Zonohedral Completion
of
Convex Polyhedra

by
Russell Towle
rustybel@foothill.net

Introduction

Zonohedra are polyhedra bounded entirely by centrally symmetrical polygons; such a polygon has an even number of sides, in opposite parallel pairs. The cube, truncated octahedron, and right regular hexagonal prism, are simple examples of zonohedra. The polygons bounding them are sometimes called zonogons, and their generalizations in higher space, zonotopes. For instance, a 4-dimensional zonotope is bounded by zonohedra. While is it usual to think of zonohedra as convex—in fact, it is common to restrict the definition of a zonohedron to include only convex solids—, there is an infinitude of non-convex, zonohedra-like solids. Some of these can be constructed using this notebook.

	Not only may zonogons and zonohedra be generalized to zonotopes, but zonotopes of whatever dimension are related, both in their constructions, and in a rich palette of combinatorial properties, to hypercubes. Some zonohedra are orthogonal shadows of hypercubes, as are some zonogons; the regular n-gons of an even number of sides are isometric shadows of n-cubes cast into a plane.
	
	Zonohedra are said to be determined by a star of n vectors, where n>2. Since all the faces of a zonohedron are bounded by edges in parallel equal pairs, fixing one's attention upon any one edge allows one to pick out a zone of faces which girdle the zonohedron. There are exactly n such zones, one for each direction in which edges occur. The magnitudes of the vectors in the star determine the lengths of the edges in the zonohedron. If the vectors in the star are of unit magnitude, then so also are all the edges of the determined zonohedron. If no three of the n vectors lie in a plane, then the zonohedron is bounded by n(n-1) parallelograms.
	
	The simplest zonotope which may constructed in an n-space is determined by n vectors which span the space, i.e., the n vectors have non-zero magnitudes and do not all lie in an (n-1)-space. These might be called “primitive” or “simplicial” zonotopes. The simplest zonogon is a parallelogram: when equilateral, it is a rhomb; when equilateral and equiangular, a square. Similarly, the simplest zonohedron is a parallelepiped: when equilateral, a rhombic hexahedron, and when all the dihedral angles are equal, the faces become squares, and the rhombic hexahedron is a cube. It is a curious fact that every convex zonohedron may be dissected into parallelepipeds, and that generally, every convex n-zonotope may be dissected into primitive n-zonotopes. For instance, the right regular hexagonal prism may be dissected into three right rhombic prisms.
	
	If a zonohedron's edges occur in n different directions, then it can be dissected into the “tetrahedral” number, of n things taken three at a time, parallelepipeds.
	
	However, if no three of the vectors in a star lie in the same plane, then the zonohedron is bounded entirely by parallelograms, and can be dissected into a tetrahedral number of non-degenerate parallelepipeds. When three or four or more, say, m vectors of the star, do lie in the same plane, the determined zonohedron is bounded in part by “higher” 2m-gons, such as hexagons, octagons, decagons, and so on. In such cases various parallelepipeds are degenerate and have zero volume.
	
	To introduce what follows, given two vectors, a and b, we “complete the parallelogram” by adding either a to b or by adding b to a; and by a similar process, given three vectors a, b, and c, not all in the same plane, we “complete the parallelepiped” by completing the parallelogram upon any two of the vectors, and then adding the third vector to each of the four points. Iterating upon this process we may refer to “completing the zonohedron” upon a set of vectors.
	
	It is worth noting that, just as one may dissect a zonohedron into parallelepipeds, so also one can construct a zonohedron from parallelepipeds. This notebook effects related constructions: for, the dissection (or construction) need not be confined to parallelepipeds. If a zonohedron is determined by, say, 10 vectors, then one may select any 9 of these, construct the determined 9-zonohedron, and then add a “rind” of parallelepipeds to this, to complete the 10-zonohedron. Or one might select two sets of 5 vectors, construct the two determined 5-zonohedra, and begin adding parallelepipeds. If the correct choices were made, one would close up the convex 10-zonohedron.
	
