Vertex figures (as Schlegel diagrams) for uniform polyterons, uniform honeycombs (Euclidean and hyperbolic). (Excluding prismatic forms, and nonwythoffian forms)
Tables are expanded for finite and infinite forms (spherical/Euclidean/hyperbolic) for completeness, not that I expect ever to include all of the hyperbolic forms! (Compare to 4-polytopes: Talk:Vertex figure/polychoron)
There are three fundamental affine Coxeter groups that generate regular and uniform tessellations on the 3-sphere:
# | Coxeter group | Coxeter graph | |
---|---|---|---|
1 | A5 | [34 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | B5 | [4,33 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | D5 | [32,1,1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
In addition there are prismatic groups:
Uniform prismatic forms:
# | Coxeter groups | Coxeter graph | |
---|---|---|---|
1 | A4 × A1 | [3,3,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | B4 × A1 | [4,3,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | F4 × A1 | [3,4,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | H4 × A1 | [5,3,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | D4 × A1 | [31,1,1] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Uniform duoprism prismatic forms:
Coxeter groups | Coxeter graph | |
---|---|---|
I2(p) × I2(q) × A1 | [p] × [q] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Uniform duoprismatic forms:
# | Coxeter groups | Coxeter graph | |
---|---|---|---|
1 | A3 × I2(p) | [3,3] × [p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | B3 × I2(p) | [4,3] × [p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3. | H3 × I2(p) | [5,3] × [p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4-space:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A~4 | [(3,3,3,3,3)] | ![]() ![]() ![]() ![]() ![]() |
2 | B~4 | [4,3,3,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | C~4 | [4,3,31,1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | D~4 | [31,1,1,1 | ![]() ![]() ![]() ![]() ![]() |
5 | F~4 | [3,4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
In addition there are prismatic groups:
Duoprismatic forms
Prismatic forms
1 | [5,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|---|---|
2 | [5,3,3,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | [5,3,3,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | [5,3,31,1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | [(4,3,3,3,3)] | ![]() ![]() ![]() ![]() ![]() ![]() |
There are 31 truncation forms for each group, or 19 subgrouped as half-families as given below (with 7 overlapped).
Summary chart: File:Uniform polyteron vertex figure chart.png
# | Operation Coxeter-Dynkin |
General {p,q,r,s} |
Spherical | Euclidean | Hyperbolic | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5-simplex [3,3,3,3] ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5-cube [4,3,3,3] ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5-orthoplex [3,3,3,4] ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[4,3,3,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,4,3,3]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,3,4,3]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,3,3,5]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,3,3]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[4,3,3,5]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,3,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,3,5]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||
1 | Regular![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{q,r,s}:(p) |
![]() {3,3,3}:(3) |
![]() {3,3,3}:(4) |
![]() {3,3,4}:(3) |
![]() {3,3,4}:(4) |
![]() {4,3,3}:(3) |
![]() {3,4,3}:(3) |
![]() {3,3,5}:(3) |
![]() {3,3,3}:(5) |
![]() {3,3,5}:(4) |
![]() {3,3,4}:(5) |
![]() {3,3,5}:(5) |
2 | Rectified![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {r,s}-prism |
![]() |
![]() |
![]() |
![]() | |||||||
3 | Birectified![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() p-s duoprism |
![]() 3-3 duoprism |
![]() 3-4 duoprism |
![]() 3-4 duoprism |
![]() 4-4 duoprism |
![]() 3-3 duoprism |
![]() 3-3 duoprism | |||||
4 | Truncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {r,s}-pyramid |
![]() |
![]() |
![]() |
![]() | |||||||
5 | Bitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
6 | Cantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() s-prism-wedge |
![]() |
![]() |
![]() |
![]() | |||||||
7 | Bicantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
8 | Runcinated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||||
9 | Stericated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {q,r}-{r,q} antiprism |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
10 | Cantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
11 | Bicantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
12 | Runcitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() wedge-pyramid |
![]() |
![]() |
![]() | ||||||||
13 | Steritruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
14 | Runcicantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
15 | Stericantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
16 | Runcicantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
17 | Stericantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
18 | Steriruncitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
19 | Omnitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Irr. 5-simplex |
![]() |
![]() |
![]() |
||||||||
20 | Alternated regular![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
t1{3,3,p} |
![]() t1{3,3,3} |
![]() t1{3,3,4} |
There are 23 forms from each family, with 15 repeated from the linear [4,3,3,s] families above.
