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Michael Trott Copyright Michael Trott 1998\ \>", "Author"], Cell[CellGroupData[{ Cell["Outline of the Procedure", "Section"], Cell[TextData[{ StyleBox["Step 1:", FontWeight->"Bold"], "\nWe start by building a regular dodecahedron with its center at the \ origin and with each vertex a unit distance from the origin." }], "Text", CellDingbat->None], Cell[TextData[{ StyleBox["Step 2:", FontWeight->"Bold"], "\nThe faces of the dodecahedron are then subdivided into right triangles, \ each of which is then scaled down a bit so that gaps are introduced between \ them." }], "Text", CellDingbat->None], Cell[TextData[{ StyleBox["Step 3:", FontWeight->"Bold"], "\nEach triangle is further subdivided so that, when the surfaces are \ transformed in the next step, the surfaces which result will remain \ comparatively smooth." }], "Text", CellDingbat->None, TextJustification->1], Cell[TextData[{ StyleBox["Step 4:", FontWeight->"Bold"], "\nAll coordinates are transformed by ", Cell[BoxData[ \(TraditionalForm \`r\&\[RightVector]\[LongRightArrow]r\&\[RightVector]\ \(f(r)\)\)]], ", where the function ", Cell[BoxData[ \(TraditionalForm\`f(r)\)]], " maps each distance from the origin onto a new distance. 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Since ", Cell[BoxData[ \(TraditionalForm\`f(r)\)]], " is scalar, this transformation does not change the direction between the \ origin and each point, just the distance; we'll call it the ", StyleBox["radial transformation", FontSlant->"Italic"], "." }], "Text", CellDingbat->None], Cell[TextData[{ StyleBox["Step 5:", FontWeight->"Bold"], "\nEach triangle is given depth by connecting it to a slightly smaller copy \ moved closer to the origin." }], "Text", CellDingbat->None], Cell[TextData[{ StyleBox["Step 6:", FontWeight->"Bold"], "\nThe faces of the resulting body are assigned colors." }], "Text", CellDingbat->None] }, Closed]], Cell[CellGroupData[{ Cell["Step 1: The Regular Dodecahedron", "Section"], Cell[TextData[{ "First we load the standard ", StyleBox["Mathematica", FontSlant->"Italic"], " package giving the properties of the Platonic solids." }], "Commentary"], Cell[BoxData[ \(Needs["\"]\)], "InputOnly"], Cell["Here is our starting point, the regular dodecahedron.", 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Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["ShrinkTriangles", "InlineInput"], " scales each triangle toward its center of gravity\[LongDash]that is, \ toward the average of its vertices\[LongDash]by a given factor. A scaling \ factor of 1 means no change; a factor of ", Cell[BoxData[ \(TraditionalForm\`1/2\)]], " halves the size." }], "Text"], Cell[BoxData[ \(ShrinkTriangles[triangles_, \[Sigma]_] := Function[x, \((Function[x, # + \[Sigma]\ \((x - #)\)] /@ x &)\)[Plus @@ x/3]] /@ triangles\)], "InputOnly"], Cell["Shrinking the triangles by a factor 0.8 gives us this:", "Commentary"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[ Graphics3D[ Polygon /@ \((\[ScriptH]\_3 = ShrinkTriangles[\[ScriptH]\_2, 0.8])\)]] \)], "Input"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOi`3001eOi`00`00OiaoW00HOi`00`00OiaoW00j Oi`00017nL0000@WnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiao VgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nK OiaoVgnLOi]oW7nKOiaoV`<0003HOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiao VgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nK 