	A more structured approach is obtained by using a convex polyhedron centered upon the origin, as a point of beginning. The vectors from the center to the vertices form a star, and the star determines some particular zonohedron. For instance, the eight vertices of a cube form a star of eight vectors, in four directions, discounting opposites. The determined zonohedron is Kepler's rhombic dodecahedron. In our construction, however, we ignore the star and fix our attention upon the six square faces of the cube. Each such face determines a sub-star of four vectors to its vertices, and we “complete” the zonohedron upon each of the six 4-stars, obtaining six equal rhombic dodecahedra radiating from a single point. This is the first “stage” or “step” in the zonohedral completion of a cube. Here, then, is a first look at the zonohedral completion function, ZcomA:

a=Polyhedron[Cube];
ZcomA[a,1]

Note that the six dodecahedra fit together in such a way, that 24 of the 72 faces are hidden in the interior, where they meet in twelve coincident pairs. The ZcomA function discards all such interior, coincident faces, leaving a set of polygons all meeting edge-to-edge, i.e., a polyhedron. In this case the polyhedron is a non-convex zonohedron, and we discern a number of “cups” or “hollows” where three rhombs meet. The second and final step in the zonohedral completion of a cube, is to fill all these cups.

a=Polyhedron[Cube];
ZcomA[a,2]

The resulting zonohedron is Kepler's rhombic dodecahedron, with each of its twelve rhombs divided into four smaller rhombs. This is usually the case, when the ZcomA function is applied to a symmetrical convex polyhedron.

	Here the construction (and dissection) does not incorporate only the four rhombic hexahedra which compose a rhombic dodecahedron. In the two steps required, we first create six dodecahedra, each of which dissects into four hexahedra; then we add eight more hexahedra, for a grand total of 32 hexahedra. This is in agreement with volumetric considerations: since a zonohedron may be dissected into parallelepipeds, and since the volumes of similar figures are as the cubes on corresponding dimensions, then, adopting the edge length of the hexahedra as our unit of measurement, we have 4 (2^3) or 32 hexahedra altogether.
	
	Thus the zonohedral completion of convex polyhedra usually gives a zonohedron which may be dissected into n-things-taken-3-at-a-time * (2^3) parallelepipeds (some of which may be degenerate, with zero volume).
	
	The ZcomA function performs somewhat as follows: given an input convex polyhedron, a zonohedron is completed upon each face. Then the main loop is entered, and coincident faces discarded. All vertices are found, and those of least magnitude selected. The edges meeting at these “deep” vertices are transformed into vectors, and zonohedra rooted upon the deep vertices are then constructed using these vectors. These faces are joined with the previously existing faces. The next iteration follows. The syntax is ZcomA[  input polyhedron, number of iterations ]. I know of no way to predict the number of iterations necessary. In even slightly complicated cases it runs into the hundreds.
	
	The function fails for many inputs, usually at later stages. For instance, it may happen that the final stage of zonohedral completion is a zonohedron bounded by higher 2m-gons. The center of such 2m-gons are often closer to the origin than the bottoms of the cups still to be filled, so ZcomA tries to create a zonohedron upon an already finished part of the final figure. Or, collinear vectors may meet at a deep vertex. Sometimes ZcomA blithely forges ahead and creates a degenerate zonohedron which fills the cup anyway (see, for example, the Bipyramid examples). At other times the cups go unfilled. Finally, certain cups should be filled by zonohedra bounded in part by higher 2m-gons, but the Z function called by ZcomA can only create zonohedra bounded by parallelograms. I would greatly welcome any help in forging a more robust zonohedral completion function.—Russell Towle, June, 1999.
	
	Note: a ZcomB functon has been added, in which all stages of completion are saved in the list named trip3. An example of DXF export of the stages is given.—R.T., March 8, 2000.

Code

Off[General::spell]
Off[General::spell1]

Helper Functions

vectors[n_Integer,pitch_,fac_Integer,ratio_]:=
Block[{e},
Table[If[Mod[i,fac]==0,e=ratio,e=1];
	N[{e Cos[Degree pitch] Cos[2Pi i/n],
	e Cos[Degree pitch] Sin[2Pi i/n],
	-e Sin[Degree pitch]}],
	{i,n}]]
	
vectors[n_:12,pitch_:35.2643,fac_:1,ratio_:1]:=
vectors[n, pitch, fac, ratio]

quad[x_,ref_]:={ref, ref+x[[1]], ref+x[[1]]+x[[2]], mark=ref+x[[2]]}

PolarZonohedron[n_Integer,pitch_,fac_Integer,ratio_]:=
Block[{v=vectors[n,pitch,fac,ratio], q},
h=Plus@@v/2;
	q=Table[mark=-h;
		Table[quad[{v[[j]],v[[1+Mod[i+j-1,n]]]},mark],{i,n-1}],{j,n}];
	Graphics3D[ Polygon /@ Flatten[q ,1] ]
]

