In eight-dimensional geometry, an **eight-dimensional polytope** or **8-polytope** is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

- Regular 8-polytopes
- Characteristics
- Uniform 8-polytopes by fundamental Coxeter groups
- Uniform prismatic forms
- The A8 family
- The B8 family
- The D8 family
- The E8 family
- Regular and uniform honeycombs
- Regular and uniform hyperbolic honeycombs
- References
- External links

A **uniform 8-polytope** is one which is vertex-transitive, and constructed from uniform 7-polytope facets.

Regular 8-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with **v** {p,q,r,s,t,u} 7-polytope facets around each peak.

There are exactly three such convex regular 8-polytopes:

- {3,3,3,3,3,3,3} - 8-simplex
- {4,3,3,3,3,3,3} - 8-cube
- {3,3,3,3,3,3,4} - 8-orthoplex

There are no nonconvex regular 8-polytopes.

The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.^{ [1] }

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.^{ [1] }

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.^{ [1] }

Uniform 8-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

# | Coxeter group | Forms | ||
---|---|---|---|---|

1 | A_{8} | [3^{7}] | 135 | |

2 | BC_{8} | [4,3^{6}] | 255 | |

3 | D_{8} | [3^{5,1,1}] | 191 (64 unique) | |

4 | E_{8} | [3^{4,2,1}] | 255 |

Selected regular and uniform 8-polytopes from each family include:

- Simplex family: A
_{8}[3^{7}] -- 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
- {3
^{7}} - 8-simplex or ennea-9-tope or enneazetton -

- {3

- 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
- Hypercube/orthoplex family: B
_{8}[4,3^{6}] -- 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
- {4,3
^{6}} - 8-cube or*octeract*- - {3
^{6},4} - 8-orthoplex or*octacross*-

- {4,3

- 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
- Demihypercube D
_{8}family: [3^{5,1,1}] -- 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
- {3,3
^{5,1}} - 8-demicube or*demiocteract*,**1**- ; also as h{4,3_{51}^{6}} . - {3,3,3,3,3,3
^{1,1}} - 8-orthoplex,**5**-_{11}

- {3,3

- 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
- E-polytope family E
_{8}family: [3^{4,1,1}] -- 255 uniform 8-polytopes as permutations of rings in the group diagram, including:
- {3,3,3,3,3
^{2,1}} - Thorold Gosset's semiregular**4**,_{21} - {3,3
^{4,2}} - the uniform**1**, ,_{42} - {3,3,3
^{4,1}} - the uniform**2**,_{41}

- {3,3,3,3,3

- 255 uniform 8-polytopes as permutations of rings in the group diagram, including:

There are many uniform prismatic families, including:

Uniform 8-polytope prism families | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