# | Operation Coxeter-Dynkin |
Linear equiv | General | Spherical | Euclidean | Hyperbolic |
---|---|---|---|---|---|---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[s,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[4,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
t1{3,3,s} |
![]() t1{3,3,3} |
t1{3,3,4} | t1{3,3,5} |
2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
|||
3 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
5 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() | |||
6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | ||||
7 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | ||||
8 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
9 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
10 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
11 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
12 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
13 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | ||||
14 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||
15 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
16 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
17 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||
18 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
19 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
20 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
21 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||
22 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
23 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
There are 9 forms:
Operation Coxeter-Dynkin |
Euclidean |
---|---|
Coxeter group | [31,1,1,1![]() ![]() ![]() ![]() ![]() |
There are 7 forms in the first cycle family, and 19 forms in the second cyclic family:
# | General | Euclidean | Hyperbolic | |
---|---|---|---|---|
[(p,3,3,3,3)]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[(3,3,3,3,3)]![]() ![]() ![]() ![]() ![]() |
[(4,3,3,3,3)]![]() ![]() ![]() ![]() ![]() ![]() | ||
1 | ![]() ![]() ![]() ![]() ![]() | |||
2 | ![]() ![]() ![]() ![]() ![]() | |||
3 | ![]() ![]() ![]() ![]() ![]() | |||
4 | ![]() ![]() ![]() ![]() ![]() | |||
5 | ![]() ![]() ![]() ![]() ![]() | |||
6 | ![]() ![]() ![]() ![]() ![]() | |||
7 | ![]() ![]() ![]() ![]() ![]() |
![]() |
Vertex figures (as Schlegel diagrams) for uniform polyterons, uniform honeycombs (Euclidean and hyperbolic). (Excluding prismatic forms, and nonwythoffian forms)
Tables are expanded for finite and infinite forms (spherical/Euclidean/hyperbolic) for completeness, not that I expect ever to include all of the hyperbolic forms! (Compare to 4-polytopes: Talk:Vertex figure/polychoron)
There are three fundamental affine Coxeter groups that generate regular and uniform tessellations on the 3-sphere:
# | Coxeter group | Coxeter graph | |
---|---|---|---|
1 | A5 | [34 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | B5 | [4,33 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | D5 | [32,1,1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
In addition there are prismatic groups:
Uniform prismatic forms:
# | Coxeter groups | Coxeter graph | |
---|---|---|---|
1 | A4 × A1 | [3,3,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | B4 × A1 | [4,3,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | F4 × A1 | [3,4,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | H4 × A1 | [5,3,3] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | D4 × A1 | [31,1,1] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Uniform duoprism prismatic forms:
Coxeter groups | Coxeter graph | |
---|---|---|
I2(p) × I2(q) × A1 | [p] × [q] × [ ] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Uniform duoprismatic forms:
# | Coxeter groups | Coxeter graph | |
---|---|---|---|
1 | A3 × I2(p) | [3,3] × [p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | B3 × I2(p) | [4,3] × [p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3. | H3 × I2(p) | [5,3] × [p] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4-space:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A~4 | [(3,3,3,3,3)] | ![]() ![]() ![]() ![]() ![]() |
2 | B~4 | [4,3,3,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | C~4 | [4,3,31,1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | D~4 | [31,1,1,1 | ![]() ![]() ![]() ![]() ![]() |
5 | F~4 | [3,4,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
In addition there are prismatic groups:
Duoprismatic forms
Prismatic forms
1 | [5,3,3,3] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|---|---|
2 | [5,3,3,4] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | [5,3,3,5] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | [5,3,31,1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | [(4,3,3,3,3)] | ![]() ![]() ![]() ![]() ![]() ![]() |
There are 31 truncation forms for each group, or 19 subgrouped as half-families as given below (with 7 overlapped).
