User:Tomruen/regular polychora
From Wikipedia, the free encyclopedia
Contents |
[edit] NOTE: Most of this information has been transfered to tables in List of regular polytopes, just a bit more compactly, with only one name each.
In addition, I extend the 3D, and 5D sections similarly there as tables. Tom Ruen 02:02, 13 January 2006 (UTC)
Regular polychora are defined by the Schläfli symbol symbol {p,q,r}
{p,q,r} can be interpreted as:
- {p}=Polygon face type (v/f)
- {q,r}=Polyhedral vertex figure
- {p,q}=Polyhedral cell type
- {r}=Polygonal edge figure (c/e)
- {r,q,p} = Dual polychora
[edit] Existence
The existence of a regular polychora {p,q,r} has three constraints:
- The cell {p,q} must be a regular polyhedron;
- The vertex figure {q,r} must be a regular polyhedron;
- The numbers p,q,r must satisfy the inequality:
- sin(π/p) sin(π/r) ≥ cos(π/q)
Constraints (1) and (2) have 9 regular solids available:
- 5 Platonic solids: {3,3} {3,4} {3,5} {4,3} {5,3}
- 4 Kepler-Poinsot_solids: {5/2,3} {5/2,5} {3,5/2} {5,5/2}
These 9 regular solids, along with a permutation search with constraint (3) enumerate the 16 existing regular polychora, of which 6 are convex and 10 are nonconvex.
See also: List_of_regular_polytopes
[edit] Table
Here are 6 convex regular polychora and 10 nonconvex forms listed below, with their component counts and forms.
The Euler characteristic for convex regular polychora is: chi=V+F-E-C=0. Nonconvex polychora may or may not follow this. (See V+E-F-C column below)
| # | Name |
Schläfli symbol {p,q,r} |
V {q,r} |
E {r} |
F {p} |
C {p,q} |
V+E-F-C | Dual {r,q,p} |
|---|---|---|---|---|---|---|---|---|
| C1 | 5-cell Pentachoron Simplex "Pen" |
{3,3,3} | 5 {3,3} |
10 {3} |
10 {3} |
5 {3,3} |
0 | (C1) Self-dual {3,3,3} |
| C2 | 8-cell Tesseract Hypercube "Tes" |
{4,3,3} | 16 {3,3} |
32 {3} |
24 {4} |
8 {4,3} |
0 | (C3) 16-cell {3,4,4} |
| C3 | 16-cell Hexadecachoron Octaplex "Hex" |
{3,3,4} | 8 {3,4} |
24 {4} |
32 {3} |
16 {3,3} |
0 | (C2) 8-cell {3,3,4} |
| C4 | 24-cell Icositetrachoron Orthocube "Ico" |
{3,4,3} | 24 {4,3} |
96 {3} |
96 {3} |
24 {3,4} |
0 | (C4) Self-dual {3,4,3} |
| C5 | 120-cell Hecatonicosachoron Dodecaplex "Hi" |
{5,3,3} | 600 {3,3} |
1200 {3} |
720 {5} |
120 {5,3} |
0 | {C6) 600-cell {3,3,5} |
| C6 | 600-cell Hexacosichoron Tetraplex "Ex" |
{3,3,5} | 120 {3,5} |
720 {5} |
1200 {3} |
600 {3,3} |
0 | (C5) 120-cell {5,3,3} |
| N1 | Great grand stellated 120-cell Great grand stellated hecatonicosachoron "Gogishi" |
{5/2,3,3} | 600 {3,3} |
1200 {3} |
720 {5/2} |
120 {5/2,3} |
0 | (N2) Grand 600-cell {3,3,5/2} |
| N2 | Grand 600-cell Grand hexacosichoron "Gax" |
{3,3,5/2} | 120 {3,5/2} |
720 {5/2} |
1200 {3} |
600 {3,3} |