# | Coxeter group | Coxeter-Dynkin diagram | |||||||||

7+1 | |||||||||||

1 | A_{7}A_{1} | [3,3,3,3,3,3]×[ ] | |||||||||

2 | B_{7}A_{1} | [4,3,3,3,3,3]×[ ] | |||||||||

3 | D_{7}A_{1} | [3^{4,1,1}]×[ ] | |||||||||

4 | E_{7}A_{1} | [3^{3,2,1}]×[ ] | |||||||||

6+2 | |||||||||||

1 | A_{6}I_{2}(p) | [3,3,3,3,3]×[p] | |||||||||

2 | B_{6}I_{2}(p) | [4,3,3,3,3]×[p] | |||||||||

3 | D_{6}I_{2}(p) | [3^{3,1,1}]×[p] | |||||||||

4 | E_{6}I_{2}(p) | [3,3,3,3,3]×[p] | |||||||||

6+1+1 | |||||||||||

1 | A_{6}A_{1}A_{1} | [3,3,3,3,3]×[ ]x[ ] | |||||||||

2 | B_{6}A_{1}A_{1} | [4,3,3,3,3]×[ ]x[ ] | |||||||||

3 | D_{6}A_{1}A_{1} | [3^{3,1,1}]×[ ]x[ ] | |||||||||

4 | E_{6}A_{1}A_{1} | [3,3,3,3,3]×[ ]x[ ] | |||||||||

5+3 | |||||||||||

1 | A_{5}A_{3} | [3^{4}]×[3,3] | |||||||||

2 | B_{5}A_{3} | [4,3^{3}]×[3,3] | |||||||||

3 | D_{5}A_{3} | [3^{2,1,1}]×[3,3] | |||||||||

4 | A_{5}B_{3} | [3^{4}]×[4,3] | |||||||||

5 | B_{5}B_{3} | [4,3^{3}]×[4,3] | |||||||||

6 | D_{5}B_{3} | [3^{2,1,1}]×[4,3] | |||||||||

7 | A_{5}H_{3} | [3^{4}]×[5,3] | |||||||||

8 | B_{5}H_{3} | [4,3^{3}]×[5,3] | |||||||||

9 | D_{5}H_{3} | [3^{2,1,1}]×[5,3] | |||||||||

5+2+1 | |||||||||||

1 | A_{5}I_{2}(p)A_{1} | [3,3,3]×[p]×[ ] | |||||||||

2 | B_{5}I_{2}(p)A_{1} | [4,3,3]×[p]×[ ] | |||||||||

3 | D_{5}I_{2}(p)A_{1} | [3^{2,1,1}]×[p]×[ ] | |||||||||

5+1+1+1 | |||||||||||

1 | A_{5}A_{1}A_{1}A_{1} | [3,3,3]×[ ]×[ ]×[ ] | |||||||||

2 | B_{5}A_{1}A_{1}A_{1} | [4,3,3]×[ ]×[ ]×[ ] | |||||||||

3 | D_{5}A_{1}A_{1}A_{1} | [3^{2,1,1}]×[ ]×[ ]×[ ] | |||||||||

4+4 | |||||||||||

1 | A_{4}A_{4} | [3,3,3]×[3,3,3] | |||||||||

2 | B_{4}A_{4} | [4,3,3]×[3,3,3] | |||||||||

3 | D_{4}A_{4} | [3^{1,1,1}]×[3,3,3] | |||||||||

4 | F_{4}A_{4} | [3,4,3]×[3,3,3] | |||||||||

5 | H_{4}A_{4} | [5,3,3]×[3,3,3] | |||||||||

6 | B_{4}B_{4} | [4,3,3]×[4,3,3] | |||||||||

7 | D_{4}B_{4} | [3^{1,1,1}]×[4,3,3] | |||||||||

8 | F_{4}B_{4} | [3,4,3]×[4,3,3] | |||||||||

9 | H_{4}B_{4} | [5,3,3]×[4,3,3] | |||||||||

10 | D_{4}D_{4} | [3^{1,1,1}]×[3^{1,1,1}] | |||||||||

11 | F_{4}D_{4} | [3,4,3]×[3^{1,1,1}] | |||||||||

12 | H_{4}D_{4} | [5,3,3]×[3^{1,1,1}] | |||||||||

13 | F_{4}×F_{4} | [3,4,3]×[3,4,3] | |||||||||

14 | H_{4}×F_{4} | [5,3,3]×[3,4,3] | |||||||||

15 | H_{4}H_{4} | [5,3,3]×[5,3,3] | |||||||||

4+3+1 | |||||||||||

1 | A_{4}A_{3}A_{1} | [3,3,3]×[3,3]×[ ] | |||||||||

2 | A_{4}B_{3}A_{1} | [3,3,3]×[4,3]×[ ] | |||||||||

3 | A_{4}H_{3}A_{1} | [3,3,3]×[5,3]×[ ] | |||||||||

4 | B_{4}A_{3}A_{1} | [4,3,3]×[3,3]×[ ] | |||||||||

5 | B_{4}B_{3}A_{1} | [4,3,3]×[4,3]×[ ] | |||||||||

6 | B_{4}H_{3}A_{1} | [4,3,3]×[5,3]×[ ] | |||||||||

7 | H_{4}A_{3}A_{1} | [5,3,3]×[3,3]×[ ] | |||||||||

