User:Tomruen/Pentachoron family
From Wikipedia, the free encyclopedia
[edit] The A4 [3,3,3] family - (5-cell)
The pictures are draw as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at pos.0 shown solid.
| Name | Picture | Coxeter-Dynkin and Schläfli symbols |
Face counts by location | Cell counts by location | Element counts | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3,2 | 3,1 | 3,0 | 2,1 | 2,0 | 1,0 | 3 (5) |
2 (10) |
1 (10) |
0 (5) |
Cells | Faces | Edges | Vertices | |||
| 5-cell |  |
   {3,3,3} |
 {3} |
 (3.3.3) |
5 | 10 | 10 | 5 | ||||||||
| truncated 5-cell |  |
t0,1{3,3,3} |
 {6} |
 {3} |
 (3.6.6) |
(3.3.3) |
10 | 30 | 40 | 20 | ||||||
| rectified 5-cell |  |
t1{3,3,3} |
{3} |
{3} |
 (3.3.3.3) |
(3.3.3) |
10 | 30 | 30 | 10 | ||||||
| cantellated 5-cell |  |
t0,2{3,3,3} |
{3} |
 {4} |
{3} |
{3} |
 (3.4.3.4) |
 (3.4.4) |
(3.3.3.3) |
20 | 80 | 90 | 30 | |||
| cantitruncated 5-cell |  |
t0,1,2{3,3,3} |
{6} |
{4} |
{6} |
{3} |
 (4.6.6) |
(3.4.4) |
(3.6.6) |
20 | 80 | 120 | 60 | |||
| runcitruncated 5-cell |  |
t0,1,3{3,3,3} |
{6} |
{3} |
{4} |
{4} |
{3} |
(3.6.6) |
 (4.4.6) |
(3.4.4) |
(3.4.3.4) |
30 | 120 | 150 | 60 | |
| *bitruncated 5-cell |  |
t1,2{3,3,3} |
{3} |
{6} |
{3} |
(3.6.6) |
(3.6.6) |
10 | 40 | 60 | 30 | |||||
| *runcinated 5-cell |  |
t0,3{3,3,3} |
{3} |
{4} |
{3} |
(3.3.3) |
(3.4.4) |
(3.4.4) |
(3.3.3) |
30 | 70 | 60 | 20 | |||
| *omnitruncated 5-cell |  |
t0,1,2,3{3,3,3} |
{6} |
{4} |
{6} |
{4} |
{4} |
{6} |
(4.6.6) |
(4.4.6) |
(4.4.6) |
(4.6.6) |
30 | 150 | 240 | 120 |
The 5-cell has diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.
The three forms marked with an asterisk have the higher extended pentachoric symmetry, of order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.
