(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you 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, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Zonohedra and Zonotopes :[font = text; inactive; preserveAspect; center] David Eppstein* Dept. of Information & Computer Science U.C. Irvine, CA, 92717 http://www.ics.uci.edu/~eppstein/ ;[s] 6:0,1;14,0;15,1;16,0;17,2;80,0;114,-1; 3:3,14,10,Times,0,12,0,0,0;2,14,10,Times,1,12,0,0,0;1,14,10,Times,2,12,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect; endGroup] A zonotope is a set of points in d-dimensional space constructed from vectors v[[i]] by taking the sum of a[[i]] v[[i]] where each a[[i]] is a scalar between 0 and 1. Different choices of scalars give different points, and the zonotope is the set of all such points. Alternately it can be viewed as a Minkowski sum of line segments connecting the origin to the endpoint of each vector. It is called a zonotope because the faces parallel to each vector form a so-called zone wrapping around the polytope. A zonohedron is just a three-dimensional zonotope. Zonohedra include many familiar polyhedra including cubes, truncated octahedra, and rhombic dodecahedra. Towle [T96] recently described Mathematica code for polar zonohedra, generated by vectors evenly spaced around a circular cone. This notebook contains code for constructing zonotopes and displaying zonohedra without such restrictions on the generators. There is some confusion in the definition of zonotopes; Wells [W91] requires the generating vectors to be in general position (all d-tuples of vectors must span the whole space), so that all the faces of the zonotope are parallelotopes. Others [BEG95,Z95] do not make this restriction. Coxeter [C73] starts with one definition but soon switches to the other. We use the unrestricted definition here. The combinatorics of the faces of a zonotope are equivalent to those of an arrangement of hyperplanes in a space of one fewer dimension, so for instance zonohedra correspond to planar line arrangements. This can be most easily seen by considering the space of d-dimensional hyperplanes tangent to the zonotope. The space of all d-dimensional unit vectors can be seen as a unit sphere, equivalent to oriented projective (d-1)-space. For any given unit vector, there is a unique hyperplane normal to the vector and tangent to the zonotope at some kind of face. One can swing a hyperplane tangent to a k-dimensional face through a collection of angles with (d-1-k) degrees of freedom; for instance, tangent planes to a facet of a polyhedron must lie on the facet itself (no degrees of freedom), tangent planes to an edge have one degree of freedom, and tangent planes to a vertex have two degrees of freedom. Thus if we divide up the the tangent space according to which zonotope face each tangent touches, we get a decomposition of the tangent space into cells of varying dimensions. The tangent plane has fewer degrees of freedom when it swings around high dimensional faces, so high dimensional faces of the zonotope correspond to low dimensional cells in the tangent space decomposition. For the zonotope corresponding to a single vector (a line segment), this decomposition of the tangent space would be given by a single hyperplane corresponding to the tangents parallel to the segment, and partitioning the rest of the tangents into two subsets, one for each endpoint of the segment. But the decomposition corresponding to a Minkowski sum is formed by overlaying the decompositions corresponding to the two summands, so the cell decomposition of a zonotope's tangent space is formed by overlaying a collection of hyperplanes. Each zone of the zonotope corresponds to a hyperplane in this arrangement. Because of this correspondence, we can construct zonotopes in the same amount of time as it takes to construct hyperplane arrangements in the tangent space, O(n^(d-1)). The arrangement tells us how the zonotope's faces connect to each other, and the actual vertex coordinates are not hard to find within the same time bound. In particular it would take O(n^2) time to construct a zonohedron from an initial set of n vectors. However our implementation uses a slower algorithm better suited to the functional nature of Mathematica programming. We first find the subsets of vectors corresponding to the faces of the zonotope. These subsets are determined incrementally, by combining each new vector with previously generated subsets, and then forming additional faces pairing the new vector with any remaining old vectors. Once we have generated all faces, we determine coordinates for their vertices (recursively, since each face is again a zonotope in one lower dimension) and lift them into position by adding appropriate subsets of the vectors. ;[s] 17:0,0;2,1;10,0;303,1;316,0;473,1;477,0;511,1;521,0;697,1;708,0;718,1;733,0;1029,1;1045,0;3759,1;3770,0;4290,-1; 2:9,14,10,Times,0,12,0,0,0;8,14,10,Times,2,12,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Implementation :[font = subsection; inactive; preserveAspect; startGroup] Coplanarity and Orientation Test :[font = text; inactive; preserveAspect] In the face generation stage, we represent a face by the indices of the vectors generating it. When we add a new vector to the zonohedron, we test it against each face and determine whether it is coplanar by computing the determinant of a matrix formed by the new vector and d-1 of the vectors already in the face; it is coplanar