8 | H_{4}B_{3}A_{1} | [5,3,3]×[4,3]×[ ] | |||||||||

9 | H_{4}H_{3}A_{1} | [5,3,3]×[5,3]×[ ] | |||||||||

10 | F_{4}A_{3}A_{1} | [3,4,3]×[3,3]×[ ] | |||||||||

11 | F_{4}B_{3}A_{1} | [3,4,3]×[4,3]×[ ] | |||||||||

12 | F_{4}H_{3}A_{1} | [3,4,3]×[5,3]×[ ] | |||||||||

13 | D_{4}A_{3}A_{1} | [3^{1,1,1}]×[3,3]×[ ] | |||||||||

14 | D_{4}B_{3}A_{1} | [3^{1,1,1}]×[4,3]×[ ] | |||||||||

15 | D_{4}H_{3}A_{1} | [3^{1,1,1}]×[5,3]×[ ] | |||||||||

4+2+2 | |||||||||||

... | |||||||||||

4+2+1+1 | |||||||||||

... | |||||||||||

4+1+1+1+1 | |||||||||||

... | |||||||||||

3+3+2 | |||||||||||

1 | A_{3}A_{3}I_{2}(p) | [3,3]×[3,3]×[p] | |||||||||

2 | B_{3}A_{3}I_{2}(p) | [4,3]×[3,3]×[p] | |||||||||

3 | H_{3}A_{3}I_{2}(p) | [5,3]×[3,3]×[p] | |||||||||

4 | B_{3}B_{3}I_{2}(p) | [4,3]×[4,3]×[p] | |||||||||

5 | H_{3}B_{3}I_{2}(p) | [5,3]×[4,3]×[p] | |||||||||

6 | H_{3}H_{3}I_{2}(p) | [5,3]×[5,3]×[p] | |||||||||

3+3+1+1 | |||||||||||

1 | A_{3}^{2}A_{1}^{2} | [3,3]×[3,3]×[ ]×[ ] | |||||||||

2 | B_{3}A_{3}A_{1}^{2} | [4,3]×[3,3]×[ ]×[ ] | |||||||||

3 | H_{3}A_{3}A_{1}^{2} | [5,3]×[3,3]×[ ]×[ ] | |||||||||

4 | B_{3}B_{3}A_{1}^{2} | [4,3]×[4,3]×[ ]×[ ] | |||||||||

5 | H_{3}B_{3}A_{1}^{2} | [5,3]×[4,3]×[ ]×[ ] | |||||||||

6 | H_{3}H_{3}A_{1}^{2} | [5,3]×[5,3]×[ ]×[ ] | |||||||||

3+2+2+1 | |||||||||||

1 | A_{3}I_{2}(p)I_{2}(q)A_{1} | [3,3]×[p]×[q]×[ ] | |||||||||

2 | B_{3}I_{2}(p)I_{2}(q)A_{1} | [4,3]×[p]×[q]×[ ] | |||||||||

3 | H_{3}I_{2}(p)I_{2}(q)A_{1} | [5,3]×[p]×[q]×[ ] | |||||||||

3+2+1+1+1 | |||||||||||

1 | A_{3}I_{2}(p)A_{1}^{3} | [3,3]×[p]×[ ]x[ ]×[ ] | |||||||||

2 | B_{3}I_{2}(p)A_{1}^{3} | [4,3]×[p]×[ ]x[ ]×[ ] | |||||||||

3 | H_{3}I_{2}(p)A_{1}^{3} | [5,3]×[p]×[ ]x[ ]×[ ] | |||||||||

3+1+1+1+1+1 | |||||||||||

1 | A_{3}A_{1}^{5} | [3,3]×[ ]x[ ]×[ ]x[ ]×[ ] | |||||||||

2 | B_{3}A_{1}^{5} | [4,3]×[ ]x[ ]×[ ]x[ ]×[ ] | |||||||||

3 | H_{3}A_{1}^{5} | [5,3]×[ ]x[ ]×[ ]x[ ]×[ ] | |||||||||

2+2+2+2 | |||||||||||

1 | I_{2}(p)I_{2}(q)I_{2}(r)I_{2}(s) | [p]×[q]×[r]×[s] | |||||||||

2+2+2+1+1 | |||||||||||

1 | I_{2}(p)I_{2}(q)I_{2}(r)A_{1}^{2} | [p]×[q]×[r]×[ ]×[ ] | |||||||||

2+2+1+1+1+1 | |||||||||||

2 | I_{2}(p)I_{2}(q)A_{1}^{4} | [p]×[q]×[ ]×[ ]×[ ]×[ ] | |||||||||

2+1+1+1+1+1+1 | |||||||||||

1 | I_{2}(p)A_{1}^{6} | [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ] | |||||||||

1+1+1+1+1+1+1+1 | |||||||||||

1 | A_{1}^{8} | [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ] |

The A_{8} family has symmetry of order 362880 (9 factorial).

There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. (128+8-1 cases) These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.

See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes.

