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Chiral Symetries In Four Space.

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Posted by Brian Kirk Parquette on January 14, 2001 21:35:26 UTC

http://members.aol.com/Polycell/uniform.htmlAbove: A three-dimensional section through the Great Prismosaurus—a uniform star-polychoron whose cells are 720 highly intersecting congruent pentagonal prisms.
The section is orthogonal to an icosahedral symmetry axis by a realm passing 15% of the way out from the center toward a vertex. This polychoron was discovered
independently by me in 1997 and (somewhat earlier) by Jonathan Bowers of Tyler, Texas, and this website is its “public debut.” It is one of 8186 currently known uniform
polychora (this count will change as Jonathan and I continue searching; we recently lost four figures to a simple miscount), many of which are simpler but most of which
are much more complicated. Jonathan has discovered more than 8000 of these during the past decade, working—in the tradition of Ludwig Schläfli, Thorold Gosset, and
Alicia Boole Stott—independently and in virtual isolation.
[Regarding Jonathan Bowers, from December 12, 1999 through January 2, 2000 I added his alternative names for the convex uniform polychora to the tables. See the
Nomenclature section for details.]
The Great Prismosaurus has 120 vertices, 1200 edges, 1800 square faces, and 720 pentagonal faces to connect its 720 pentagonal-prism cells together. Sixty cells
come together at each vertex. The squares also define a uniform compound of 75 tesseracts, while the pentagons are also the faces of the regular star-polychoron
{5,5/2,3}, the great grand hecatonicosachoron. Both of these figures of course have the same sets of vertices and edges as the Great Prismosaurus.
Each edge is surrounded alike by nine prisms, which makes the Great Prismosaurus not only uniform but quasi-regular; and since it has only one kind of cell, it is also
isochoric. If you look closely at the figure, you will see that nine intersecting face planes meet at every corner—the sections of the nine prisms that surround each edge in
the polychoron. The prisms are quite large relative to the size of the polychoron itself, and they pass quite close to the center. The dihedral angle between two prisms at a
common pentagonal face is 36°, and if you travel from prism to prism across the parallel pentagonal bases, you’ll pass through five before returning to your starting point,
after going twice around the polychoron. If you travel from prism to prism around a common edge, you’ll circle the edge four times in passing through the nine prisms that
share that edge. In the section shown, all the vertices are sections of various edges, and all the faces are sections of various cells: the sectioning realm bypasses the
polychoron’s own vertices.
I estimate that this section of the Great Prismosaurus (Greekish name: pentagonal-prismatic heptacosiicosachoron) has on the order of 400,000 facelets, which a
polyhedron model-maker would have to cut out and paste together to make a paper model. Many are very small—at or below the limits of the picture resolution for a
picture this size.
The Great Prismosaurus has a simpler conjugate: a uniform polychoron whose cells are 720 pentagrammatic prisms. It is the simpler of the two polychora because it has
far fewer self-intersections, so we call it the Small Prismosaurus; its Greekish name would be pentagrammatic-prismatic heptacosiicosachoron. It has the same
vertices, edges, and square faces as the Great Prismosaurus, but the pentagons are discarded and replaced by the pentagrams of the regular star-polychoron {5/2,5,3},
the stellated hecatonicosachoron (which has the same vertices and edges as the great grand hecatonicosachoron and both Prismosauri).
The most recent additions to the roster of uniform polychora are the Prismosaurus Hybrids: a sequence of six prismlike uniform polychora that fit in between the Small
and Great Prismosaurus. In the first Hybrid, 120 pentagrammatic prisms (ten girdles of twelve) of the Small Prismosaurus are exchanged for 120 pentagonal prisms of the
Great Prismosaurus. In the second, another set of 120 prisms is exchanged. There are two ways to uniformly exchange three sets of 120, there is one way to exchange
four sets of 120, and one final way to exchange five sets. If you exchange all six sets of 120 prisms of the Small Prismosaurus, you get the Great Prismosaurus, of course.
The Hybrids are less symmetric than the Prismosauri themselves, but they are nevertheless still uniform. Each Hybrid is chiral—it comes in left-handed and right-handed
forms—as is its vertex figure. The Hybrids are six of ten known polychora I call swirlprisms (and Norman Johnson calls chiroprisms) that I discovered during August and
September 1999. They are the only known chiral uniform polychora that also have chiral vertex figures. We expect to turn up more of these as we get the hang of working
with chiral symmetries in four-space. To see five views of each Hybrid Prismosaurus vertex figure, constructed by Jonathan Bowers, click here: Hybrid swirlprism vertex
figures. To see pictures of the chiral vertex figures of the first four swirlprisms, click here: Swirlprism vertex figures.
The Great Prismosaurus picture above was produced by Bruce L. Chilton of Tonawanda, New York using his four-dimensional sectioning program and the Imagine
three-dimensional graphic display system. He is the one who named it the Prismosaurus. The original colors were inverted for this composition by Joel McVey of San
Diego, California.
I added this picture to this website August 18, 1998. In succeeding months, I will begin to post information on the other nonconvex uniform polychora, as well as more basic
and introductory matter related to the geometry of four-dimensional space. Meanwhile, information on all the convex uniform polychora may be found below. Ultimately,
I’d like to create an illustrated Web page for each different uniform polychoron, convex and nonconvex, but this would take about 15 years’ work and require about a
gigabyte of disk space!
Name and Location:
George Olshevsky
3305 Adams Avenue #221
San Diego, California 92116

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