Summary chart: File:Uniform polyteron vertex figure chart.png
# | Operation Coxeter-Dynkin |
General {p,q,r,s} |
Spherical | Euclidean | Hyperbolic | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5-simplex [3,3,3,3] ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5-cube [4,3,3,3] ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5-orthoplex [3,3,3,4] ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[4,3,3,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,4,3,3]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,3,4,3]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,3,3,5]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,3,3]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[4,3,3,5]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,3,4]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,3,5]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||
1 | Regular![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{q,r,s}:(p) |
![]() {3,3,3}:(3) |
![]() {3,3,3}:(4) |
![]() {3,3,4}:(3) |
![]() {3,3,4}:(4) |
![]() {4,3,3}:(3) |
![]() {3,4,3}:(3) |
![]() {3,3,5}:(3) |
![]() {3,3,3}:(5) |
![]() {3,3,5}:(4) |
![]() {3,3,4}:(5) |
![]() {3,3,5}:(5) |
2 | Rectified![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {r,s}-prism |
![]() |
![]() |
![]() |
![]() | |||||||
3 | Birectified![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() p-s duoprism |
![]() 3-3 duoprism |
![]() 3-4 duoprism |
![]() 3-4 duoprism |
![]() 4-4 duoprism |
![]() 3-3 duoprism |
![]() 3-3 duoprism | |||||
4 | Truncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {r,s}-pyramid |
![]() |
![]() |
![]() |
![]() | |||||||
5 | Bitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
6 | Cantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() s-prism-wedge |
![]() |
![]() |
![]() |
![]() | |||||||
7 | Bicantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
8 | Runcinated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||||
9 | Stericated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {q,r}-{r,q} antiprism |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
10 | Cantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
11 | Bicantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
12 | Runcitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() wedge-pyramid |
![]() |
![]() |
![]() | ||||||||
13 | Steritruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
14 | Runcicantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
15 | Stericantellated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
16 | Runcicantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
17 | Stericantitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
18 | Steriruncitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() | ||||||||
19 | Omnitruncated![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Irr. 5-simplex |
![]() |
![]() |
![]() |
||||||||
20 | Alternated regular![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
t1{3,3,p} |
![]() t1{3,3,3} |
![]() t1{3,3,4} |
There are 23 forms from each family, with 15 repeated from the linear [4,3,3,s] families above.
# | Operation Coxeter-Dynkin |
Linear equiv | General | Spherical | Euclidean | Hyperbolic |
---|---|---|---|---|---|---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[s,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[3,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[4,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[5,3,31,1![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
t1{3,3,s} |
![]() t1{3,3,3} |
t1{3,3,4} | t1{3,3,5} |
2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
|||
3 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
5 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() | |||
6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | ||||
7 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | ||||
8 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
9 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
10 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
11 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
12 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | |||
13 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | ||||
14 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||
15 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
16 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
17 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||
18 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
19 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
20 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
21 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||
22 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
23 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
There are 9 forms:
Operation Coxeter-Dynkin |
Euclidean |
---|---|
Coxeter group | [31,1,1,1![]() ![]() ![]() ![]() ![]() |
There are 7 forms in the first cycle family, and 19 forms in the second cyclic family:
# | General | Euclidean | Hyperbolic | |
---|---|---|---|---|
[(p,3,3,3,3)]![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[(3,3,3,3,3)]![]() ![]() ![]() ![]() ![]() |
[(4,3,3,3,3)]![]() ![]() ![]() ![]() ![]() ![]() | ||
1 | ![]() ![]() ![]() ![]() ![]() | |||
2 | ![]() ![]() ![]() ![]() ![]() | |||
3 | ![]() ![]() ![]() ![]() ![]() | |||
4 | ![]() ![]() ![]() ![]() ![]() | |||
5 | ![]() ![]() ![]() ![]() ![]() | |||
6 | ![]() ![]() ![]() ![]() ![]() | |||
7 | ![]() ![]() ![]() ![]() ![]() |
![]() |