PolarZonohedron[n_:12, pitch_:35.2643, fac_:1,ratio_:1] := 
PolarZonohedron[n, pitch, fac, ratio]

trans[shape_, vec_List] :=
    Block[{tempvec = N[vec]},
	shape /. { poly:Polygon[_] :> Map[(tempvec + #)&, poly, {2}]}
    ] /; Length[vec] == 3

dePair[poly_]:=
Block[{a},
a=Sort[((Chop[Rationalize[(cent@#),10^-3]])@#&/@ (poly)),
OrderedQ[{Head@#1,Head@#2}]&];
Nest[
	(
	(DeleteCases[#,Null,{1}]&) @ (Flatten[#,1]&) @
	((If[SameQ@@(Head/@#),Null,#]&) /@
		Partition[Join[RotateRight[#],{Null,Null,Null}],2])
	)&,a,2]/.({___}[li_List]:>li)
]

mag[v_] := Sqrt[v.v]
bag[v_] := Plus@@(v^2)(*mock magnitude*)
tag[v_] := Round[Plus@@v](*mock face center*)

Z[v_List,spot_]:=
Block[{n=Length[v],fin,s,a,b,c,gon},
fin=hemi[n];(*make a list of indices for half the faces*)
Flatten[Table[(*now create all the faces in a Table*)
(*1. create a single face*)
gon=quad[{v[[ fin[[i,1]] ]],v[[ fin[[i,2]] ]]},spot];
(*2. get surface normal*)
s=surf[ gon ];
(*find the vectors not involved in the face*)
a=Complement[v, v[[ fin[[i]] ]] ];
(*sort them according to the dot product with surface normal*)
b=Select[a,Dot[s,#]<0&];
c=Select[a,Dot[s,#]>0&];
(*now add the sets of vectors to the face, creating two faces*)
{Map[(Plus @@ b)+#&,gon],Map[(Plus @@ c)+#&,gon]},
{i,Length[fin]}],1]
]

hemi[n_Integer]:=
	Block[{i,j},Flatten[Table[{i,j},{i,n-1},{j,i+1,n}],1]]



mycross[ {ax_, ay_, az_}, {bx_, by_, bz_} ] :=
	{ay bz - az by, az bx - ax bz, ax by - ay bx}

norm[f_List] := unit[mycross[ f[[2]] - f[[1]], f[[3]] - f[[2]] ]]

surf[f_List] := mycross[ f[[2]] - f[[1]], f[[3]] - f[[2]] ]

memq[p_,v_]:=Table[same[p[[i]], v ], {i,Length[p]}]

same[v1_,v2_]:= Chop[v1-v2]=={0,0,0}

mag::usage = "mag[v_] returns the norm or magnitude of
the n-vector 'v'.";
mag[v_]:= N[Sqrt[v.v]]

unit::usage = "unit[v_] returns the unit vector in
the direction of the n-vector 'v'.";
unit[v_]:=v/mag[v]

cent::usage = "cent[v_] returns the centroid of the list
of n-vectors 'v'.";
cent[v_]:=Plus@@v/Length[v]

vunion::usage = "vunion[v] returns the union of a list of n-vectors.";

vunion[v_]:=
Union[Chop[v],
SameTest->((Sign[#1]==Sign[#2]) && (Chop[#1-#2]==Table[0,{Length[v[[1]]]}]) &)]

<<Graphics`Polyhedra`

Completion Function A

ZcomA[poly_,iterator_Integer]:=
Block[{zaq,vertmin,deepverts,deepfaces,newv,trip2,trip1},

zaq=N[Cases[poly,Polygon[x_]:>x,Infinity]];
Print["there are ",Length[zaq]," zonohedra added"];
Print["they are of types ",Union[Map[Length[#]&,zaq]] ];

trip1=Flatten[
	Table[Z[ zaq[[i]], {0,0,0} ],{i,Length[zaq]}],1];

