Schläfli-Hess polychoron
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In geometry, Schläfli-Hess polychora are the complete set of 10 regular (nonconvex) star polychora (4-polytopes).
They are an extension to the 6 regular convex 4-polytopes, allowing star polygons as faces, edge figures and vertex figures. Nine are based on the vertices of the 120-cell, {5,3,3}, and one on the 600-cell, {3,3,5}.
They are 4-dimensional analogues to the four regular (nonconvex) star polyhedra, the Kepler-Poinsot solids, and in fact are constructed with cells made from these.
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[edit] 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).
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 the names for the Kepler-Poinsot solids. He offered these operational definitions:
- stellation - replaces edges by longer edges in same lines.
- greatening - replaces the faces by large ones in same planes.
- aggrandizement - replaces the cells by large ones in same 3-spaces.
[edit] Table of elements
| Name |
Schläfli Symbol {p,q,r} |
Cells {p,q} |
Faces {p} |
Edges {r} |
Vertices {q,r} |
χ | Dual {r,q,p} |
|---|---|---|---|---|---|---|---|
| Great grand stellated 120-cell | {5/2,3,3} | 120 {5/2,3} |
720 {5/2} |
1200 {3} |
600 {3,3} |
0 | Grand 600-cell |
| Grand 600-cell | {3,3,5/2} | 600 {3,3} |
1200 {3} |
720 {5/2} |
120 {3,5/2} |
0 | Great grand stellated 120-cell |
| Great stellated 120-cell | {5/2,3,5} | 120 {5/2,3} |
720 {5/2} |
720 {5} |
120 {3,5} |
0 | Grand 120-cell |
| Grand 120-cell | {5,3,5/2} | 120 {5,3} |
720 {5} |
720 {5/2} |
120 {3,5/2} |
0 | Great stellated 120-cell |
| Grand stellated 120-cell | {5/2,5,5/2} | 120 {5/2,5} |
720 {5/2} |
720 {5/2} |
120 {5,5/2} |
0 | Self-dual |
| Small stellated 120-cell | {5/2,5,3} | 120 {5/2,5} |
720 {5/2} |
1200 {3} |
120 {5,3} |
-480 | Icosahedral 120-cell |
| Icosahedral 120-cell | {3,5,5/2} | 120 {3,5} |
1200 {3} |
720 {5/2} |
120 {5,5/2} |
480 | Small stellated 120-cell |
| Great icosahedral 120-cell | {3,5/2,5} | 120 {3,5/2} |
1200 {3} |
720 {5} |
120 {5/2,5} |
480 | Great grand 120-cell |
| Great grand 120-cell | {5,5/2,3} | 120 {5,5/2} |
720 {5} |
1200 {3} |
120 {5/2,3} |
-480 | Great icosahedral 120-cell |
| Great 120-cell | {5,5/2,5} | 120 {5,5/2} |
720 {5} |
720 {5} |
120 {5/2,5} |
0 | Self-dual |
[edit] See also
[edit] External links
[edit] 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 [1].
- H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
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