exactly when this determinant is zero, and if so it gets added to the list of indices for that face. In the face placement stage, we instead represent a face by the coordinates of its vertices, so we need code to translate between the two representations. Once we have found the coordinates of a face recursively as a lower-dimensional zonotope, we lift two copies of it in place by using a similar determinant computation to determine which vectors to add to each copy. :[font = text; inactive; preserveAspect] In order to compute lower-dimensional faces recursively, we need a way of testing signs of determinants when there are fewer than d vectors involved. We do this by including enough extra vectors to make up a d*d matrix. The extra vectors are chosen on the moment curve (x,x^2,x^3,...), so that they are independent of each other and (hopefully) of the inputs. ;[s] 3:0,0;258,1;271,0;362,-1; 2:2,14,10,Times,0,12,0,0,0;1,14,10,Times,2,12,0,0,0; :[font = input; initialization; preserveAspect; endGroup] *) ZSignTest[vv_,f_] := Det [ Part[vv,f] ~Join~ Table[(i+200)^j, {i,Length[vv[[1]]]-Length[f]}, {j,Length[vv[[1]]]}]] (* :[font = subsection; inactive; preserveAspect; startGroup] Face Generation Stage :[font = text; inactive; preserveAspect] As described earlier, we generate faces (as subsets of vectors) incrementally, adding one vector at a time to our zonotope. Adding one vector has two parts: including it in the faces with which it is coplanar, and then making new faces not coplanar to any existing faces. :[font = input; initialization; preserveAspect] *) ZOldFaces[vv_,ff_,i_,d_] := (ZAddToFace[vv, #, i, d])& /@ ff (* :[font = text; inactive; preserveAspect] This routine tests whether to add a new vector to an existing face. :[font = input; initialization; preserveAspect] *) ZAddToFace[vv_,f_,i_,d_] := If[ Length[f] < (d-1) || ZSignTest[vv, Append[Take[f,d-1],i]] == 0, Append[f,i], f ] (* :[font = text; inactive; preserveAspect] To generate the new faces, we make all (d-1)-tuples of indices that have index i as their largest element, and that are not subsets of the indices in existing faces. If d is 2, this set of tuples is either the singleton {i} or nothing at all, depending on whether i is in any face. Otherwise, we combine tuples generated recursively for all j{12,3,5}, Boxed -> False] (* :[font = section; inactive; preserveAspect; startGroup] Examples :[font = subsection; inactive; preserveAspect; startGroup] The Cube :[font = input; preserveAspect; startGroup] Zonohedron[{{1,0,0},{0,1,0},{0,0,1}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 257; pictureHeight = 286; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.1167 L 0 1.1167 L closepath clip newpath p P p .002 w .951 1.00749 m .25291 1.06546 L .25949 .40916 L p .693 .549 .686 r F P s P p .002 w .25949 .40916 m .93968 .34252 L .951 1.00749 L p .693 .549 .686 r F P s P p .002 w .06375 .12965 m .25949 .40916 L .25291 1.06546 L p .871 .697 .663 r F P s P p .002 w .78558 .05401 m .93968 .34252 L .25949 .40916 L p .697 .666 .815 r F P s P p .002 w .25949 .40916 m .06375 .12965 L .78558 .05401 L p .697 .666 .815 r F P s P p .002 w .78558 .05401 m .93968 .34252 L .951 1.00749 L p .913 .726 .636 r F P s P p .002 w .25291 1.06546 m .05128 .82196 L .06375 .12965 L p .871 .697 .663 r F P s P p .002 w .79331 .75593 m .951 1.00749 L .25291 1.06546 L p .693 .682 .835 r F P s P p .002 w .25291 1.06546 m .05128 .82196 L .79331 .75593 L p .693 .682 .835 r F P s P p .002 w .951 1.00749 m .79331 .75593 L .78558 .05401 L p .913 .726 .636 r F P s P p .002 w .79331 .75593 m .05128 .82196 L .06375 .12965 L p .692 .547 .684 r F P s P p .002 w .06375 .12965 m .78558 .05401 L .79331 .75593 L p .692 .547 .684 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] Prisms :[font = input; preserveAspect; startGroup] Zonohedron[{{1,Sqrt[3],0},{1,-Sqrt[3],0},{2,0,0},{0,0,2}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 236; endGroup] %! 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%%Creator: Mathematica %%AspectRatio: .7031 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations 0.0243854 0.975418 -0.124275 0.975418 [ [ 0 0 0 0 ] [ 1 .7031 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 .7031 L 0 .7031 L closepath clip newpath p P p .002 w .19436 .59196 m .19779 .31192 L .4553 .36957 L p .739 .587 .686 r F P s P p .002 w .4553 .36957 m .45469 .64639 L .19436 .59196 L p .739 .587 .686 r F P s P p .002 w .7458 .62092 m .45469 .64639 L .4553 .36957 L p .693 .549 .686 r F P s P p .002 w .4553 .36957 m .7433 .34259 L .7458 .62092 L p .693 .549 .686 r F P s P p .002 w .7433 .34259 m .4553 .36957 L .19779 .31192 L p .696 .67 .82 r F P s P p .002 w .90952 .52815 m .90516 .24436 L .7433 .34259 L p .613 .485 .682 r F P s P p .002 w .7433 .34259 m .7458 .62092 L .90952 .52815 L p .613 .485 .682 r F P s P p .002 w .90516 .24436 m .7433 .34259 L p .19779 .31192 L .696 .67 .82 r F P s P p .002 w .7458 .62092 m .45469 .64639 L .19436 .59196 L p .694 .676 .828 r F P s P p .002 w .11489 .1988 m .19779 .31192 L .19436 .59196 L p .871 .697 .663 r F P s P p .002 w .84002 .12757 m .90516 .24436 L p .19779 .31192 L .696 .67 .82 r F P s P p .002 w .57304 .06304 m .84002 .12757 L p .19779 .31192 L .696 .67 .82 r F P s P p .002 w .26727 .09369 m .57304 .06304 L p .19779 .31192 L .696 .67 .82 r F P s P p .002 w .19779 .31192 m .11489 .1988 L .26727 .09369 L p .696 .67 .82 r F P s P p .002 w .90952 .52815 m .7458 .62092 L p .19436 .59196 L .694 .676 .828 r F P s P p .002 w .19436 .59196 m .11044 .48511 L .11489 .1988 L p .871 .697 .663 r F P s P p .002 w .84375 .41779 m .90952 .52815 L p .19436 .59196 L .694 .676 .828 r F P s P p .002 w .57374 .35678 m .84375 .41779 L p .19436 .59196 L .694 .676 .828 r F P s P p .002 w .26446 .38576 m .57374 .35678 L p .19436 .59196 L .694 .676 .828 r F P s P p .002 w .19436 .59196 m .11044 .48511 L .26446 .38576 L p .694 .676 .828 r F P s P p .002 w .84002 .12757 m .90516 .24436 L .90952 .52815 L p .913 .726 .636 r F P s P p .002 w .26446 .38576 m .26727 .09369 L .11489 .1988 L p .598 .469 .676 r F P s P p .002 w .11489 .1988 m .11044 .48511 L .26446 .38576 L p .598 .469 .676 r F P s P p .002 w .90952 .52815 m .84375 .41779 L .84002 .12757 L p .913 .726 .636 r F P s P p .002 w .57374 .35678 m .57304 .06304 L .84002 .12757 L p .742 .587 .683 r F P s P p .002 w .84002 .12757 m .84375 .41779 L .57374 .35678 L p .742 .587 .683 r F P s P p .002 w .57374 .35678 m .26446 .38576 L .26727 .09369 L p .692 .547 .684 r F P s P p .002 w .26727 .09369 m .57304 .06304 L .57374 .35678 L p .692 .547 .684 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Truncated Octahedron :[font = text; inactive; preserveAspect] The octahedron itself is not a zonohedron, since its faces are triangular and do not form zones. However if one uses the edges of the octahedron as generators, one gets this zonohedron, in which the triangular faces of the octahedron have been truncated to hexagons and squares added to connect them. The twelve octahedron edges come in six pairs, so there are six generators. This shape can fill space without leaving any gaps. :[font = input; preserveAspect; startGroup] Zonohedron[{{1,1,0},{1,-1,0},{1,0,1},{1,0,-1}, {0,1,1},{0,1,-1}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 259; pictureHeight = 289; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.1167 L 0 1.1167 L closepath clip newpath p P p .002 w .59864 .86832 m .4265 .71766 L .59752 .53806 L p .693 .549 .686 r F P s P p .002 w .59752 .53806 m .771 .68645 L .59864 .86832 L p .693 .549 .686 r F P s P p .002 w .55521 .30843 m .59752 .53806 L .4265 .71766 L p .731 .61 .723 r F P s P p .002 w .771 .68645 m .59752 .53806 L .55521 .30843 L p .646 .553 .735 r F P s P p .002 w .33772 .25628 m .55521 .30843 L p .4265 .71766 L .731 .61 .723 r F P s P p .002 w .1624 .43893 m .33772 .25628 L p .4265 .71766 L .731 .61 .723 r F P s P p .002 w .4265 .71766 m .20892 .67066 L .1624 .43893 L p .731 .61 .723 r F P s P p .002 w .55636 .9778 m .59864 .86832 L .4265 .71766 L p .747 .536 .609 r F P s P p .002 w .33301 .93238 m .55636 .9778 L p .4265 .71766 L .747 .536 .609 r F P s P p .002 w .15773 .7758 m .33301 .93238 L p .4265 .71766 L .747 .536 .609 r F P s P p .002 w .4265 .71766 m .20892 .67066 L .15773 .7758 L p .747 .536 .609 r F P s P p .002 w .90816 .60625 m .771 .68645 L p .55521 .30843 L .646 .553 .735 r F P s P p .002 w .86746 .37018 m .90816 .60625 L p .55521 .30843 L .646 .553 .735 r F P s P p .002 w .55521 .30843 m .68791 .22082 L .86746 .37018 L p .646 .553 .735 r F P s P p .002 w .68791 .22082 m .55521 .30843 L .33772 .25628 L p .697 .666 .815 r F P s P p .002 w .771 .68645 m .59864 .86832 L .55636 .9778 L p .494 .265 .496 r F P s P p .002 w .90816 .60625 m .771 .68645 L p .55636 .9778 L .494 .265 .496 r F P s P p .002 w .10967 .53975 m .1624 .43893 L .20892 .67066 L p .871 .697 .663 r F P s P p .002 w .20892 .67066 m .15773 .7758 L .10967 .53975 L p .871 .697 .663 r F P s P p .002 w .33772 .25628 m .1624 .43893 L .10967 .53975 L p .396 .136 .398 r F P s P p .002 w .33772 .25628 m .46764 .16656 L .68791 .22082 L p .697 .666 .815 r F P s P p .002 w .46764 .16656 m .33772 .25628 L p .10967 .53975 L .396 .136 .398 r F P s P p .002 w .83029 .47106 m .86746 .37018 L .68791 .22082 L p .752 .529 .594 r F P s P p .002 w .60416 .41805 m .83029 .47106 L p .68791 .22082 L .752 .529 .594 r F P s P p .002 w .42109 .26282 m .60416 .41805 L p .68791 .22082 L .752 .529 .594 r F P s P p .002 w .68791 .22082 m .46764 .16656 L .42109 .26282 L p .752 .529 .594 r F P s P p .002 w .83029 .47106 m .86746 .37018 L .90816 .60625 L p .913 .726 .636 r F P s P p .002 w .87228 .71162 m .90816 .60625 L p .55636 .9778 L .494 .265 .496 r F P s P p .002 w .90816 .60625 m .87228 .71162 L .83029 .47106 L p .913 .726 .636 r F P s P p .002 w .55636 .9778 m .69269 .90147 L .87228 .71162 L p .494 .265 .496 r F P s P p .002 w .69269 .90147 m .55636 .9778 L .33301 .93238 L p .693 .682 .835 r F P s P p .002 w .33301 .93238 m .15773 .7758 L .10967 .53975 L p .639 .55 .738 r F P s P p .002 w .42109 .26282 m .46764 .16656 L p .10967 .53975 L .396 .136 .398 r F P s P p .002 w .46642 .85416 m .33301 .93238 L p .10967 .53975 L .639 .55 .738 r F P s P p .002 w .41982 .61131 m .46642 .85416 L p .10967 .53975 L .639 .55 .738 r F P s P p .002 w .10967 .53975 m .23828 .45354 L .41982 .61131 L p .639 .55 .738 r F P s P p .002 w .10967 .53975 m .23828 .45354 L .42109 .26282 L p .396 .136 .398 r F P s P p .002 w .33301 .93238 m .46642 .85416 L .69269 .90147 L p .693 .682 .835 r F P s P p .002 w .83029 .47106 m .87228 .71162 L .69269 .90147 L p .733 .613 .724 r F P s P p .002 w .60416 .41805 m .83029 .47106 L p .69269 .90147 L .733 .613 .724 r F P s P p .002 w .41982 .61131 m .60416 .41805 L p .69269 .90147 L .733 .613 .724 r F P s P p .002 w .69269 .90147 m .46642 .85416 L .41982 .61131 L p .733 .613 .724 r F P s P p .002 w .41982 .61131 m .23828 .45354 L .42109 .26282 L p .692 .547 .684 r F P s P p .002 w .42109 .26282 m .60416 .41805 L .41982 .61131 L p .692 .547 .684 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Rhombic Dodecahedron :[font = text; inactive; preserveAspect] This is the dual to the cuboctahedron, an Archimedean solid formed by combining the six squares of a cube with the eight triangles of an octahedron. The cuboctahedron itself is not a zonohedron because of its triangular faces. :[font = input; preserveAspect; startGroup] Zonohedron[{{1,1,1},{1,-1,1},{1,1,-1},{1,-1,-1}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 337; pictureHeight = 376; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.1167 L 0 1.1167 L closepath clip newpath p P p .002 w .72983 .45577 m .59808 .70212 L .38251 .48907 L p .697 .588 .729 r F P s P p .002 w .16008 .60625 m .38251 .48907 L .59808 .70212 L p .739 .587 .685 r F P s P p .002 w .59808 .70212 m .3808 .82261 L .16008 .60625 L p .739 .587 .685 r F P s P p .002 w .86985 .53975 m .72983 .45577 L .59808 .70212 L p .611 .483 .681 r F P s P p .002 w .59808 .70212 m .73273 .79153 L .86985 .53975 L p .611 .483 .681 r F P s P p .002 w .73273 .79153 m .59808 .70212 L .3808 .82261 L p .674 .452 .576 r F P s P p .002 w .38251 .48907 m .51212 .23862 L .72983 .45577 L p .697 .588 .729 r F P s P p .002 w .28914 .35265 m .51212 .23862 L .38251 .48907 L p .779 .693 .768 r F P s P p .002 w .38251 .48907 m .16008 .60625 L .28914 .35265 L p .779 .693 .768 r F P s P p .002 w .86985 .53975 m .72983 .45577 L .51212 .23862 L p .374 .512 .868 r F P s P p .002 w .51212 .23862 m .647 .31716 L .86985 .53975 L p .374 .512 .868 r F P s P p .002 w .647 .31716 m .51212 .23862 L .28914 .35265 L p .667 .433 .556 r F P s P p .002 w .3808 .82261 m .51212 .91699 L .73273 .79153 L p .674 .452 .576 r F P s P p .002 w .51212 .91699 m .3808 .82261 L .16008 .60625 L p .189 .438 .867 r F P s P p .002 w .16008 .60625 m .2861 .69524 L .51212 .91699 L p .189 .438 .867 r F P s P p .002 w .42046 .43586 m .28914 .35265 L .16008 .60625 L p .6 .472 .677 r F P s P p .002 w .16008 .60625 m .2861 .69524 L .42046 .43586 L p .6 .472 .677 r F P s P p .002 w .64885 .66208 m .86985 .53975 L .73273 .79153 L p .787 .706 .775 r F P s P p .002 w .73273 .79153 m .51212 .91699 L .64885 .66208 L p .787 .706 .775 r F P s P p .002 w .28914 .35265 m .42046 .43586 L .647 .31716 L p .667 .433 .556 r F P s P p .002 w .42046 .43586 m .647 .31716 L .86985 .53975 L p .741 .587 .684 r F P s P p .002 w .86985 .53975 m .64885 .66208 L .42046 .43586 L p .741 .587 .684 r F P s P p .002 w .64885 .66208 m .51212 .91699 L .2861 .69524 L p .696 .588 .731 r F P s P p .002 w .2861 .69524 m .42046 .43586 L .64885 .66208 L p .696 .588 .731 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Extended Rhombic Dodecahedron :[font = text; inactive; preserveAspect] This zonotope is noteworthy for (like the cube, hexagonal prism, truncated octahedron, and rhombic dodecahedron) being able to fill space without leaving any gaps. The one here is drawn with different angles from the rhombic dodecahedron itself, to make the hexagonal sides regular. :[font = input; preserveAspect; startGroup] Zonohedron[{{0,Sqrt[3],1},{Sqrt[3],0,1}, {0,-Sqrt[3],1},{-Sqrt[3],0,1}, {0,0,2}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 252; pictureHeight = 408; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.62226 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.27324 1.56013 0.0390032 1.56013 [ [ 0 0 0 0 ] [ 1 1.62226 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.62226 L 0 1.62226 L closepath clip newpath p P p .002 w .59107 .5735 m .25414 .79101 L .16523 .47412 L p .727 .621 .739 r F P s P p .002 w .16523 .47412 m .25414 .79101 L .25192 1.1658 L p .879 .702 .659 r F P s P p .002 w .25192 1.1658 m .25414 .79101 L .59107 .5735 L p .693 .548 .686 r F P s P p .002 w .16523 .47412 m .50682 .25354 L .59107 .5735 L p .727 .621 .739 r F P s P p .002 w .93181 .72704 m .59107 .5735 L .50682 .25354 L p .658 .578 .755 r F P s P p .002 w .59256 1.32601 m .25192 1.1658 L p .59107 .5735 L .693 .548 .686 r F P s P p .002 w .93556 1.10472 m .59256 1.32601 L p .59107 .5735 L .693 .548 .686 r F P s P p .002 w .59107 .5735 m .93181 .72704 L .93556 1.10472 L p .693 .548 .686 r F P s P p .002 w .06992 .52797 m .16523 .47412 L p .25192 1.1658 L .879 .702 .659 r F P s P p .002 w .06595 .91453 m .06992 .52797 L p .25192 1.1658 L .879 .702 .659 r F P s P p .002 w .25192 1.1658 m .15911 1.23535 L .06595 .91453 L p .879 .702 .659 r F P s P p .002 w .59256 1.32601 m .25192 1.1658 L .15911 1.23535 L p .745 .45 .482 r F P s P p .002 w .50682 .25354 m .85157 .40634 L .93181 .72704 L p .658 .578 .755 r F P s P p .002 w .85157 .40634 m .93181 .72704 L .93556 1.10472 L p .903 .719 .644 r F P s P p .002 w .41943 .30229 m .50682 .25354 L .16523 .47412 L p .525 .902 .73 r F P s P p .002 w .16523 .47412 m .06992 .52797 L .41943 .30229 L p .525 .902 .73 r F P s P p .002 w .85157 .40634 m .50682 .25354 L .41943 .30229 L p .744 .407 .412 r F P s P p .002 w .85781 1.17349 m .93556 1.10472 L .59256 1.32601 L p 0 0 0 r F P s P p .002 w .59256 1.32601 m .50682 1.39993 L .85781 1.17349 L p 0 0 0 r F P s P p .002 w .15911 1.23535 m .50682 1.39993 L .59256 1.32601 L p .745 .45 .482 r F P s P p .002 w .77145 .45911 m .85157 .40634 L p .93556 1.10472 L .903 .719 .644 r F P s P p .002 w .77387 .84872 m .77145 .45911 L p .93556 1.10472 L .903 .719 .644 r F P s P p .002 w .93556 1.10472 m .85781 1.17349 L .77387 .84872 L p .903 .719 .644 r F P s P p .002 w .41943 .30229 m .77145 .45911 L .85157 .40634 L p .744 .407 .412 r F P s P p .002 w .50682 1.39993 m .15911 1.23535 L .06595 .91453 L p .652 .578 .759 r F P s P p .002 w .06595 .91453 m .06992 .52797 L .41943 .30229 L p .692 .547 .685 r F P s P p .002 w .41782 1.07849 m .06595 .91453 L p .41943 .30229 L .692 .547 .685 r F P s P p .002 w .77387 .84872 m .41782 1.07849 L p .41943 .30229 L .692 .547 .685 r F P s P p .002 w .41943 .30229 m .77145 .45911 L .77387 .84872 L p .692 .547 .685 r F P s P p .002 w .85781 1.17349 m .50682 1.39993 L .41782 1.07849 L p .728 .624 .741 r F P s P p .002 w .06595 .91453 m .41782 1.07849 L .50682 1.39993 L p .652 .578 .759 r F P s P p .002 w .41782 1.07849 m .77387 .84872 L .85781 1.17349 L p .728 .624 .741 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Truncated Cuboctahedron :[font = text; inactive; preserveAspect] As can be seen from its set of generators, this is the Minkowski sum of a cube and a truncated octahedron. It is also known as the Great Rhombicuboctahedron. :[font = input; preserveAspect; startGroup] Zonohedron[{{1,1,0},{1,-1,0},{1,0,1},{1,0,-1}, {0,1,1},{0,1,-1}, {Sqrt[2],0,0},{0,Sqrt[2],0},{0,0,Sqrt[2]}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 257; pictureHeight = 287; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.1167 L 0 1.1167 L closepath clip newpath p P p .002 w .65621 .31739 m .68681 .48703 L .50831 .50404 L p .697 .587 .729 r F P s P p .002 w .50831 .50404 m .47664 .33519 L .65621 .31739 L p .697 .587 .729 r F P s P p .002 w .38234 .63609 m .50831 .50404 L .68681 .48703 L p .693 .549 .686 r F P s P p .002 w .47664 .33519 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.47664 .33519 L p .26763 .22388 L .697 .666 .815 r F P s P p .002 w .38146 .8079 m .22043 .77398 L .22241 .60092 L p .739 .586 .686 r F P s P p .002 w .4757 1.0011 m .50824 .91986 L .38146 .8079 L p .747 .537 .611 r F P s P p .002 w .31153 .96807 m .4757 1.0011 L p .38146 .8079 L .747 .537 .611 r F P s P p .002 w .18306 .85287 m .31153 .96807 L p .38146 .8079 L .747 .537 .611 r F P s P p .002 w .38146 .8079 m .22043 .77398 L .18306 .85287 L p .747 .537 .611 r F P s P p .002 w .18847 .43093 m .22241 .60092 L .22043 .77398 L p .871 .697 .663 r F P s P p .002 w .91657 .53589 m .81535 .59601 L p .65621 .31739 L .647 .553 .735 r F P s P p .002 w .88686 .36275 m .91657 .53589 L p .65621 .31739 L .647 .553 .735 r F P s P p .002 w .65621 .31739 m .75501 .25326 L .88686 .36275 L p .647 .553 .735 r F P s P p .002 w .75501 .25326 m .65621 .31739 L p .26763 .22388 L .697 .666 .815 r F P s P p .002 w .91657 .53589 m .81535 .59601 L .81742 .76925 L p .614 .486 .682 r F P s P p .002 w .31676 .29725 m .18847 .43093 L .13711 .35965 L p .777 .689 .766 r F P s P p .002 w .13711 .35965 m .26763 .22388 L .31676 .29725 L p .777 .689 .766 r F P s P p .002 w .66005 .9856 m .6897 .90428 L .50824 .91986 L p .675 .457 .582 r F P s P p .002 w .50824 .91986 m .4757 1.0011 L .66005 .9856 L p .675 .457 .582 r F P s P p .002 w .13711 .35965 m .18847 .43093 L p .22043 .77398 L .871 .697 .663 r F P s P p .002 w .0977 .43284 m .13711 .35965 L p .22043 .77398 L .871 .697 .663 r F P s P p .002 w .09476 .61184 m .0977 .43284 L p .22043 .77398 L .871 .697 .663 r F P s P p .002 w .13073 .78751 m .09476 .61184 L p .22043 .77398 L .871 .697 .663 r F P s P p .002 w .22043 .77398 m .18306 .85287 L .13073 .78751 L p .871 .697 .663 r F P s P p .002 w .81742 .76925 m .91937 .71126 L .91657 .53589 L p .614 .486 .682 r F P s P p .002 w .81742 .76925 m .6897 .90428 L .66005 .9856 L p .503 .277 .504 r F P s P p .002 w .91937 .71126 m .81742 .76925 L p .66005 .9856 L .503 .277 .504 r F P s P p .002 w .88686 .36275 m .75501 .25326 L .71274 .17841 L p .41 .524 .862 r F P s P p .002 w .71274 .17841 m .75501 .25326 L p .26763 .22388 L .697 .666 .815 r F P s P p .002 w .88686 .36275 m .91657 .53589 L .91937 .71126 L p .913 .726 .636 r F P s P p .002 w .26763 .22388 m .13711 .35965 L .0977 .43284 L p .377 .112 .38 r F P s P p .002 w .5492 .13769 m .71274 .17841 L p .26763 .22388 L .697 .666 .815 r F P s P p .002 w .26763 .22388 m .36277 .15697 L .5492 .13769 L p .697 .666 .815 r F P s P p .002 w .36277 .15697 m .26763 .22388 L p .0977 .43284 L .377 .112 .38 r F P s P p .002 w .4757 1.0011 m .31153 .96807 L .26098 .90414 L p .693 .682 .835 r F P s P p .002 w .66005 .9856 m .4757 1.0011 L p .26098 .90414 L .693 .682 .835 r F P s P p .002 w .71274 .17841 m .84648 .28916 L .88686 .36275 L p .41 .524 .862 r F P s P p .002 w .84648 .28916 m .88686 .36275 L p .91937 .71126 L .913 .726 .636 r F P s P p .002 w .81863 .36235 m .84648 .28916 L p .91937 .71126 L .913 .726 .636 r F P s P p .002 w .82084 .5438 m .81863 .36235 L p .91937 .71126 L .913 .726 .636 r F P s P p .002 w .85226 .72285 m .82084 .5438 L p .91937 .71126 L .913 .726 .636 r F P s P p .002 w .89323 .79036 m .91937 .71126 L p .66005 .9856 L .503 .277 .504 r F P s P p .002 w .91937 .71126 m .89323 .79036 L .85226 .72285 L p .913 .726 .636 r F P s P p .002 w .31153 .96807 m .18306 .85287 L .13073 .78751 L p .096 .394 .85 r F P s P p .002 w .66005 .9856 m .76158 .92974 L .89323 .79036 L p .503 .277 .504 r F P s P p .002 w .76158 .92974 m .66005 .9856 L p .26098 .90414 L .693 .682 .835 r F P s P p .002 w .13073 .78751 m .26098 .90414 L .31153 .96807 L p .096 .394 .85 r F P s P p .002 w .81863 .36235 m .84648 .28916 L .71274 .17841 L p .753 .528 .591 r F P s P p .002 w .65189 .32228 m .81863 .36235 L p .71274 .17841 L .753 .528 .591 r F P s P p .002 w .5162 .20834 m .65189 .32228 L p .71274 .17841 L .753 .528 .591 r F P s P p .002 w .71274 .17841 m .5492 .13769 L .5162 .20834 L p .753 .528 .591 r F P s P p .002 w .5162 .20834 m .5492 .13769 L .36277 .15697 L p .665 .426 .548 r F P s P p .002 w .36277 .15697 m .32671 .22765 L .5162 .20834 L p .665 .426 .548 r F P s P p .002 w .32671 .22765 m .36277 .15697 L p .0977 .43284 L .377 .112 .38 r F P s P p .002 w .89323 .79036 m .76158 .92974 L .71825 .8645 L p .79 .711 .777 r F P s P p .002 w .71825 .8645 m .85226 .72285 L .89323 .79036 L p .79 .711 .777 r F P s P p .002 w .19209 .36782 m .0977 .43284 L .09476 .61184 L p .596 .468 .676 r F P s P p .002 w .0977 .43284 m .19209 .36782 L .32671 .22765 L p .377 .112 .38 r F P s P p .002 w .26098 .90414 m .13073 .78751 L .09476 .61184 L p .638 .55 .738 r F P s P p .002 w .71825 .8645 m .76158 .92974 L p .26098 .90414 L .693 .682 .835 r F P s P p .002 w .09476 .61184 m .18979 .54908 L .19209 .36782 L p .596 .468 .676 r F P s P p .002 w .35865 .8458 m .26098 .90414 L p .09476 .61184 L .638 .55 .738 r F P s P p .002 w .32346 .66639 m .35865 .8458 L p .09476 .61184 L .638 .55 .738 r F P s P p .002 w .09476 .61184 m .18979 .54908 L .32346 .66639 L p .638 .55 .738 r F P s P p .002 w .55023 .82897 m .71825 .8645 L p .26098 .90414 L .693 .682 .835 r F P s P p .002 w .26098 .90414 m .35865 .8458 L .55023 .82897 L p .693 .682 .835 r F P s P p .002 w .19209 .36782 m .32671 .22765 L .5162 .20834 L p .692 .547 .684 r F P s P p .002 w .18979 .54908 m .19209 .36782 L p .5162 .20834 L .692 .547 .684 r F P s P p .002 w .65189 .32228 m .81863 .36235 L .82084 .5438 L p .742 .587 .683 r F P s P p .002 w .82084 .5438 m .85226 .72285 L .71825 .8645 L p .733 .613 .724 r F P s P p .002 w .32346 .66639 m .18979 .54908 L p .5162 .20834 L .692 .547 .684 r F P s P p .002 w .51627 .64866 m .32346 .66639 L p .5162 .20834 L .692 .547 .684 r F P s P p .002 w .65291 .50512 m .51627 .64866 L p .5162 .20834 L .692 .547 .684 r F P s P p .002 w .5162 .20834 m .65189 .32228 L .65291 .50512 L p .692 .547 .684 r F P s P p .002 w .65291 .50512 m .82084 .5438 L p .71825 .8645 L .733 .613 .724 r F P s P p .002 w .51627 .64866 m .65291 .50512 L p .71825 .8645 L .733 .613 .724 r F P s P p .002 w .71825 .8645 m .55023 .82897 L .51627 .64866 L p .733 .613 .724 r F P s P p .002 w .51627 .64866 m .55023 .82897 L .35865 .8458 L p .696 .588 .731 r F P s P p .002 w .35865 .8458 m .32346 .66639 L .51627 .64866 L p .696 .588 .731 r F P s P p .002 w .82084 .5438 m .65291 .50512 L .65189 .32228 L p .742 .587 .683 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Truncated Rhombic Dodecahedron :[font = text; inactive; preserveAspect] This is the Minkowski sum of a cube and a rhombic dodecahedron. The hexagons can not be regular, although they do have all sides the same length, since three regular hexagons would meet in a flat solid angle instead of the corners here. Although this shape does not fill space on its own, it can be combined with cubes in a regular space-filling pattern. :[font = input; preserveAspect; startGroup] Zonohedron[{{1,1,1},{1,1,-1},{1,-1,1},{1,-1,-1}, {Sqrt[3],0,0},{0,Sqrt[3],0},{0,0,Sqrt[3]}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 260; pictureHeight = 290; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.1167 L 0 1.1167 L closepath clip newpath p P p .002 w .70317 .79305 m .49416 .81151 L .4943 .61232 L p .693 .549 .686 r F P s P p .002 w .4943 .61232 m .70167 .59307 L .70317 .79305 L p .693 .549 .686 r F P s P p .002 w .34467 .46449 m .4943 .61232 L .49416 .81151 L p .739 .587 .686 r F P s P p .002 w .70167 .59307 m .4943 .61232 L .34467 .46449 L p .697 .588 .729 r F P s P p .002 w .19105 .54484 m .34467 .46449 L p .49416 .81151 L .739 .587 .686 r F P s P p .002 w .79405 .42112 m .70167 .59307 L p .34467 .46449 L .697 .588 .729 r F P s P p .002 w .64344 .27053 m .79405 .42112 L p .34467 .46449 L .697 .588 .729 r F P s P p .002 w .34467 .46449 m .4337 .29156 L .64344 .27053 L p .697 .588 .729 r F P s P p .002 w .34467 .46449 m .19105 .54484 L .13135 .46411 L p .778 .691 .767 r F P s P p .002 w .4337 .29156 m .34467 .46449 L p .13135 .46411 L .778 .691 .767 r F P s P p .002 w .79405 .42112 m .70167 .59307 L .70317 .79305 L p .613 .484 .681 r F P s P p .002 w .18849 .7468 m .19105 .54484 L p .49416 .81151 L .739 .587 .686 r F P s P p .002 w .49416 .81151 m .3418 .89667 L .18849 .7468 L p .739 .587 .686 r F P s P p .002 w .70317 .79305 m .49416 .81151 L .3418 .89667 L p .675 .454 .579 r F P s P p .002 w .79893 .85703 m .70317 .79305 L p .3418 .89667 L .675 .454 .579 r F P s P p .002 w .89162 .47827 m .79405 .42112 L p .70317 .79305 L .613 .484 .681 r F P s P p .002 w .8947 .68294 m .89162 .47827 L p .70317 .79305 L .613 .484 .681 r F P s P p .002 w .70317 .79305 m .79893 .85703 L .8947 .68294 L p .613 .484 .681 r F P s P p .002 w .59244 .18439 m .64344 .27053 L .4337 .29156 L p .697 .666 .815 r F P s P p .002 w .4337 .29156 m .3789 .20624 L .59244 .18439 L p .697 .666 .815 r F P s P p .002 w .3789 .20624 m .4337 .29156 L p .13135 .46411 L .778 .691 .767 r F P s P p .002 w .79405 .42112 m .64344 .27053 L .59244 .18439 L p .392 .518 .865 r F P s P p .002 w .89162 .47827 m .79405 .42112 L p .59244 .18439 L .392 .518 .865 r F P s P p .002 w .13135 .46411 m .19105 .54484 L .18849 .7468 L p .871 .697 .663 r F P s P p .002 w .18849 .7468 m .12826 .66935 L .13135 .46411 L p .871 .697 .663 r F P s P p .002 w .3418 .89667 m .18849 .7468 L .12826 .66935 L p .147 .418 .861 r F P s P