Print["the total number of faces is ",Dimensions[trip1][[1]]];

(*"File Header">>testfile;*)
	
Do[
trip1=dePair[trip1];

(*{trip1}>>>testfile;*)

Print["the number of exterior faces is ",Length[trip1]];
Print[" "];

Show[Graphics3D[Polygon /@ trip1 ],
	Boxed->False];
	
If[k==iterator,Break[]];

vertmin=Min[Map[mag[#]&,Level[trip1,{-2}]]];
deepverts=vunion[Select[Level[trip1,{-2}],Abs[mag[#]-vertmin]<.001&]];
(*Print["Length deepverts ",Length[deepverts]];*)

deepfaces=
	Table[
		Select[trip1, MemberQ[memq[#, deepverts[[i]] ],True]&],
	{i,Length[deepverts]}];
(*Print["Length deepfaces ",Length[Flatten[deepfaces,1]]];*)

i=1;newv={};
While[i<=Length[deepverts],
q=deepfaces[[i]];
AppendTo[newv,
  vunion[Flatten[
	Table[{q[[j,2]]-q[[j,1]],q[[j,4]]-q[[j,1]]},
		{j,Length[q]}],1]]
   ];
i++
];

Print["there are ",Length[newv]," zonohedra added"];
Print["they are of types ",Union[Map[Length[#]&,newv]] ];

trip2=Flatten[Table[Z[ newv[[i]], deepverts[[i]] ],{i,Length[deepverts]}],1];
	
trip1=Join[trip1,trip2];(*combine the previous stages*)
Print["the total number of faces is ",Dimensions[trip1][[1]]],(*how many rhombs now?*)

{k,iterator}];
]

Completion Function B: export files

ZcomB[poly_,iterator_Integer]:=
Block[{zaq,vertmin,deepverts,deepfaces,newv,trip2,trip1},

zaq=N[Cases[poly,Polygon[x_]:>x,Infinity]];
Print["there are ",Length[zaq]," zonohedra added"];
Print["they are of types ",Union[Map[Length[#]&,zaq]] ];

trip2=
	Table[Z[ zaq[[i]], {0,0,0} ],{i,Length[zaq]}];

trip1=Flatten[trip2,1];

Print["the total number of faces is ",Dimensions[trip1][[1]]];

trip3={};


	
Do[

AppendTo[trip3,trip2];

trip1=dePair[trip1];

(*{trip1}>>>testfile;*)

Print["the number of exterior faces is ",Length[trip1]];
Print[" "];

Show[Graphics3D[Polygon /@ trip1 ],
	Boxed->False];
	
If[k==iterator,Break[]];

vertmin=Min[Map[mag[#]&,Level[trip1,{-2}]]];
deepverts=vunion[Select[Level[trip1,{-2}],Abs[mag[#]-vertmin]<.001&]];
(*Print["Length deepverts ",Length[deepverts]];*)

deepfaces=
	Table[
		Select[trip1, MemberQ[memq[#, deepverts[[i]] ],True]&],
	{i,Length[deepverts]}];
(*Print["Length deepfaces ",Length[Flatten[deepfaces,1]]];*)

i=1;newv={};
While[i<=Length[deepverts],
q=deepfaces[[i]];
AppendTo[newv,
  vunion[Flatten[
	Table[{q[[j,2]]-q[[j,1]],q[[j,4]]-q[[j,1]]},
		{j,Length[q]}],1]]
   ];
i++
];

Print["there are ",Length[newv]," zonohedra added"];
Print["they are of types ",Union[Map[Length[#]&,newv]] ];

trip2=Table[Z[ newv[[i]], deepverts[[i]] ],{i,Length[deepverts]}];
	
trip1=Join[trip1,Flatten[trip2,1]];(*combine the previous stages*)
Print["the total number of faces is ",Dimensions[trip1][[1]]],(*how many rhombs now?*)

{k,iterator}];
]