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ImageMargins->{{43, 0}, {0, 1}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Step 3: Subdividing the Triangles", "Section"], Cell[TextData[{ StyleBox["Subdivide", "InlineInput"], " subdivides any triangle\[LongDash]not just right triangles\[LongDash]into \ ", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], " smaller triangles." }], "Commentary"], Cell[BoxData[ \(Subdivide[triangles_, m_] := \n\t Flatten[Apply[ Function[{p, q, r}, MapThread[Append, \({\(Flatten[Join[#1, Rest[#1]], 1] &\)[ \(Partition[#1, 2, 1] &\) /@ Drop[#1, \(-1\)]], \n\t\t\t\t\t\tFlatten[ Join[Rest[#1], \(Drop[Rest[#1], \(-1\)] &\) /@ Drop[#1, \(-1\)]], 1]} &\)[ Table[\ p + \((i\ \((q - p)\) + j\ \((r - p)\))\)/m, {j, 0, m}, {i, 0, m - j}]]]], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t\t\t\ triangles, {1}], 1]\)], "InputOnly"], Cell["\<\ For example, we can extract the list of triangles making one face of the \ dodecahedron.\ \>", "Commentary"], Cell[BoxData[ \(\(\(\[ScriptH]\_4 = Take[\[ScriptH]\_3, {41, 80}]\)\(;\)\)\)], "InputOnly"], Cell[TextData[{ "Here is the face after subdividing each triangle into ", Cell[BoxData[ \(TraditionalForm\`4\^2 \[Equal] 16\)]], " smaller triangles." }], "Commentary"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[ Graphics3D[ Polygon\ /@ \ \((\[ScriptH]\_\(5 a\) = Subdivide[\[ScriptH]\_4, 4])\)\ ]]\)], "Input"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggnL0P008gnL00<007nLOi`027nL00<007nLOi`0FWnL0000>GnLOi]oW7nK OiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW0020000S7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nK001oVgnLOi]oW7nKOiaoVgnLOi]oW7nK001oVgnLOi]oW7nKOiao VgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nK 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1}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["SubdivideEdges", "InlineInput"], " takes the edges of the larger triangles\[LongDash]the ones produced in \ step 2\[LongDash]and subdivides them." }], "Commentary"], Cell[BoxData[ \(SubdivideEdges[triangles_, m_] := \n\t Apply[Function[{p, q, r}, Join @@ Apply[ Table[#1 + i/m \((#2 - #1)\), {i, 0, m}] &, \n\t\t\t\ \ \ \ \t\t{{p, q}, {q, r}, {r, p}}, {1}]], triangles, {1}]\)], "InputOnly"], Cell["In this diagram the points of subdivision are marked.", "Commentary"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[ Graphics3D[ {{Hue[0], Line /@ \((\ \[ScriptH]\_\(5 b\) = SubdivideEdges[\[ScriptH]\_4, 3])\)}, \ \ \ \n\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {PointSize[ .02], Map[Point, \ \[ScriptH]\_\(5 b\), \ {2}]}}\ ]]\)], "Input"], Cell[GraphicsData["Bitmap", "\<\ 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OiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW05oV`00\ \>"], "Graphics", Evaluatable->False, ImageSize->{200.438, 287.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Steps 4 and 5: Radial Transformation and Thickening", "Section"], Cell["\<\ The next task is to create a function for calculating the radial \ transformation.