(*^
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	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
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:[font = section; inactive; dontPreserveAspect]
Polyhedra

:[font = text; inactive; preserveAspect]
We will need the homogeneous transformation matrices
:[font = input; Cclosed; preserveAspect; startGroup]

homogenize[vec:{_,_}]:=Append[vec,1]
deHomogenize[vec:{_,_,_}]:=Drop[vec/(Last[vec]),-1]
homogenize[vec:{_,_,_}]:=Append[vec,1]
deHomogenize[vec:{_,_,_,_}]:=Drop[vec/(Last[vec]),-1]
normalize[vec:_List]:= vec/Sqrt[Apply[Plus,vec^2]]


rotationMatrix2D[theta_] := Transpose[
	{{Cos[theta], Sin[theta], 0},
	 {-Sin[theta], Cos[theta], 0},
	 {0,0,1}}]
	 
rotationMatrix2D[theta_,{cx_,cy_}] := Transpose[
	{{Cos[theta], Sin[theta], 0},
	 {-Sin[theta], Cos[theta], 0},
	 {(1-Cos[theta])cx +(Sin[theta])cy,
	 (1-Cos[theta])cy -(Sin[theta])cx,1}}]
		
scalingMatrix2D[{sx_,sy_}] := Transpose[ 
		{ {sx,0,0},{0,sy,0},{0,0,1} }]
		
scalingMatrix2D[{sx_,sy_},{cx_,cz_}] := Transpose[ 
		{ {sx,0,0},{0,sy,0},{(1-sx)cx, (1-sy)cy, 1} }]
		
translationMatrix2D[{tx_,ty_}] := Transpose[ 
		{ {1,0,0},{0,1,0},{tx,ty,1} }]


xRotationMatrix3D[phi_] := Transpose[	{	{1, 0, 0, 0},
					{0, Cos[phi], Sin[phi],0},
					{0,-Sin[phi], Cos[phi],0},
					{0, 0, 0, 1}	}]

yRotationMatrix3D[psi_] := Transpose[	{	{Cos[psi], 0, -Sin[psi], 0},
					{0, 1, 0, 0},
					{Sin[psi], 0, Cos[psi], 0},
					{0, 0, 0, 1}	}]

zRotationMatrix3D[theta_]:= Transpose[{	{Cos[theta], Sin[theta], 0,0},
					{-Sin[theta], Cos[theta], 0, 0},
					{0, 0, 1, 0},
					{0, 0, 0, 1}	}]

		
scalingMatrix3D[{sx_,sy_,sz_}] := Transpose[ 
		{ {sx,0,0,0},{0,sy,0,0},{0,0,sz,0},{0,0,0,1} }]
	
scalingMatrix3D[{sx_,sy_,sz_},{cx_,cy_,cz}] := Transpose[ 
		{ {sx,0,0,0},{0,sy,0,0},{0,0,sz,0},
		{(1-sx)cx, (1-sy)cy,(1-sz)cz, 1} }]

translationMatrix3D[{tx_,ty_,tz_}] := Transpose[ 
		{ {1,0,0,0},{0,1,0,0},{0,0,1,0},{tx,ty,tz,1} }]


:[font = message; inactive; preserveAspect]
General::spell1: 
   Possible spelling error: new symbol name "rotationMatrix2D"
     is similar to existing symbol "RotationMatrix2D".
:[font = message; inactive; preserveAspect]
General::spell1: 
   Possible spelling error: new symbol name "xRotationMatrix3D"
     is similar to existing symbol "RotationMatrix3D".
:[font = message; inactive; preserveAspect]
General::spell: 
   Possible spelling error: new symbol name "yRotationMatrix3D"
     is similar to existing symbols {RotationMatrix3D, xRotationMatrix3D}.
:[font = message; inactive; preserveAspect; endGroup]
General::spell: 
   Possible spelling error: new symbol name "zRotationMatrix3D"
     is similar to existing symbols 
    {RotationMatrix3D, xRotationMatrix3D, yRotationMatrix3D}.
:[font = subsection; inactive; dontPreserveAspect]
Polygon Representation
:[font = text; inactive; dontPreserveAspect]
We will define a tetrahedron to illustrate the representation of 3D polygons and their grouping into graphic objects.
The most flexible data structure is obtained by setting up a list (or array) of vertice coordinates,  with a companion list of face indexes,  indicating for each face the corresponding subset of vertices. 
Tetrahedra have four vertices defined by four triples {x,y,z} (quadruples {x,y,z,1} in homogeneous coordinate).  This tetrahedron is centered at the origin. 
:[font = input; dontPreserveAspect; startGroup]

verticelist=
{{0, 0, 3^(1/2)}, {0, (8/3)^(1/2), -3^(-1/2)}, 
  {-2^(1/2), -(2/3)^(1/2), -3^(-1/2)}, 
  {2^(1/2), -(2/3)^(1/2), -3^(-1/2)}}//N
:[font = output; output; inactive; preserveAspect]
{{0, 0, 1.732050807568877294}, 
 