Pyramids

Pyramid[n_Integer,pitch_] :=
Block[{c=Cos[Degree pitch],d=Sin[Degree pitch]},
	Graphics3D[{N[Table[
	Polygon[{{c Cos[2Pi i/n], c Sin[2Pi i/n], 0},
			 {c Cos[2Pi (i+1)/n], c Sin[2Pi (i+1)/n], 0},
			 {0, 0, d}}], {i, n}]],
	Polygon[Table[N[{c Cos[2Pi i/n], c Sin[2Pi i/n], 0}],
	{i, n, 1, -1}]]}]]/; n > 2
Pyramid[n_:12, pitch_:45] := Pyramid[n, pitch]

Bipyramids

Bipyramid[n_Integer,pitch_] :=
  Graphics3D[Flatten[
	N[Table[
		{Polygon[
		{{Cos[Degree pitch] Cos[2Pi i/n],
		Cos[Degree pitch] Sin[2Pi i/n], 0},
		{Cos[Degree pitch] Cos[2Pi (i+1)/n],
		Cos[Degree pitch] Sin[2Pi (i+1)/n], 0},
		{0, 0, Sin[Degree pitch]}}],
		Polygon[
		{{0, 0, -Sin[Degree pitch]},
		{Cos[Degree pitch] Cos[2Pi (i+1)/n],
		Cos[Degree pitch] Sin[2Pi (i+1)/n], 0},
		{Cos[Degree pitch] Cos[2Pi i/n],
		Cos[Degree pitch] Sin[2Pi i/n], 0}}]},
	{i, n}]],1]] /; n > 2
Bipyramid[n_:12, pitch_:45] := Bipyramid[n, pitch]

Prisms

Prism[n_Integer, pitch_] :=
Block[{c,d},
c=Cos[Degree*pitch];
d=Sin[Degree*pitch];
	Graphics3D[{N[Table[(*lateral faces*)
	Polygon[{{c Cos[2Pi i/n], c Sin[2Pi i/n], -d},
			 {c Cos[2Pi (i+1)/n], c Sin[2Pi (i+1)/n], -d},
			 {c Cos[2Pi (i+1)/n], c Sin[2Pi (i+1)/n], d},
			 {c Cos[2Pi i/n], c Sin[2Pi i/n], d}}],
		{i, n}]],
	Polygon[Table[N[{c Cos[2Pi i/n],c Sin[2Pi i/n],-d}],
	{i, n}]],(*lower base*)
	Polygon[Table[N[{c Cos[2Pi i/n],c Sin[2Pi i/n],d}],
	{i, n, 1, -1}](*upper base*)
	]}
	](*end Graphics3D*)
	](*end Block*)
Prism[n_:12, pitch_:35.2643] := Prism[n, pitch]

Antiprisms

Antiprism[n_Integer,pitch_] :=
Block[{c,d},
c=Cos[Degree*pitch];
d=Sin[Degree*pitch];
	Graphics3D[
	Flatten[N[{Flatten[Table[(*lateral faces*)
	{Polygon[{{c Cos[2*Pi*i/n], c Sin[2*Pi*i/n], -d},
			 {c Cos[2*Pi*(i+1)/n], c Sin[2*Pi*(i+1)/n], -d},
			 {c Cos[(Pi/n)+(2*Pi*i/n)], c Sin[(Pi/n)+(2*Pi*i/n)], d}}],
	Polygon[{{c Cos[(Pi/n)+(2*Pi*(i+1)/n)], c Sin[(Pi/n)+(2*Pi*(i+1)/n)], d},
			 {c Cos[(Pi/n)+(2*Pi*i/n)], c Sin[(Pi/n)+(2*Pi*i/n)], d},
			 {c Cos[(2*Pi*(i+1)/n)], c Sin[(2*Pi*(i+1)/n)], -d}}]},
				{i, n}],2],
	Polygon[Table[{c Cos[(Pi/n)+(2*Pi*i/n)],c Sin[(Pi/n)+(2*Pi*i/n)],d},
	{i, n}]],(*upper base*)
	Polygon[Table[{c Cos[2*Pi*i/n],c Sin[2*Pi*i/n],-d},
	{i, n, 1, -1}]](*lower base*)
	}],1]
	](*end Graphics 3D*)
	](*end Block*)
Antiprism[n_:12, pitch_:35.2643] := Antiprism[n, pitch]