\ \>", "Text"], Cell[BoxData[ \(\(\(attract[x_, \[Eta]_, \[Phi]_, triangles_] := \n\t Module[{xMin, xMax, X, \[Delta]x}, \n\t\t{xMin, xMax} = \@\({\[Phi]\ Max[#], Min[#]} &\)[\(# . # &\) /@ Level[triangles, \ {\(-2\)}]]; X = \((xMin + xMax)\)/2; \[Delta]x = xMax - X; Evaluate //@ Re[If[x < X, \@\(1 - \((x - X)\)\^2/\[Delta]x\^2\), \[Eta]\ \((1 - \@\(1 - \((x - X)\)\^2/\[Delta]x\^2\))\) + 1]]]\)\(;\)\)\)], "InputOnly"], Cell["\<\ As the following plot shows, this function has one of three \ different behaviors, depending on the initial distance of a point from the \ origin. For points nearest the origin, the transformation draws them closer \ still. All points beyond a certain distance from the origin are moved to a \ fixed limit. And points between these two ranges are moved away from the \ origin.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Plot[attract[x, \ 0.6, 0.96, \[ScriptH]\_2], {x, 0.795, 1}, \n\t\ \ \ \ \ \ \ \ \ PlotRange \[Rule] All, Frame \[Rule] True, Axes \[Rule] False, PlotStyle \[Rule] {Hue[0]}]\)\(; \)\)\)], "Input"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgGnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi/007nKOi`0 07nLOi]oW7nKOiaoVgnLOi]oW002000017nKOiaoVgnL0P0007UoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL001oW7nK001o VgnLOi]oW7nKOiaoVgnL001oW7nK001oVgnLOi]oW7nKOiaoVgnLOi]oW7nK OiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOi`007nLOi/007nKOiaoVgnLOi]oW7nKOi`007nL Oi/007nKOiaoV`020000>gnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL 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VgnL001oW7nKOiaoVgnLOi]oW7nKOiaoV`5oV`000WnL0P002WnL0`0017nL 00<007nLOi`02GnL00<007nLOi`02WnL00<007nLOi`02GnL00<007nLOi`0 2GnL00<007nLOi`02WnL00<007nLOi`02GnL00<007nLOi`02GnL00<007nL Oi`02WnL00<007nLOi`02GnL00<007nLOi`02GnL00<007nLOi`02WnL00<0 07nLOi`02GnL00<007nLOi`02GnL00<007nLOi`02WnL00<007nLOi`02GnL 00<007nLOi`02GnL00<007nLOi`02WnL00<007nLOi`02GnL00<007nLOi`0 2GnL00<007nLOi`02WnL00<007nLOi`02GnL00<007nLOi`00gnL0@000GnL 00005GnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W03o000;0001Oi/00001\ \>"], "Graphics", Evaluatable->False, ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell["\<\ Each polygon has zero thickness, so the ThickenTransform builds \ solid 3D shapes by matching each polygon with a slightly offset, slightly \ smaller copy of itself, and then joining the corresponding edges with new \ polygons.\ \>", "Text"], Cell[BoxData[ \(ThickenTransform[f_, triangles_, polys_, edges_, \[Xi]_, \[Eta]_] := \n\tModule[\n\t\t\t{l = Map[f, edges, {\(-2\)}], p = Map[f, polys, {\(-2\)}], outerPolys, innerPolys, radialPolys, outerEdges, innerEdges, radialEdges}, \n\t\t\t{outerPolys, innerPolys} = {Polygon /@ p, Polygon /@ Map[\[Xi]\ # &, p, {\(-2\)}]}; \n\ \ \t\tinnerPolys = Polygon /@ Map[\[Xi]\ # &, p, {\(-2\)}]; \n\ \ \t\tradialPolys = Polygon /@ \(\((Join[#\[LeftDoubleBracket]1\[RightDoubleBracket], Reverse[#\[LeftDoubleBracket]2\[RightDoubleBracket]]] &)\) /@ \n\t\t\t\t\t\ \ \ \ \ \ \ Transpose[ {Flatten[\(Partition[#, 2, 1] &\) /@ l, 1], \ \ \n\t\t\t\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ Flatten[ \(Partition[#, 2, 1] &\) /@ Map[\[Xi]\ # &, l, {\(-2\)}], 1]}]\); \ \ \n\t\t\ \ {outerEdges, innerEdges} = {Line /@ l, Line /@ Map[\[Xi]\ # &, l, {\(-2\)}]}; \n\t\t\ radialEdges = Line /@ Transpose[ \({#, Map[\[Xi]\ # &, #, {\(-2\)}]} &\)[\n\t\t\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Map[ f, Flatten[triangles, 1], {\(-2\)}]]]; {outerPolys, innerPolys, radialPolys, outerEdges, innerEdges, radialEdges}]\)], "InputOnly"], Cell["Here is the result of radial transformation and thickening.", "Commentary"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Show[ Graphics3D[ ThickenTransform[\n\ \ \ \t\t\t\tEvaluate[ attract[Sqrt[# . #], 