A_{8} uniform polytopes | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

# | Coxeter-Dynkin diagram | Truncation indices | Johnson name | Basepoint | Element counts | |||||||

7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | |||||

1 | t_{0} | 8-simplex (ene) | (0,0,0,0,0,0,0,0,1) | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | |

2 | t_{1} | Rectified 8-simplex (rene) | (0,0,0,0,0,0,0,1,1) | 18 | 108 | 336 | 630 | 576 | 588 | 252 | 36 | |

3 | t_{2} | Birectified 8-simplex (bene) | (0,0,0,0,0,0,1,1,1) | 18 | 144 | 588 | 1386 | 2016 | 1764 | 756 | 84 | |

4 | t_{3} | Trirectified 8-simplex (trene) | (0,0,0,0,0,1,1,1,1) | 1260 | 126 | |||||||

5 | t_{0,1} | Truncated 8-simplex (tene) | (0,0,0,0,0,0,0,1,2) | 288 | 72 | |||||||

6 | t_{0,2} | Cantellated 8-simplex | (0,0,0,0,0,0,1,1,2) | 1764 | 252 | |||||||

7 | t_{1,2} | Bitruncated 8-simplex | (0,0,0,0,0,0,1,2,2) | 1008 | 252 | |||||||

8 | t_{0,3} | Runcinated 8-simplex | (0,0,0,0,0,1,1,1,2) | 4536 | 504 | |||||||

9 | t_{1,3} | Bicantellated 8-simplex | (0,0,0,0,0,1,1,2,2) | 5292 | 756 | |||||||

10 | t_{2,3} | Tritruncated 8-simplex | (0,0,0,0,0,1,2,2,2) | 2016 | 504 | |||||||

11 | t_{0,4} | Stericated 8-simplex | (0,0,0,0,1,1,1,1,2) | 6300 | 630 | |||||||

12 | t_{1,4} | Biruncinated 8-simplex | (0,0,0,0,1,1,1,2,2) | 11340 | 1260 | |||||||

13 | t_{2,4} | Tricantellated 8-simplex | (0,0,0,0,1,1,2,2,2) | 8820 | 1260 | |||||||

14 | t_{3,4} | Quadritruncated 8-simplex | (0,0,0,0,1,2,2,2,2) | 2520 | 630 | |||||||

15 | t_{0,5} | Pentellated 8-simplex | (0,0,0,1,1,1,1,1,2) | 5040 | 504 | |||||||

16 | t_{1,5} | Bistericated 8-simplex | (0,0,0,1,1,1,1,2,2) | 12600 | 1260 | |||||||

17 | t_{2,5} | Triruncinated 8-simplex | (0,0,0,1,1,1,2,2,2) | 15120 | 1680 | |||||||

18 | t_{0,6} | Hexicated 8-simplex | (0,0,1,1,1,1,1,1,2) | 2268 | 252 | |||||||

19 | t_{1,6} | Bipentellated 8-simplex | (0,0,1,1,1,1,1,2,2) | 7560 | 756 | |||||||

20 | t_{0,7} | Heptellated 8-simplex | (0,1,1,1,1,1,1,1,2) | 504 | 72 | |||||||

21 | t_{0,1,2} | Cantitruncated 8-simplex | (0,0,0,0,0,0,1,2,3) | 2016 | 504 | |||||||

22 | t_{0,1,3} | Runcitruncated 8-simplex | (0,0,0,0,0,1,1,2,3) | 9828 | 1512 | |||||||

23 | t_{0,2,3} | Runcicantellated 8-simplex | (0,0,0,0,0,1,2,2,3) | 6804 | 1512 | |||||||

24 | t_{1,2,3} | Bicantitruncated 8-simplex | (0,0,0,0,0,1,2,3,3) | 6048 | 1512 | |||||||

25 | t_{0,1,4} | Steritruncated 8-simplex | (0,0,0,0,1,1,1,2,3) | 20160 | 2520 | |||||||

26 | t_{0,2,4} | Stericantellated 8-simplex | (0,0,0,0,1,1,2,2,3) | 26460 | 3780 | |||||||

27 | t_{1,2,4} | Biruncitruncated 8-simplex | (0,0,0,0,1,1,2,3,3) | 22680 | 3780 | |||||||

28 | t_{0,3,4} | Steriruncinated 8-simplex | (0,0,0,0,1,2,2,2,3) | 12600 | 2520 | |||||||

29 | t_{1,3,4} | Biruncicantellated 8-simplex | (0,0,0,0,1,2,2,3,3) | 18900 | 3780 | |||||||

30 | t_{2,3,4} | Tricantitruncated 8-simplex | (0,0,0,0,1,2,3,3,3) | 10080 | 2520 | |||||||

31 | t_{0,1,5} | Pentitruncated 8-simplex | (0,0,0,1,1,1,1,2,3) | 21420 | 2520 | |||||||

32 | t_{0,2,5} | Penticantellated 8-simplex | (0,0,0,1,1,1,2,2,3) | 42840 | 5040 | |||||||

