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                    Mathematica-Compatible Notebook

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(*CacheID: 232*)


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Notebook[{
Cell[TextData["Zonohedrification"], "Title",
  Evaluatable->False,
  CellHorizontalScrolling->False,
  TextAlignment->Center],

Cell["\<\
George W. Hart,                               August, 1996
http://www.georgehart.com
\
\>", "Text",
  Evaluatable->False],

Cell[TextData["Reference"], "Section",
  Evaluatable->False,
  FontFamily->"Helv",
  FontSize->14,
  FontWeight->"Bold"],

Cell[TextData[
"\"Zonohedrification\" submitted to Mathematica Journal \nfor publication, by \
George W. Hart, August 1996."], "Input",
  PageWidth->Infinity],

Cell[TextData["Discussion"], "Section",
  Evaluatable->False,
  FontFamily->"Helv",
  FontSize->14,
  FontWeight->"Bold"],

Cell[TextData[
"Runs under Mathematica Kernel 2.1, Front end 2.2\nSee above reference for \
explanation of algorithm."], "Input",
  PageWidth->Infinity],

Cell[TextData["Implementation"], "Section",
  Evaluatable->False],

Cell[BoxData[
    \(SetDirectory["\<c:/mathemat/packages/polyhedr\>"]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(<< "\<Graphics`Polyhedra`\>"\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(builtInPoly[polyhedron_] := 
      Map[\(Vertices[polyhedron]\)\[LeftDoubleBracket]#1
            \[RightDoubleBracket]&, Faces[polyhedron], {2}]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(vertices[poly_] := Union[Flatten[poly, 1]]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[poly_] := Show[Graphics3D[Polygon/@poly], Boxed \[Rule] False]\)], 
  "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(cross[{ax_, ay_, az_}, {bx_, by_, bz_}] := {ay\ bz - az\ by, 
        az\ bx - ax\ bz, ax\ by - ay\ bx}\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(\(tolerance = 0.000001; \)\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(collinear[v1_, v2_] := 
      And@@\(\((Abs[#1] < tolerance&)\)/@cross[v1, v2]\)\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(setStar[vlist_] := 
      Module[{selected = {}}, 
        Scan[Function[v, 
            If[v \[NotEqual] {0, 0, 0} && 
                Select[selected, collinear[v, #1]&] == {}, 
              AppendTo[selected, v]]], vlist]; 
        Print[Length[selected], "\< zonal directions.\>"]; 
        gStar = selected\/2]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(pToXYZ[p_] := Plus@@MapThread[Times, {gStar, p}]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(faceNormal[p_] := 
      With[{indices = Position[p, 0]}, 
        cross[gStar
            \[LeftDoubleBracket]indices\[LeftDoubleBracket]1, 1
              \[RightDoubleBracket]\[RightDoubleBracket], 
          gStar\[LeftDoubleBracket]indices\[LeftDoubleBracket]2, 1
              \[RightDoubleBracket]\[RightDoubleBracket]]]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(endPoints[p_] := {\((If[#1 == 0, \(-1\), #1]&)\)/@p, 
        \((If[#1 == 0, 1, #1]&)\)/@p}\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(allPairs[n_] := 
      With[{indices = Table[i, {i, n}]}, 
        Select[Flatten[Outer[List, indices, indices], 
            1], #1\[LeftDoubleBracket]1\[RightDoubleBracket] < #1
                \[LeftDoubleBracket]2\[RightDoubleBracket]&]]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(makeFace[{i_, j_}] := 
      With[{normal = 
            cross[gStar\[LeftDoubleBracket]i\[RightDoubleBracket], 
              gStar\[LeftDoubleBracket]j\[RightDoubleBracket]]}, 
        \((approxSign[normal . #1]&)\)/@gStar]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(approxSign[x_] := If[Abs[x] < tolerance, 0, Sign[x]]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(allFaces := 
      With[{halfTheFaces = makeFace/@allPairs[Length[gStar]]}, 
        halfTheFaces \[Union] \(-halfTheFaces\)]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(makeEdge[i_, p_, faceNormal_] := 
      With[{normal = 
            cross[gStar\[LeftDoubleBracket]i\[RightDoubleBracket], 
              faceNormal]}, 
        MapThread[
          If[#1 \[NotEqual] 0, #1, approxSign[#2 . normal]]&, {p, gStar}]]
        \)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(makeEdges[p_] := 
      With[{normal = faceNormal[p]}, 
        With[{halfEdges = 
              \((makeEdge[#1, p, normal]&)\)/@Flatten[Position[p, 0]]}, 
          halfEdges \[Union] \((2\ p - #1&)\)/@halfEdges]]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(makePolygon[p_] := 
      \((pToXYZ[#1]&)\)/@
        With[{edges = makeEdges[p]}, 
          With[{vertices = Union[Flatten[endPoints/@edges, 1]]}, 
            Select[cycleSort[Join[edges, vertices]], Count[#1, 0] == 0&]]]
          \)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(allEdges := Union[Flatten[makeEdges/@allFaces, 1]]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(\(lineDrawing := \((Line[pToXYZ/@endPoints[#1]]&)\)/@allEdges; \)\)], 
  "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(cycleSort[list_] := 
      Module[{result = {list\[LeftDoubleBracket]1\[RightDoubleBracket]}, 
          remaining = Rest[list]}, 
        While[remaining \[NotEqual] {}, 
          choices = 
            Select[remaining, 
              incident[#1, 
                  result\[LeftDoubleBracket]1\[RightDoubleBracket]]&]; 
          If[choices \[NotEqual] {}, 
            next = choices\[LeftDoubleBracket]1\[RightDoubleBracket], 
            Print["\<zono-error!:\>", list]; Break[]]; 
          PrependTo[result, next]; 
          remaining = Complement[remaining, {next}]]; result]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(incident[p1_, p2_] := Plus@@Abs[p1 - p2] == 1\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(Null\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(zonohedrify[poly_] := 
      Module[{plist}, setStar[vertices[poly]]; plist = allFaces; 
        printSummary[plist]; makePolygon/@plist]\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(printSummary[p2list_] := 
      With[{maxSize = Max[\((Count[#1, 0]&)\)/@p2list]}, 
        Do[With[{howMany = Length[Select[p2list, Count[#1, 0] == i&]]}, 
            If[howMany > 0, 
              Print["\<     \>", 2\ i, "\<-gons:  \>", howMany]]], {i, 
            maxSize, 2, \(-1\)}]; Print["\<Total faces:  \>", Length[p2list]]]
        \)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(\(polarStar[n_] := 
      N[Table[{Sin[\(2\ \[Pi]\ i\)\/n], Cos[\(2\ \[Pi]\ i\)\/n], 1}, {i, 1, 
            n}]]; \)\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(faceCenters[poly_] := \((Plus@@#1\/Length[#1]&)\)/@poly\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(edgeMidpoints[poly_] := 
      Union[Flatten[\((1\/2\ \((#1 + RotateLeft[#1])\)&)\)/@poly, 1]]\)], 
  "Input",
  PageWidth->Infinity,
  InitializationCell->True],