0.6, 0.94, \[ScriptH]\_2]\ #/Sqrt[# . #]] &, \n\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[ScriptH]\_4, \[ScriptH]\_\(5\ a\), \[ScriptH]\_\(5\ b\), 0.88, 0.6]], \n\t\ \ \ \ \ \ \ \ \ \ \ PlotRange \[Rule] All, Axes \[Rule] True]\)\(;\)\)\)], "Input"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiao VgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nK OiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiao 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h3, h4a, h4b, h5}, \nh1 = SolidToTriangles[solid]; \nh2 = Triangulate[h1, n]; \n h3 = ShrinkTriangles[h2, \[Sigma]]; \nh4a = Subdivide[h3, m]; \n h4b = SubdivideEdges[h3, m]; \n h5 = Apply[ ThickenTransform[ GG = Evaluate[ attract[Sqrt[# . #], \[Eta], \[Phi], h2]\ #/ Sqrt[# . #]] &, ##, \[Xi], 0.6] &, \n Transpose[ {Partition[h3, Length[h3]/l], Partition[h4a, Length[h4a]/ l], \ \ \ \ \ \ \ \ \n\t\t\t\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ Partition[h4b, Length[h3]/l]}], {1}]]\)], "Input"], Cell["\<\ This example generates a slightly simplified version of the final \ cover image.\ \>", "Commentary"], Cell["CoverImage[Dodecahedron,{3,4,0.96,.85},0.6,0.955];", "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(With[ {colors\ = {0.63, 0.97, 0.7, 0.7, 0.38, 0.15, 0.7, 0.7, 0.7, 0.85, 0.6, 0.7}}, \n\t Show[Graphics3D[ {EdgeForm[], Thickness[0.00023], \ \n\t\t\t\t\ \ \ \ \ \ \ Transpose[ {\(SurfaceColor[Hue[#, 0.85, 1], Hue[#, 0.2, 1], 3.2] &\) /@ colors, \ \n\t\t\t\t\t\t\ \ \ \ \ \ \ \ CoverImage[ 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Since the faces \ of the dodecahedron have been divided into a great many polygons, the \ calculation of the graphic may be beyond the memory capacity of your \ computer.\ \>", "Commentary"], Cell[BoxData[ \(\(With[ {colors\ = {0.63, 0.97, 0.7, 0.7, 0.38, 0.15, 0.7, 0.7, 0.7, 0.85, 0.6, 0.7}}, \n\t Show[Graphics3D[ {EdgeForm[], Thickness[0.00001], Transpose[ {\(SurfaceColor[Hue[#, 0.85, 1], Hue[#, 0.2, 1], 3.2] &\) /@ colors, CoverImage[Dodecahedron, {6, 6, 0.96, 0.85}, 0.6, 0.955]}]}], \n\t\tBoxed \[Rule] False, PlotRange \[Rule] All, ViewPoint \[Rule] {2, \(-3\), 3.3}]]; \)\)], "Input", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["Some Variants", "Section"], Cell["\<\ The above implementation can be used with other Platonic solids, \ radial transformations, and coloring schemes. Here is a cube (hexahedron) with its vertices pushed far outwards and without \ a fixed outer limit. Each face is assigned a randomly chosen color.\ \>", "Commentary"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(With[{colors\ = Table[Random[], {6}]}, \n\t Show[Graphics3D[ {EdgeForm[], Thickness[0.00023], Transpose[ {\(SurfaceColor[Hue[#, 0.35, 0.8], Hue[#, 0.28, 0.9], 2.7] &\) /@ colors, CoverImage[Hexahedron, {4, 4, 0.9, 0.9}, 1.4, 1]}]}], \n\ \ \ \ \ \ \ \ \ \t\tBoxed \[Rule] False, PlotRange \[Rule] All]]\)\(;\)\)\)], "Input"], Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0`0005UoVgnLOi]oW7nK OiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL Oi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiao VgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nK005oW000Z7nL00@0 02S]000a3`<005YoW0000:QoW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nK OiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]o W7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnLOi]oW7nKOiaoVgnL 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