p .002 w .64698 .9448 m .79893 .85703 L p .3418 .89667 L .675 .454 .579 r F P s P p .002 w .3418 .89667 m .4316 .96311 L .64698 .9448 L p .675 .454 .579 r F P s P p .002 w .4316 .96311 m .3418 .89667 L p .12826 .66935 L .147 .418 .861 r F P s P p .002 w .79893 .85703 m .64698 .9448 L .59465 .86972 L p .788 .708 .776 r F P s P p .002 w .8947 .68294 m .79893 .85703 L p .59465 .86972 L .788 .708 .776 r F P s P p .002 w .13135 .46411 m .22129 .28552 L .3789 .20624 L p .778 .691 .767 r F P s P p .002 w .59244 .18439 m .3789 .20624 L .22129 .28552 L p .666 .43 .552 r F P s P p .002 w .84473 .39492 m .89162 .47827 L p .59244 .18439 L .392 .518 .865 r F P s P p .002 w .84473 .39492 m .89162 .47827 L .8947 .68294 L p .913 .726 .636 r F P s P p .002 w .22129 .28552 m .13135 .46411 L .12826 .66935 L p .598 .47 .677 r F P s P p .002 w .59244 .18439 m .68848 .2384 L .84473 .39492 L p .392 .518 .865 r F P s P p .002 w .68848 .2384 m .59244 .18439 L p .22129 .28552 L .666 .43 .552 r F P s P p .002 w .59465 .86972 m .64698 .9448 L .4316 .96311 L p .693 .682 .835 r F P s P p .002 w .4316 .96311 m .37526 .88877 L .59465 .86972 L p .693 .682 .835 r F P s P p .002 w .37526 .88877 m .4316 .96311 L p .12826 .66935 L .147 .418 .861 r F P s P p .002 w .8947 .68294 m .84748 .60295 L .84473 .39492 L p .913 .726 .636 r F P s P p .002 w .84748 .60295 m .8947 .68294 L p .59465 .86972 L .788 .708 .776 r F P s P p .002 w .12826 .66935 m .2161 .73302 L .37526 .88877 L p .147 .418 .861 r F P s P p .002 w .31102 .34192 m .22129 .28552 L p .12826 .66935 L .598 .47 .677 r F P s P p .002 w .30934 .55208 m .31102 .34192 L p .12826 .66935 L .598 .47 .677 r F P s P p .002 w .12826 .66935 m .2161 .73302 L .30934 .55208 L p .598 .47 .677 r F P s P p .002 w .53122 .32004 m .68848 .2384 L p .22129 .28552 L .666 .43 .552 r F P s P p .002 w .22129 .28552 m .31102 .34192 L .53122 .32004 L p .666 .43 .552 r F P s P p .002 w .68848 .2384 m .84473 .39492 L .84748 .60295 L p .742 .587 .684 r F P s P p .002 w .53122 .32004 m .68848 .2384 L p .84748 .60295 L .742 .587 .684 r F P s P p .002 w .59465 .86972 m .37526 .88877 L .2161 .73302 L p .696 .588 .731 r F P s P p .002 w .53138 .53107 m .53122 .32004 L p .84748 .60295 L .742 .587 .684 r F P s P p .002 w .84748 .60295 m .69166 .68989 L .53138 .53107 L p .742 .587 .684 r F P s P p .002 w .59465 .86972 m .69166 .68989 L .84748 .60295 L p .788 .708 .776 r F P s P p .002 w .69166 .68989 m .59465 .86972 L p .2161 .73302 L .696 .588 .731 r F P s P p .002 w .53138 .53107 m .69166 .68989 L p .2161 .73302 L .696 .588 .731 r F P s P p .002 w .2161 .73302 m .30934 .55208 L .53138 .53107 L p .696 .588 .731 r F P s P p .002 w .53138 .53107 m .30934 .55208 L .31102 .34192 L p .692 .547 .684 r F P s P p .002 w .31102 .34192 m .53122 .32004 L .53138 .53107 L p .692 .547 .684 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Truncated Small Rhombicuboctahedron :[font = text; inactive; preserveAspect] This is the Minkowski sum of a cube, a truncated octahedron, and a rhombic dodecahedron. Some of the octagonal faces are non-regular. :[font = input; preserveAspect; startGroup] Zonohedron[{{1,1,0},{1,-1,0},{1,0,1},{1,0,-1}, {0,1,1},{0,1,-1}, {Sqrt[2/3],Sqrt[2/3],Sqrt[2/3]}, {Sqrt[2/3],Sqrt[2/3],-Sqrt[2/3]}, {Sqrt[2/3],-Sqrt[2/3],Sqrt[2/3]}, {Sqrt[2/3],-Sqrt[2/3],-Sqrt[2/3]}, {Sqrt[2],0,0},{0,Sqrt[2],0},{0,0,Sqrt[2]}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 257; pictureHeight = 287; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 0 0 m 1 0 L 1 1.1167 L 0 1.1167 L closepath clip newpath p P p .002 w .54187 .5781 m .46309 .6608 L .3824 .58124 L p .717 .588 .709 r F P s P p .002 w .3824 .58124 m .46143 .49846 L .54187 .5781 L p .717 .588 .709 r F P s P p .002 w .65355 .56765 m .54187 .5781 L .46143 .49846 L p .697 .587 .729 r F P s P p .002 w .46309 .6608 m .54187 .5781 L .65355 .56765 L p .693 .549 .686 r F P s P p .002 w .46309 .6608 m .46289 .76814 L .38122 .81337 L p .739 .587 .686 r F P s P p .002 w .3824 .58124 m .46309 .6608 L p .38122 .81337 L .739 .587 .686 r F P s P p .002 w .46289 .76814 m .46309 .6608 L p .65355 .56765 L .693 .549 .686 r F P s P p .002 w .70281 .47541 m .65355 .56765 L p .46143 .49846 L .697 .587 .729 r F P s P p .002 w .6837 .36905 m .70281 .47541 L p .46143 .49846 L .697 .587 .729 r F P s P p .002 w .6026 .28808 m .6837 .36905 L p .46143 .49846 L .697 .587 .729 r F P s P p .002 w .48981 .29935 m .6026 .28808 L p .46143 .49846 L .697 .587 .729 r F P s P p .002 w .46143 .49846 m .44142 .39278 L .48981 .29935 L p .697 .587 .729 r F P s P p .002 w .44142 .39278 m .46143 .49846 L .3824 .58124 L p .731 .61 .723 r F P s P p .002 w .73352 .63597 m .65355 .56765 L .70281 .47541 L p .668 .552 .714 r F P s P p .002 w .54217 .83762 m .46289 .76814 L p .65355 .56765 L .693 .549 .686 r F P s P p .002 w .655 .82773 m .54217 .83762 L p .65355 .56765 L .693 .549 .686 r F P s P p .002 w .73446 .74386 m .655 .82773 L p .65355 .56765 L .693 .549 .686 r F P s P p .002 w .65355 .56765 m .73352 .63597 L .73446 .74386 L p .693 .549 .686 r F P s P p .002 w .28246 .55896 m .3824 .58124 L p .38122 .81337 L .739 .587 .686 r F P s P p .002 w .3415 .36942 m .44142 .39278 L p .3824 .58124 L .731 .61 .723 r F P s P p .002 w .26157 .45283 m .3415 .36942 L p .3824 .58124 L .731 .61 .723 r F P s P p .002 w .3824 .58124 m .28246 .55896 L .26157 .45283 L p .731 .61 .723 r F P s P p .002 w .54217 .83762 m .46289 .76814 L .38122 .81337 L p .72 .544 .652 r F P s P p .002 w .38966 .27537 m .48981 .29935 