Show[{Antiprism[5],Graphics3D[Line /@ Map[{{0,0,0},#}&, 1.2*-vectors[5] ]]},
Axes->True];

Archimedean Polygon Lists

arch={{Polygon[{{0, 0, 1.1055416}, {0.85280287, 0, 0.70352647}, \
{0.99493668, 0.47140452, -0.10050378}, {0.28426762, 0.94280904, \
-0.50251891}, {-0.56853524, 0.94280904, -0.10050378}, {-0.71066905, \
0.47140452, 0.70352647}}], Polygon[{{0, 0, 1.1055416}, {-0.71066905, \
0.47140452, 0.70352647}, {-1.0896926, 0.15713484, -0.10050378}, \
{-0.75804699, -0.62853936, -0.50251891}, {-0.047377937, -1.0999439, \
-0.10050378}, {0.33164556, -0.78567420, 0.70352647}}], Polygon[{{0, 0, \
1.1055416}, {0.33164556, -0.78567420, 0.70352647}, {0.85280287, 0, \
0.70352647}}], Polygon[{{0.85280287, 0, 0.70352647}, {0.33164556, \
-0.78567420, 0.70352647}, {-0.047377937, -1.0999439, -0.10050378}, \
{0.094755874, -0.62853936, -0.90453403}, {0.61591318, 0.15713484, \
-0.90453403}, {0.99493668, 0.47140452, -0.10050378}}], \
Polygon[{{-0.71066905, 0.47140452, 0.70352647}, {-0.56853524, 0.94280904, \
-0.10050378}, {-1.0896926, 0.15713484, -0.10050378}}], \
Polygon[{{0.99493668, 0.47140452, -0.10050378}, {0.61591318, 0.15713484, \
-0.90453403}, {0.28426762, 0.94280904, -0.50251891}}], \
Polygon[{{-1.0896926, 0.15713484, -0.10050378}, {-0.56853524, 0.94280904, \
-0.10050378}, {0.28426762, 0.94280904, -0.50251891}, {0.61591318, \
0.15713484, -0.90453403}, {0.094755874, -0.62853936, -0.90453403}, \
{-0.75804699, -0.62853936, -0.50251891}}], Polygon[{{-0.047377937, \
-1.0999439, -0.10050378}, {-0.75804699, -0.62853936, -0.50251891}, \
{0.094755874, -0.62853936, -0.90453403}}]}, {Polygon[{{0, 0, 1.1547005}, \
{1.0000000, 0, 0.57735027}, {0.33333333, 0.94280904, 0.57735027}}], \
Polygon[{{0, 0, 1.1547005}, {0.33333333, 0.94280904, 0.57735027}, \
{-0.66666667, 0.94280904, 0}, {-1.0000000, 0, 0.57735027}}], Polygon[{{0, \
0, 1.1547005}, {-1.0000000, 0, 0.57735027}, {-0.33333333, -0.94280904, \
0.57735027}}], Polygon[{{0, 0, 1.1547005}, {-0.33333333, -0.94280904, \
0.57735027}, {0.66666667, -0.94280904, 0}, {1.0000000, 0, 0.57735027}}], \
Polygon[{{1.0000000, 0, 0.57735027}, {0.66666667, -0.94280904, 0}, \
{1.0000000, 0, -0.57735027}}], Polygon[{{1.0000000, 0, 0.57735027}, \
{1.0000000, 0, -0.57735027}, {0.33333333, 0.94280904, -0.57735027}, \
{0.33333333, 0.94280904, 0.57735027}}], Polygon[{{0.33333333, 0.94280904, \
0.57735027}, {0.33333333, 0.94280904, -0.57735027}, {-0.66666667, \
0.94280904, 0}}], Polygon[{{-1.0000000, 0, 0.57735027}, {-0.66666667, \
0.94280904, 0}, {-1.0000000, 0, -0.57735027}}], Polygon[{{-1.0000000, 0, \
0.57735027}, {-1.0000000, 0, -0.57735027}, {-0.33333333, -0.94280904, \
-0.57735027}, {-0.33333333, -0.94280904, 0.57735027}}], \
Polygon[{{-0.33333333, -0.94280904, 0.57735027}, {-0.33333333, \
-0.94280904, -0.57735027}, {0.66666667, -0.94280904, 0}}], \
Polygon[{{1.0000000, 0, -0.57735027}, {0.66666667, -0.94280904, 0}, \
{-0.33333333, -0.94280904, -0.57735027}, {0, 0, -1.1547005}}], \
Polygon[{{1.0000000, 0, -0.57735027}, {0, 0, -1.1547005}, {0.33333333, \
0.94280904, -0.57735027}}], Polygon[{{-0.66666667, 0.94280904, 0}, \
{0.33333333, 0.94280904, -0.57735027}, {0, 0, -1.1547005}, {-1.0000000, 0, \
-0.57735027}}], Polygon[{{-1.0000000, 0, -0.57735027}, {0, 0, -1.1547005}, \
{-0.33333333, -0.94280904, -0.57735027}}]}, {Polygon[{{0, 0, 1.0540926}, \
{0.63245553, 0, 0.84327404}, {0.84327404, 0.47140452, 0.42163702}, \
{0.42163702, 0.94280904, 0.21081851}, {-0.21081851, 0.94280904, \
0.42163702}, {-0.42163702, 0.47140452, 0.84327404}}], Polygon[{{0, 0, \
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Examples

