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)
Spherical
editThere 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] |
Euclidean
editThere 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
- B~2xB~2: [4,4]x[4,4] = [4,3,3,4] = (Same as tesseractic honeycomb family)
- B~2xH~2: [4,4]x[6,3]
- H~2xH~2: [6,3]x[6,3]
- A~2xB~2: [3[3]]]x[4,4] (Same forms as [6,3]x[4,4])
- A~2xH~2: [3[3]]]x[6,3] (Same forms as [6,3]x[6,3])
- A~2xA~2: [3[3]]]x[3[3]] (Same forms as [6,3]x[6,3])
Prismatic forms
- B~3xI~1: [4,3,4]x[∞]
- D~3xI~1: [4,31,1]x[∞]
- A~3xI~1: [3[4]]x[∞]
Hyperbolic
edit1 | [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)] |
Linear Coxeter graphs
editThere 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 |
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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 |
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5 | Bitruncated |
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6 | Cantellated |
s-prism-wedge |
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7 | Bicantellated |
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8 | Runcinated |
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9 | Stericated |
{q,r}-{r,q} antiprism |
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10 | Cantitruncated |
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11 | Bicantitruncated |
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12 | Runcitruncated |
wedge-pyramid |
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13 | Steritruncated |
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14 | Runcicantellated |
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15 | Stericantellated |
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16 | Runcicantitruncated |
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17 | Stericantitruncated |
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18 | Steriruncitruncated |
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19 | Omnitruncated |
Irr. 5-simplex |
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20 | Alternated regular |
t1{3,3,p} | t1{3,3,3} |
t1{3,3,4} |
Bifurcating Coxeter graphs
editThere 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 | ||||||
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23 |
Trifurcating Coxeter graphs
editThere are 9 forms:
Operation Coxeter-Dynkin |
Euclidean |
---|---|
Coxeter group | [31,1,1,1] |
Cyclic Coxeter graphs
editThere 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 | ||||
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7 |