Schläfli-Hess polychoron

In four dimensional geometry, Schläfli-Hess polychora are the complete set of 10 regular self-intersecting star polychora (Four dimensional polytopes). They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. They are all represented by a Schläfli symbol {p,q,r} including pentagrammic ("5/2") elements.

Like the set of regular nonconvex polyhedra, the Kepler-Poinsot polyhedra, the Schläfli-Hess polychora, are the set of regular nonconvex polychora. Allowing for regular star polygons as faces, "edge figures" and vertex figures, these 10 polychora add to the set of six regular convex 4-polytopes. All may be derived as stellations of the 120-cell "{5,3,3}" or the 600-cell "{3,3,5}".

History

Four of the were found by Ludwig Schläfli while the other six were skipped because he would not allow forms that failed the Euler characteristic on cells or vertex figures (For zero-hole toruses: F+V-E=2). That excludes cells and vertex figures as {5,5/2}, and {5/2,5}.

Edmund Hess (1843-1903) published the complete list in his 1883 German book "Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder".

Their names given here were given by John Horton Conway, extending Cayley's names for the Kepler-Poinsot solids, along with "stellated" and "great", he adds a "grand" modifier.
He offered these operational definitions:
#stellation - replaces edges by longer edges in same lines. (Example: a pentagon stellates into a pentagram)
#greatening - replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron)
#aggrandizement - replaces the cells by large ones in same 3-spaces.

Table of elements

Note:
* There are 2 unique vertex arrangements, matching the 120-cell, and 600-cell.
* There are 4 unique edge arrangements which are shown as "wireframes" orthographic projections.
* There are 7 unique face arrangements shown as "solids" (face-colored) orthographic projections.

The cell polyhedra, face polygons are given by their Schläfli symbol. Also the regular "polygonal edge figures", regular and "polyhedral vertex figures" are given similarly.

Existence

The existence of a regular polychoron $\{p,q,r\}$ is constrained by the existence of the regular polyhedra $\{p,q\},\; \{q,r\}$ and a dihedral angle contraint:
*

The 10 star polytopes above are the only solutions that exist.

There are four nonconvex Schläfli symbols {p, q,r} that have valid cells {p, q} and vertex figures {q, r}, and pass the dihedral test, but fail to produce finite figures: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}.

See also

* List of regular polytopes
* Convex regular 4-polytope - Set of convex regular polychoron
* Kepler-Poinsot solids - regular star polyhedron
* Star polygon - regular star polygons

References

* Edmund Hess, (1883) "Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder" [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001] .
* Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 10) H.S.M. Coxeter, "Star Polytopes and the Schlafli Function f(α,β,γ)" [Elemente der Mathematik 44 (2) (1989) 25-36]
*Coxeter, "Regular Polytopes", 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q,r} in four dimensions, pp. 292-293)
*H. S. M. Coxeter, "Regular Complex Polytopes", 2nd. ed., Cambridge University Press 1991. ISBN 978-0521394901. [http://www.amazon.com/dp/0521394902]
* Peter McMullen and Egon Schulte, "Abstract Regular Polytopes", 2002, [http://assets.cambridge.org/052181/4960/sample/0521814960ws.pdf PDF]
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, "The Symmetry of Things" 2008, ISBN 978-1-56881-220-5 (Chapter 24, Regular Star-polytopes, pp. 404-408)

External links

*Mathworld | urlname=Kepler-PoinsotSolid | title=10 "Kepler-Poinsot solids" in 4-dimensions
*Mathworld | urlname=Polychoron | title=Polychoron
*
** GlossaryForHyperspace | anchor=Hexacosichoron | title= Hexacosichoron
** GlossaryForHyperspace | anchor=Stellation | title=Stellation
** GlossaryForHyperspace | anchor=Greatening | title=Greatening
** GlossaryForHyperspace | anchor=Aggrandizement | title=Aggrandizement
* [http://members.aol.com/hedrondude/regulars.html Jonathan Bowers, 16 regular polychora]
* [http://www.polytope.net/hedrondude/regulars.htm Regular polychora]
* [http://mathforum.org/library/drmath/view/54786.html Discussion on names]
* [http://www.mathematik.uni-regensburg.de/Goette/sterne Reguläre Polytope]
* [http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora]
* [http://www.software3d.com/Stella.php#stella4D Stella4D] Stella (software) produces interactive views of all 1849 known uniform polychora including the 64 convex forms and the infinite prismatic families. Was used to create images for this page.