Platonic Solids

Tetrahedron

Unlike the rest of the Platonic solids (and all convex zonohedra), the tetrahedron, a simplex, does not have central symmetry; vertices and faces do not occur in diametrically opposed pairs. Hence the determined zonohedron (Kepler's rhombic dodecahedron) has undivided faces. Since the tetrahedron is a simplex, only one step is required in its zonohedral completion. From some positions, the rhombic dodecahedron looks like a cube, so a second view is provided.

a=Polyhedron[Tetrahedron];
ZcomA[a,1]

Octahedron

The eight triangular faces of the octahedron give eight cubes emanating from a single point, fitting together to compose a larger cube of twice the edge length. Only one step is required.

a=Polyhedron[Octahedron][[1]];
ZcomA[a,1]

Cube

The six square faces of the cube give six rhombic dodecahedra emanating from a single point. Eight “cups” arise between the dodecahedra; the second step fills these with rhombic hexahedra. Kepler's rhombic dodecahedron results, each of its twelve faces divided into four smaller faces.

a=Polyhedron[Cube];
ZcomA[a,2]

Icosahedron

The twenty triangles of the icosahedron give twenty Golden rhombic hexahedra emanating from a single point. These give twelve cups where five rhombs meet; twelve n=5 polar zonohedra are added in the second step. 20 more cups are filled with Golden rhombic hexahedra in the final step, giving Kepler's rhombic triacontahedron. Each face of the triacontahedron (a Golden rhomb) is divided into four smaller rhombs.

a=Polyhedron[Icosahedron];
ZcomA[a,3]

Dodecahedron

The twelve pentagons of the dodecahedron give twelve n=5 polar zonohedra emanating from a single point. Seven stages are required to effect the zonohedral completion, giving a rhombic enneacontahedron, bounded by 90 rhombs of two types. The relationship between the two Platonic solids with icosahedral symmetry, the icosahedron and dodecahedron, with the cube and octahedron, is dimly reflected in the enneacontahedron, wherein the larger of the tow rhombs is identical to the rhombs bounding Kepler's rhombic dodecahedron.

a=Polyhedron[Dodecahedron];
ZcomA[a,7]

Polar Zonohedra

PolarZonohedron[ 4 ]

The twelve faces of Kepler's rhombic dodecahedron give twelve 4-zonohedra emanating from a single point. Six steps are required to effect completion, giving a truncated rhombic dodecahedron.

a=PolarZonohedron[4];(*gives truncated rhombic dodecahedron*)
ZcomA[a,6]

PolarZonohedron[ 5 ]

The twenty faces of a rhombic icosahedron give twenty 4-zonohedra emanating from a single point. The algorithm fails beyond the 16th stage.

a=PolarZonohedron[5];(*fails beyond k=16*)
ZcomA[a,16]

Prisms

Prism[ 6 ]