  {0, 1.632993161855452065, -0.5773502691896257645}, 
 
  {-1.414213562373095049, -0.8164965809277260328, -0.5773502691896257645}, 
 
  {1.414213562373095049, -0.8164965809277260328, -0.5773502691896257645}}
;[o]
{{0, 0, 1.73205}, {0, 1.63299, -0.57735}, {-1.41421, -0.816497, -0.57735}, 
 
  {1.41421, -0.816497, -0.57735}}
:[font = text; inactive; dontPreserveAspect]
At this point we could plot the vertices;  We first define a display function that shows the axes and scale:
:[font = input; dontPreserveAspect]
display[object_] := Show[Graphics3D[{PointSize[0.04],
object,
RGBColor[1,0,0],PointSize[0.02],Point[{0,0,0}],
Line[{{0,0,4},{0,0,0},{0,4,0},{0,0,0},{4,0,0}}]}],
Axes->Automatic, PlotRange->{{-3,3},{-3,3},{-3,3}},
ViewPoint->{1.170, -1.730, 0.650},AxesLabel->{"X","Y","Z"}]
;[s]
5:0,0;161,1;178,0;271,1;272,0;273,-1;
2:3,14,10,Courier,1,12,0,0,0;2,17,12,New York,1,12,0,0,0;
:[font = input; dontPreserveAspect; startGroup]
display[ Map[Point,verticelist]]
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Graphics3D["<<>>"]
;[o]
-Graphics3D-
:[font = subsection; inactive; dontPreserveAspect]
Polyhedron Construction 
:[font = text; inactive; dontPreserveAspect]
The facesmust be specified next. Note that each sublist is ordered consistently, with right-handed circulation pointing inside the volume.
:[font = input; dontPreserveAspect; startGroup]
facelist ={{1, 2, 3}, {1, 3, 4}, {1, 4, 2}, {2, 4, 3}}

:[font = output; output; inactive; dontPreserveAspect; endGroup]
{{1, 2, 3}, {1, 3, 4}, {1, 4, 2}, {2, 4, 3}}
;[o]
{{1, 2, 3}, {1, 3, 4}, {1, 4, 2}, {2, 4, 3}}
:[font = text; inactive; dontPreserveAspect]
To render the volume,  we need to extend facelist by changing the indexes in explicit list of edges to pass to the appropriate primitive (here: Polygon). 
:[font = input; dontPreserveAspect; endGroup]
edgelist= Map[Function[verticelist[[#]]], facelist];

:[font = text; inactive; dontPreserveAspect]
We may now render the volume by wrapping the "Polygon" primitive around the edgelist: 

:[font = input; dontPreserveAspect; startGroup]
display[Map[Polygon,edgelist]]
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Applying Geometric Transformations
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We can now use this volume as a primitive object and transform it using homogeneous coordinate transformations for which we have already defined a set of scaling, translation and rotation matrices. In defining the transformed edgelists we will have to insert the conversion to and from homogeneous coordinates. 
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We will try some transformations:
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display[Map[Polygon,newEdgelist]]
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MathPictureEnd

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%!
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:[font = text; inactive; dontPreserveAspect]
We can similarly define a dodecahedron. The faces are pentagons:
:[font = input; dontPreserveAspect; startGroup]
facelist =  {{3, 1, 2, 5, 6}, {12, 11, 5, 2, 7}, {17, 13, 6, 5, 11}, 
      {14, 9, 3, 6, 13}, {10, 4, 1, 3, 9}, {8, 7, 2, 1, 4}, 
      {4, 10, 16, 15, 8}, {9, 14, 19, 16, 10}, 
      {13, 17, 20, 19, 14}, {11, 12, 18, 20, 17},
       {7, 8, 15, 18, 12}, {18, 15, 16, 19, 20}}

verticelist=
    {{0.1171133902139618, -1.000511576276664, -0.3621589273242113}, 
   {-0.2984036824225265, -0.5817988070943417, -0.847563231579627}, 
   {0.7957196239318588, -0.6599726122576239, -0.2777850442631916}, 
   {-0.2354424399154574, -0.995440477478119, 0.3155362956874772}, 
   {0.1233988775001608, 0.01751887980296279, -1.063185706833948}, 
   {0.7996042687105845, -0.03079518881785797, -0.7110434210240867}, 
   {-0.907763186347153, -0.3179489854175332, -0.4698643668832807}, 
   {-0.868850998503861, -0.5735935968779865, 0.2489706732667203}, 
   {0.862565511217654, -0.4444368592016375, 0.4520561062430151}, 
   {0.2252723078505214, -0.6517674020412752, 0.81874886058316}, 
   {-0.2252723078505258, 0.6517674020412736, -0.818748860583157}, 
   {-0.862565511217653, 0.4444368592016384, -0.4520561062430158}, 
   {0.868850998503857, 0.5735935968779876, -0.2489706732667215}, 
   {0.907763186347152, 0.317948985417531, 0.469864366883284}, 
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