0 | (N1) Great grand stellated 120-cell {5/2, 3, 3} |
| N3 | Great stellated 120-cell Great stellated hecatonicosachoron "Gishi" |
{5/2,3,5} | 120 {3,5} |
720 {5} |
720 {5/2} |
120 {5/2,3} |
0 | (N4) Grand 120-cell {5, 3, 5/2} |
| N4 | Grand 120-cell Grand hecatonicosachoron "Gahi" |
{5,3,5/2} | 120 {3,5/2} |
720 {5/2} |
720 {5} |
120 {5,3} |
0 | (N3) Great stellated 120-cell {5/2, 3, 5} |
| N5 | Grand stellated 120-cell Grand stellated hecatonicosachoron "Gashi" |
{5/2,5,5/2} | 120 {5,5/2} |
720 {5/2} |
720 {5/2} |
120 {5/2,5} |
0 | (N5) Self-dual {5/2,5,5/2} |
| N6 | Small stellated 120-cell Small stellated hecatonicosachoron "Sishi" |
{5/2,5,3} | 120 {5,3} |
1200 {3} |
720 {5/2} |
120 {5/2,5} |
-480 | (N7) Icosahedral 120-cell {3, 5, 5/2} |
| N7 | Icosahedral 120-cell Faceted hexacosichoron "Fix" |
{3,5,5/2} | 120 {5,5/2} |
720 {5/2} |
1200 {3} |
120 {3,5} |
480 | (N6) Small stellated 120-cell {5/2, 5, 3} |
| N8 | Great icosahedral 120-cell Great faceted hexacosichoron "Gofix" |
{3,5/2,5} | 120 {5/2,5} |
720 {5} |
1200 {3} |
120 {3,5/2} |
480 | (N9) Great grand 120-cell {5, 5/2, 3} |
| N9 | Great grand 120-cell Great grand hecatonicosachoron "Gaghi" |
{5,5/2,3} | 120 {5/2,3} |
1200 {3} |
720 {5} |
120 {5, 5/2} |
-480 | (N8) Great icosahedral 120-cell {3, 5/2, 5} |
| N10 | Great 120-cell Great hecatonicosachoron "Gohi" |
{5,5/2,5} | 120 {5/2,5} |
720 {5} |
720 {5} |
120 {5, 5/2} |
0 | (N10) Self-dual {5, 5/2, 5} |
[edit] Table2
| Name |
Image | Schläfli {p,q,r} Coxeter-Dynkin |
Cells {p,q} |
Faces {p} |
Edges {r} |
Vertices {q,r} |
χ | Symmetry group | Dual {r,q,p} |
|---|---|---|---|---|---|---|---|---|---|
| Small stellated 120-cell |  |
{5/2,5,3}     |
120 {5/2,5}  |
720 {5/2}  |
1200 {3}  |
120 {5,3}  |
-480 | H4 | Icosahedral 120-cell |
| Great 120-cell |  |
{5,5/2,5} |
120 {5,5/2}  |
720 {5}  |
720 {5} |
120 {5/2,5} |
0 | H4 | Self-dual |
| Great stellated 120-cell |  |
{5/2,3,5} |
120 {5/2,3}  |
720 {5/2} |
720 {5} |
120 {3,5}  |
0 | H4 | Grand 120-cell |
| Grand 120-cell | |
{5,3,5/2} |
120 {5,3} |
720 {5} |
720 {5/2} |
120 {3,5/2}  |
0 | H4 | 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 | H4 | Self-dual |
| Great grand 120-cell | |
{5,5/2,3} |
120 {5,5/2} |
720 {5} |
1200 {3} |
120 {5/2,3} |
-480 | H4 | Great icosahedral 120-cell |
| Great grand stellated 120-cell |  |
{5/2,3,3} |
120 {5/2,3} |
720 {5/2} |
1200 {3} |
600 {3,3}  |
0 | H4 | Grand 600-cell |
| Icosahedral 120-cell | |
{3,5,5/2} |
120 {3,5} |
1200 {3} |
720 {5/2} |
120 {5,5/2} |
480 | H4 | 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 | H4 | Great grand 120-cell |
| Grand 600-cell | |
{3,3,5/2} |
600 {3,3} |
1200 {3} |
720 {5/2} |
120 {3,5/2} |
0 | H4 | Great grand stellated 120-cell |