The six lateral faces of a right hexagonal prism give six 4-zonohedra; the two hexagonal faces give two n=6 polar zonohedra; these eight zonohedra emanate from a single point, and in four further steps, the completion of the prism gives an n=6 polar zonohedron.

a=Prism[6];(*gives n=6 polar zonohedron*)
ZcomA[a,5]

Prism[ 5 ]

The five lateral faces of a right pentagonal prism give five 4-zonohedra; the two pentagonal faces give two n=5 polar zonohedra; these seven zonohedra emanate from a single point, and in three further steps, the completion of the prism gives an n=10 polar zonohedron. This is because the vertices of the pentagonal prism do not occur in antipodal pairs; hence the determined star has ten vectors.

a=Prism[5];(*gives n=10 polar zonohedron*)
ZcomA[a,4]

Prism[ 12 ]

The twelve lateral faces of a right dodecagonal prism give twelve 4-zonohedra; the two dodecagonal faces give two n=12 polar zonohedra; these fourteen zonohedra emanate from a single point, and in an unknown number of further steps, the completion of the prism gives an n=12 polar zonohedron. The algorithm fails beyond step 15.

a=Prism[12];(*gives n=12 polar zonohedron*)
ZcomA[a,15]

Antiprisms

Antiprism[ 7 ]

The fourteen lateral faces of a right heptagonal antiprism give fourteen rhombic hexahedra; the two heptagonal faces give two n=7 polar zonohedra; these sixteen zonohedra emanate from a single point, and in five further steps, the completion of the antiprism gives an n=7 polar zonohedron.

a=Antiprism[7];(*gives n=7 polar zonohedron*)
ZcomA[a,6]

Antiprism[ 6 ]

The twelve lateral faces of a right hexagonal antiprism give twelve rhombic hexahedra; the two hexagonal faces give two n=6 polar zonohedra; these fourteen zonohedra emanate from a single point, and in five further steps, the completion of the antiprism gives an n=12 polar zonohedron. Here again, since the vertices of a right regular n-antiprism, when n is even, do not occur in antipodal pairs, the determined star has 2n vectors.

a=Antiprism[6];(*gives n=12 polar zonohedron*)
ZcomA[a,6]

Bipyramids

Bipyramid[ 5 ]

The ten triangular faces of a right pentagonal bipyramid give ten rhombic hexahedra which emanate from a single point, and in one further step, the completion of the bipyramid gives a decagonal prism.

a=Bipyramid[5,55];(*gives a decagonal prism*)
ZcomA[a,2]

Bipyramid[ 6 ]

The twelve triangular faces of a right hexagonal bipyramid give twelve rhombic hexahedra which emanate from a single point, and in one further step, the completion of the bipyramid gives a hexagonal prism.

a=Bipyramid[6,55];(*gives a hexagonal prism*)
ZcomA[a,2]

Archimedean Solids

The arch[[ i ]] and (number of steps)

1    Truncated Tetrahedron (6)
2    Cuboctahedron (4)
3    Truncated Octahedron (fails beyond 13)
4    Truncated Cube (fails beyond 19)
5    Rhombicuboctahedron (18)
6    Truncated Cuboctahedron (>50)
7    Snub Cube (fails beyond 22)
8    Icosidodecahedron (fails beyond 19)
9    Truncated Icosahedron (fails at 112)
10   Truncated Dodecahedron (>50?)
11   Rhombicosidodecahedron (>50)
12   Truncated Icosidodecahedron (>50)
13   Snub Dodecahedron (>50)

a=Graphics3D[ arch[[5]] ];
Show[a,
Boxed->False];

ZcomA[a,18]

Export Stages to DXF

a=Polyhedron[Icosahedron];
ZcomB[a,3]

(*how many stages*)
Length[trip3]

(*were the stages truly recorded? a test*)
Show[Graphics3D[ Map[Polygon[#]&, trip3[[1]],{2}] ] ];

(*read in DXF export function*)
<<Utilities`DXF`

(*where to write the files?*)
SetDirectory["Macintosh HD:Desktop Folder:"]

(*write them all as icosa1, icosa2, etc.*)
Do[
z=Graphics3D[ Map[Polygon[#]&, trip3[[i]],{2}] ];
WriteDXF[ StringJoin["icosa",ToString[i]], z ],
{i,Length[trip3]}]