Cell[BoxData[
    \(unit[v_] := v\/\@\(v . v\)\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(\(viewStar[star_] := 
      Show[Graphics3D[\((Line[{{0, 0, 0}, #1}]&)\)/@star, 
          Boxed \[Rule] False]]; \)\)], "Input",
  PageWidth->Infinity,
  InitializationCell->True,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[TextData["Use the functions"], "Section",
  Evaluatable->False],

Cell[BoxData[
    \(view[builtInPoly[Icosahedron]]\)], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(view[zonohedrify[builtInPoly[Octahedron]]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[zonohedrify[{edgeMidpoints[builtInPoly[Cube]]}]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(\(truncatedCuboctahedron = 
      zonohedrify[{
          unit/@Join[faceCenters[builtInPoly[Cube]], 
              edgeMidpoints[builtInPoly[Cube]]]}]; \)\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[truncatedCuboctahedron]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[zonohedrify[{edgeMidpoints[builtInPoly[Icosahedron]]}]]\)], "Input",\

  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[zonohedrify[builtInPoly[Cube]]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[{
    \(\(star = 
      N[{{1, 0, 0}, {0, 1, 0}, {\@2\/2, \@2\/2, 0}, {\@2\/2, \(-\(\@2\/2\)\), 
            0}, {0, 0, 1}}]; \)\), 
    \(view[zonohedrify[{star}]]\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[Nest[zonohedrify, builtInPoly[Octahedron], 3]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[{
    \(\(star = 
      N[{{1, 0, 1}, {\(-1\), 0, 1}, {0, 1, 1}, {0, \(-1\), 1}, {0, 0, 2}}]; 
    \)\), 
    \(view[zonohedrify[{star}]]\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[{
    \(\(rhombicEnneacontahedron = zonohedrify[builtInPoly[Dodecahedron]]; 
    \)\), 
    \(view[rhombicEnneacontahedron]\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[{
    \(\(rhombicTriacontahedron = zonohedrify[builtInPoly[Icosahedron]]; 
    \)\), 
    \(view[rhombicTriacontahedron]\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[{
    \(\(RTstar = gStar; \)\), 
    \(view[zonohedrify[{Drop[RTstar, \(-1\)]}]]\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[zonohedrify[{Drop[RTstar, \(-2\)]}]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[zonohedrify[{Drop[RTstar, \(-3\)]}]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[{
    \(\(starFig3a = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {\(-\(1\/2\)\), 1\/2, 
          1\/2}}; \)\), 
    \(view[zonohedrify[{starFig3a}]]\)}], "Input",
  PageWidth->Infinity],

Cell[BoxData[{
    \(\(starFig3b = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 1, 0}}; \)\), 
    \(view[zonohedrify[{starFig3b}]]\)}], "Input",
  PageWidth->Infinity],

Cell[BoxData[{
    \(\(setStar[starFig3a]; \)\), 
    \(Show[Graphics3D[lineDrawing, Boxed \[Rule] False]]\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[{
    \(\(setStar[starFig3b]; \)\), 
    \(Show[Graphics3D[lineDrawing, Boxed \[Rule] False]]\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(viewStar[starFig3a]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(viewStar[starFig3b]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[zonohedrify[rhombicTriacontahedron]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[
      zonohedrify[{
          unit/@Join[vertices[builtInPoly[Icosahedron]], 
              faceCenters[builtInPoly[Icosahedron]], 
              edgeMidpoints[builtInPoly[Icosahedron]]]}]]\)], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(view[zonohedrify[truncatedCuboctahedron]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
  FontSize->12,
  FontWeight->"Plain",
  FontSlant->"Plain",
  FontTracking->"Plain",
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[BoxData[
    \(view[zonohedrify[{polarStar[10]}]]\)], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(view[
      zonohedrify[{
          unit/@Join[vertices[builtInPoly[Icosahedron]], 
              edgeMidpoints[builtInPoly[Icosahedron]]]}]]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier",
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