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Cell["The Cover Image ", "Title"],
Cell[TextData[{
"Making the Escher\[Hyphen]style hyperbolic dodecahedron symbolizing ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" 5\n"
}], "Subtitle",
FontSize->16],
Cell["\<\
Michael Trott
Copyright Michael Trott 2003\
\>", "Author"],
Cell[CellGroupData[{
Cell["Outline of the Procedure", "Section"],
Cell[TextData[{
StyleBox["Step 1:",
FontWeight->"Bold"],
"\nWe construct a regular dodecahedron centered at the origin."
}], "Text",
CellDingbat->None],
Cell[TextData[{
StyleBox["Step 2:",
FontWeight->"Bold"],
"\nThe faces of the dodecahedron are then subdivided into right triangles \
(similar to \
M.\[NonBreakingSpace]C.\[NonBreakingSpace]Escher\[CloseCurlyQuote]s ",
StyleBox["Quadratlimit", "TI"],
" work), each of which is then scaled down 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 resulting surfaces are relatively smooth."
}], "Text",
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TextJustification->1],
Cell[TextData[{
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Cell[TextData[{
"The function ",
StyleBox["MapTo3DTriangle", "InlineInput"],
" maps this triangulation onto all triangles of the above triangulation of \
the dodecahedron."
}], "Commentary"],
Cell[BoxData[
\(MapTo3DTriangle[Polygon[{p1_, p2_, p3_}]] :=
Module[{mp = \((p1 + p2)\)/2, dirx, diry}, diry = p3 - mp;
dirx = p1 - mp;
Map[p3 + #\[LeftDoubleBracket]1\[RightDoubleBracket]\ dirx + #\
\[LeftDoubleBracket]2\[RightDoubleBracket]\ diry &,
N[TriangulatedTriangle], {\(-2\)}]]\)], "Input"],
Cell[CellGroupData[{
Cell[BoxData[
\(Show[
Graphics3D[\[ScriptH]\_2 =
Polygon /@ \((N[
First /@ Flatten[
MapTo3DTriangle /@ Take[\[ScriptH]\_1, All]]])\)],
PlotRange \[Rule] All, Axes \[Rule] False]; \)], "Input"],
Cell[GraphicsData["PostScript", "\<\
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\(TriangleBorders[{plato_, \
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p1i, p2i, p3i,
ppl}, {p1i, p2i,
p3i} = \((mp +
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p3}; \[IndentingNewLine]ppl =
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Max[\(\@\(# . #\) &\) /@
Apply[Subtract,
Partition[{p1i, p2i, p3i, p1i}, 2,
1], {1}]]]; \[IndentingNewLine]{Apply[
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j\[RightDoubleBracket],
pn\[LeftDoubleBracket]i + 1, j\[RightDoubleBracket],
pn\[LeftDoubleBracket]i + 1,
j + 1\[RightDoubleBracket],
pn\[LeftDoubleBracket]i,
j + 1\[RightDoubleBracket]}], {i, ppl}, {j,
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iy\ \((q1i - q1)\)\)\/ppq + \(ix\ iy\ \((q2i - q2)\)\)\/\(ppl\ ppq\), {ix, 0,
ppl}, {iy, 0, ppq}]]], {{p1, p1i, p2i, p2}, {p2, p2i,
p3i, p3}, {p3, p3i, p1i, p1}}, {1}],
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+ \(i\ \((#\[LeftDoubleBracket]2\[RightDoubleBracket] - \
#\[LeftDoubleBracket]1\[RightDoubleBracket])\)\)\/ppl, {i, 0,
ppl}] &)\) /@ {{p1i, p2i}, {p2i, p3i}, {p3i,
p1i}}\)]}]\)], "Input"],
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Cell[BoxData[
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Graphics3D[{EdgeForm[], \ \[ScriptH]\_3 =
Transpose[\(TriangleBorders[{Dodecahedron,
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.51334 .84053 m
.5175 .85351 L
s
.709 .208 .046 r
.57372 .24281 m .58718 .25289 L .57449 .24418 L F
.57372 .24281 m .58957 .25369 L .58718 .25289 L F
0 g
.58084 .24686 m
.5856 .24513 L
s
.5619 .23553 m
.57449 .24418 L
s
.57137 .24036 m
.58084 .24686 L
s
.273 .313 .722 r
.75938 .33415 m .75732 .35674 L .7584 .33818 L F
.75938 .33415 m .75803 .35738 L .75732 .35674 L F
0 g
.75625 .37509 m
.75732 .35674 L
s
0 0 0 r
.41273 .27257 m .4211 .26301 L .41583 .26959 L F
0 g
.4211 .26301 m
.42631 .2565 L
s
0 .109 .635 r
.26803 .68768 m .26843 .69994 L .26729 .6889 L F
.26803 .68768 m .26946 .7015 L .26843 .69994 L F
0 g
.26843 .69994 m
.26957 .71089 L
s
.709 .208 .046 r
.65301 .28047 m .67034 .28736 L .65153 .28111 L F
0 g
.67034 .28736 m
.67034 .28736 L
s
0 .109 .635 r
.27603 .72302 m .27879 .71563 L .27575 .72029 L F
0 g
.27275 .72489 m
.27575 .72029 L
s
0 .109 .635 r
.27603 .72302 m .27984 .71721 L .27879 .71563 L F
.244 0 .208 r
.70325 .74656 m .70303 .75317 L .70008 .74949 L F
0 g
.69716 .74585 m
.70008 .74949 L
s
.24