L .44142 .39278 L p .725 .623 .744 r F P s P p .002 w .44142 .39278 m .3415 .36942 L .38966 .27537 L p .725 .623 .744 r F P s P p .002 w .70281 .47541 m .78341 .54363 L .73352 .63597 L p .668 .552 .714 r 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.56472 .36506 L .56496 .47892 L p .692 .547 .684 r F P s P p .002 w .56496 .47892 m .56472 .36506 L p .75592 .58845 L .742 .587 .683 r F P s P p .002 w .75592 .58845 m .65137 .56471 L .56496 .47892 L p .742 .587 .683 r F P s P p .002 w .27649 .50653 m .36011 .5793 L .30923 .67738 L p .664 .549 .714 r F P s P p .002 w .65137 .56471 m .56667 .65376 L .47998 .56805 L p .718 .588 .71 r F P s P p .002 w .47998 .56805 m .56496 .47892 L .65137 .56471 L p .718 .588 .71 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Rhombic Triacontahedron :[font = text; inactive; preserveAspect] This is the dual of the icosidodecahedron, one of the Archimedian solids that (like the cuboctahedron) is not a zonohedron as it has odd faces. :[font = input; preserveAspect; startGroup] Zonohedron[{{1,0,-GoldenRatio},{1,0,GoldenRatio}, {0,-GoldenRatio,1},{0,GoldenRatio,1}, {-GoldenRatio,1,0},{GoldenRatio,1,0}}] :[font = postscript; PostScript; 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.23148 .72472 L .15717 .81616 L p .558 .477 .72 r F P s P p .002 w .15717 .81616 m .12172 .55564 L .19549 .45769 L p .558 .477 .72 r F P s P p .002 w .64775 .41382 m .67937 .25568 L .45693 .14921 L p .713 .523 .631 r F P s P p .002 w .45693 .14921 m .42094 .30346 L .64775 .41382 L p .713 .523 .631 r F P s P p .002 w .42094 .30346 m .45693 .14921 L .23615 .30022 L p .639 .45 .611 r F P s P p .002 w .23615 .30022 m .19549 .45769 L .42094 .30346 L p .639 .45 .611 r F P s P p .002 w .23148 .72472 m .45541 .83903 L .37516 .92875 L p .631 .583 .782 r F P s P p .002 w .45541 .83903 m .68223 .68377 L .87286 .75253 L p .738 .641 .752 r F P s P p .002 w .87286 .75253 m .64993 .90515 L .45541 .83903 L p .738 .641 .752 r F P s P p .002 w .83974 .48748 m .87286 .75253 L .68223 .68377 L p .755 .618 .708 r F P s P p .002 w .67937 .25568 m .64775 .41382 L .83974 .48748 L p .773 .583 .642 r F P s P p .002 w .68223 .68377 m .64775 .41382 L .83974 .48748 L p .755 .618 .708 r F P s P p .002 w .41998 .56968 m .45541 .83903 L .23148 .72472 L p .674 .561 .719 r F P s P p .002 w .23148 .72472 m .19549 .45769 L .41998 .56968 L p .674 .561 .719 r F P s P p .002 w .19549 .45769 m .42094 .30346 L .64775 .41382 L p .692 .547 .684 r F P s P p .002 w .64775 .41382 m .41998 .56968 L .19549 .45769 L p .692 .547 .684 r F P s P p .002 w .64775 .41382 m .68223 .68377 L .45541 .83903 L p .713 .588 .715 r F P s P p .002 w .45541 .83903 m .41998 .56968 L .64775 .41382 L p .713 .588 .715 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Truncated Icosidodecahedron :[font = text; inactive; preserveAspect] This is also known as the Great Rhombicosidodecahedron. It may not be obvious, but the generators below can be grouped into five triples of orthogonal vectors, showing that this is the Minkowski sum of five cubes. :[font = input; preserveAspect; startGroup] Zonohedron[{ {1,GoldenRatio,GoldenRatio-1},{1,-GoldenRatio,GoldenRatio-1}, {1,-GoldenRatio,1-GoldenRatio},{1,GoldenRatio,1-GoldenRatio}, {GoldenRatio,1-GoldenRatio,1},{GoldenRatio,1-GoldenRatio,-1}, {GoldenRatio,GoldenRatio-1,-1},{GoldenRatio,GoldenRatio-1,1}, {GoldenRatio-1,1,GoldenRatio},{GoldenRatio-1,-1,-GoldenRatio}, {GoldenRatio-1,1,-GoldenRatio},{GoldenRatio-1,-1,GoldenRatio}, {2,0,0},{0,2,0},{0,0,2}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 339; pictureHeight = 378; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] 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.58269 L .725 .574 .684 r F P s P p .002 w .7093 .58269 m .61813 .6008 L .52624 .56074 L p .725 .574 .684 r F P s P p .002 w .3864 .66193 m .32993 .57918 L .37129 .48729 L p .694 .566 .706 r F P s P p .002 w .37129 .48729 m .4694 .47788 L p .3864 .66193 L .694 .566 .706 r F P s P p .002 w .52624 .56074 m .4846 .6529 L .3864 .66193 L p .694 .566 .706 r F P s P p .002 w .4694 .47788 m .52624 .56074 L p .3864 .66193 L .694 .566 .706 r F P s P p .002 w .52624 .56074 m .61813 .6008 L .57648 .69279 L p .713 .588 .715 r F P s P p .002 w .57648 .69279 m .4846 .6529 L .52624 .56074 L p .713 .588 .715 r F P s P p P p P % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] The Truncated Small Rhombicosidodecahedron :[font = text; inactive; preserveAspect] This shape is the Minkowski sum of the great rhombicosidodecahedron and the rhombic triacontahedron. As drawn here, the edges have two slightly different lengths. :[font = input; preserveAspect; startGroup] Zonohedron[{{1,0,-GoldenRatio},{1,0,GoldenRatio}, {0,-GoldenRatio,1},{0,GoldenRatio,1}, {-GoldenRatio,1,0},{GoldenRatio,1,0}, {1,GoldenRatio,GoldenRatio-1},{1,-GoldenRatio,GoldenRatio-1}, {1,-GoldenRatio,1-GoldenRatio},{1,GoldenRatio,1-GoldenRatio}, {GoldenRatio,1-GoldenRatio,1},{GoldenRatio,1-GoldenRatio,-1}, {GoldenRatio,GoldenRatio-1,-1},{GoldenRatio,GoldenRatio-1,1}, {GoldenRatio-1,1,GoldenRatio},{GoldenRatio-1,-1,-GoldenRatio}, {GoldenRatio-1,1,-GoldenRatio},{GoldenRatio-1,-1,GoldenRatio}, {2,0,0},{0,2,0},{0,0,2}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 257; pictureHeight = 287] %! %%Creator: Mathematica %%AspectRatio: 1.1167 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0338461 1.09193 0.0272983 1.09193 [ [ 0 0 0 0 ] [ 1 1.1167 0 0 ] ] MathScale % 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