvHa/Ib38Hok9peyq0A2mvBzkRIPpo4aVV IDrs6GfZTiDdwO+h2AWk7/y7rwWiD/zWDgfRckcUE3uA9KKnn2b2A2mFYja1 ySA+p03BUSAdIenTcAxIS2rPmX0cSB/7ycZ1AkhrfF3lkXMeSDtrM+cB6Rk/ HguC6Fk/6qXzgbSH5cRwEL23/rtdAZAWmmSTXQKki0vvhawF0u46WZpPgbQd 2w0TEG0h3/vkA5CuCjttfewCkGb08ADR4n0rPaX+vHJs6nporwCkVVZzpIHo 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RowBox[{ RowBox[{"fs", "[", RowBox[{"x", "+", SubscriptBox["u", OverscriptBox["x", "p"]]}], "]"}], "-", RowBox[{"fs", "[", RowBox[{"x", "-", SubscriptBox["u", OverscriptBox["x", "p"]]}], "]"}]}], ",", "p"}], "]"}]}]]}]}]], "Input", CellChangeTimes->{3.6218475512855587`*^9, 3.670460971907063*^9}], Cell["\<\ The function findaRootOne is Newton's method when iterated. The function \ findaRootTwo gives exactly the same result as dividing fs[x] by its symbolic \ first derivative at x and subtracting that from x if fs[x] is a polynomial \ and Length[x] is greater than half the highest exponent of x in fs[x]. The \ function findaRootOne is faster than findaRootTwo. Sometimes it's not easy to \ find a good value for delta for findaRootOne. \ \>", "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{3.463724700485054*^9}, FontSize->14], Cell[BoxData[ RowBox[{"Clear", "[", "f", "]"}]], "Input", CellChangeTimes->{3.6218475512890368`*^9, 3.6704609719118013`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ SuperscriptBox["x", "2"], "-", "x", "-", "1"}]}]], "Input", CellChangeTimes->{3.62184755129253*^9, 3.670460971916643*^9}], Cell[TextData[{ Cell[BoxData[ FormBox[ SubscriptBox["B", "n"], TraditionalForm]]], " numbers are fields when they have an odd prime number n of coordinates if \ exact rational numbers are used when n is greater than 3. Floating point \ coordinates for ", Cell[BoxData[ FormBox[ SubscriptBox["B", "5"], TraditionalForm]]], " numbers can get too close to certain irrational quantities so that \ multiplication of two non zero B numbers gives zeros in every coordinate. \ But, floating point arithmetic is very fast compared to exact rational \ arithmetic, so lets take a chance with the N function." }], "Text", CellDingbat->"\[FilledUpTriangle]", CellChangeTimes->{3.463724719118927*^9, 3.463728504479179*^9}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"firstGuess", "=", RowBox[{"N", "[", RowBox[{"B", "@@", RowBox[{"mz", "[", RowBox[{"{", RowBox[{"1", ",", RowBox[{"-", "1"}], ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{3.621847551296033*^9, 3.670460971921666*^9}], Cell[BoxData[ RowBox[{"B", "[", RowBox[{"1.`", ",", RowBox[{"-", "1.`"}], ",", RowBox[{"-", "1.`"}], ",", "1.`", ",", "0.`"}], "]"}]], "Output", CellChangeTimes->CompressedData[" 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All of these examples are just given to show what \ Bucky Fuller might have meant by \"bringing nature into a corner\" and using \ \"four dimensional positional fixes\". Can you \"zero in\" on a solution to \ an equation quickly by following the four lines that connect the respective \ vertexes of two adjacent tetrahedrons in the sequence of tetrahedrons, that \ have exact rational synergetics coordinates, to a volume zero tetrahedron?" }], "Text", CellDingbat->"\[FilledUpTriangle]", FontSize->14] }, Open ]] }, Open ]] }, AutoGeneratedPackage->None, WindowSize->{913, 691}, WindowMargins->{{Automatic, 138}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PrivateNotebookOptions->{"VersionedStylesheet"->{"Default.nb"[8.] -> False}}, ShowSelection->True, CellLabelAutoDelete->True, FrontEndVersion->"10.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (December 4, \ 2014)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "Info3671178844-9125241"->{ Cell[97692, 1953, 127, 2, 40, "Print", CellTags->"Info3671178844-9125241"], Cell[97822, 1957, 1772, 51, 86, 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