*Mathematica* is the world's premier mathematical software. Not
only does it have a wealth of built-in, highly optimized mathematical
functions, but it is also a programming language, and thus, one
can contrive one's own functions. It has beautiful PostScript
graphical output, and allows typeset-quality mathematical notation.
This very remarkable program was pioneered by Stephen Wolfram,
and developed with the help of many others, notably Roman Maeder
and Theodore Gray.

My own introduction to *Mathematica* was
in 1988, I believe, when word came that it would be ported to
the Macintosh operating system. I saw the debut of the Mac version
at MacWorld Expo in San Francisco, in 1989. Its 3D graphics were
of special appeal to me, but for years I lacked a computer powerful
enough to run the program. Finally, about 1995, I got my foot
in the door.

At that time I had been programming in BASIC
for several years, beginning with a Commodore Plus 4 in 1987,
and continuing with a Mac Plus, and then a Mac Classic. I had
devised algorithms to construct the polar zonohedra, of arbitrary
complexity and proportion, and perform parallel projections onto
an arbitrary plane. I could make animations of rotations, and
also devised algorithms to construct what I call "expansions"
of polar zonohedra. *Mathematica* posed quite a challenge
at first.

Pursuing the challenge, I continued working with polar zonohedra, and branched out into curious variants I discovered, naming them spirallohedra, and eventually tackled the regular polytopes in four dimensions. Several of my notebooks may be downloaded below. You will need Stuffit Expander (Mac or Windows version) to decompress the notebooks.

An n=5 rhombic spirallohedron.

A close-packing of rhombic spirallohedra.

Download the Spirallohedra notebook (needs Stuffit Expander to decompress)

The dodecahedral expansion of a polar zonohedron.

A skew dodecahedral expansion of a polar zonohedron.

Download the Expansion notebook (needs Stuffit Expander to decompress)

A parallelogrammic/rhombic helicoid. The faces have been sorted by surface area and colors assigned to each subset.

Download the Helicoid notebook (needs Stuffit Expander to decompress)

3D zonotiles; above, an edge=3 zonotile based upon the truncated icosidodecahedron, in which only some of the zonohedral tiles were displayed. leaving gaps and holes. Below, the same zonotile, with an equatorial region of the zonohedra displayed. Note the truncated icosidodecahedron (an Archimedean solid) in the center.

Download the 3D Zonotile notebook (needs Stuffit Expander to decompress)

Above: A hidden-detail-removed, cube-first
projection of the truncated 24-cell, {3,4,3}. Note that this induces
a tiling of a zonohedron by smaller zonohedra: a 3D zonotile.
Created using my *Regular Polytopes* notebook, rendered in
POV-Ray.

A vertex-first projection of the regular four-dimensional star polytope, {3,3,5/2}, into a 3-space.

Download the Regular Polytopes notebook (needs Stuffit Expander to decompress)

A "star" polar zonohedron, in which the usual cyclic ordering which characterizes a convex polar zonohedron (or convex regular polygon) was abandoned, and the ordering which characterizes a regular star polygon substituted. When the ordering is such that a compound of polygons would occur, instead of a star polygon, "bead" zonohedra arise.

Download the StarBead notebook (needs Stuffit Expander to decompress)

Download the Zonohedral Completion notebook (needs Stuffit Expander to decompress)

Download the 2D Zonotiles notebook (needs Stuffit Expander to decompress)

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