33 | t_{1,2,5} | Bisteritruncated 8-simplex | (0,0,0,1,1,1,2,3,3) | 35280 | 5040 | |||||||

34 | t_{0,3,5} | Pentiruncinated 8-simplex | (0,0,0,1,1,2,2,2,3) | 37800 | 5040 | |||||||

35 | t_{1,3,5} | Bistericantellated 8-simplex | (0,0,0,1,1,2,2,3,3) | 52920 | 7560 | |||||||

36 | t_{2,3,5} | Triruncitruncated 8-simplex | (0,0,0,1,1,2,3,3,3) | 27720 | 5040 | |||||||

37 | t_{0,4,5} | Pentistericated 8-simplex | (0,0,0,1,2,2,2,2,3) | 13860 | 2520 | |||||||

38 | t_{1,4,5} | Bisteriruncinated 8-simplex | (0,0,0,1,2,2,2,3,3) | 30240 | 5040 | |||||||

39 | t_{0,1,6} | Hexitruncated 8-simplex | (0,0,1,1,1,1,1,2,3) | 12096 | 1512 | |||||||

40 | t_{0,2,6} | Hexicantellated 8-simplex | (0,0,1,1,1,1,2,2,3) | 34020 | 3780 | |||||||

41 | t_{1,2,6} | Bipentitruncated 8-simplex | (0,0,1,1,1,1,2,3,3) | 26460 | 3780 | |||||||

42 | t_{0,3,6} | Hexiruncinated 8-simplex | (0,0,1,1,1,2,2,2,3) | 45360 | 5040 | |||||||

43 | t_{1,3,6} | Bipenticantellated 8-simplex | (0,0,1,1,1,2,2,3,3) | 60480 | 7560 | |||||||

44 | t_{0,4,6} | Hexistericated 8-simplex | (0,0,1,1,2,2,2,2,3) | 30240 | 3780 | |||||||

45 | t_{0,5,6} | Hexipentellated 8-simplex | (0,0,1,2,2,2,2,2,3) | 9072 | 1512 | |||||||

46 | t_{0,1,7} | Heptitruncated 8-simplex | (0,1,1,1,1,1,1,2,3) | 3276 | 504 | |||||||

47 | t_{0,2,7} | Hepticantellated 8-simplex | (0,1,1,1,1,1,2,2,3) | 12852 | 1512 | |||||||

48 | t_{0,3,7} | Heptiruncinated 8-simplex | (0,1,1,1,1,2,2,2,3) | 23940 | 2520 | |||||||

49 | t_{0,1,2,3} | Runcicantitruncated 8-simplex | (0,0,0,0,0,1,2,3,4) | 12096 | 3024 | |||||||

50 | t_{0,1,2,4} | Stericantitruncated 8-simplex | (0,0,0,0,1,1,2,3,4) | 45360 | 7560 | |||||||

51 | t_{0,1,3,4} | Steriruncitruncated 8-simplex | (0,0,0,0,1,2,2,3,4) | 34020 | 7560 | |||||||

52 | t_{0,2,3,4} | Steriruncicantellated 8-simplex | (0,0,0,0,1,2,3,3,4) | 34020 | 7560 | |||||||

53 | t_{1,2,3,4} | Biruncicantitruncated 8-simplex | (0,0,0,0,1,2,3,4,4) | 30240 | 7560 | |||||||

54 | t_{0,1,2,5} | Penticantitruncated 8-simplex | (0,0,0,1,1,1,2,3,4) | 70560 | 10080 | |||||||

55 | t_{0,1,3,5} | Pentiruncitruncated 8-simplex | (0,0,0,1,1,2,2,3,4) | 98280 | 15120 | |||||||

56 | t_{0,2,3,5} | Pentiruncicantellated 8-simplex | (0,0,0,1,1,2,3,3,4) | 90720 | 15120 | |||||||

57 | t_{1,2,3,5} | Bistericantitruncated 8-simplex | (0,0,0,1,1,2,3,4,4) | 83160 | 15120 | |||||||

58 | t_{0,1,4,5} | Pentisteritruncated 8-simplex | (0,0,0,1,2,2,2,3,4) | 50400 | 10080 | |||||||

59 | t_{0,2,4,5} | Pentistericantellated 8-simplex | (0,0,0,1,2,2,3,3,4) | 83160 | 15120 | |||||||

60 | t_{1,2,4,5} | Bisteriruncitruncated 8-simplex | (0,0,0,1,2,2,3,4,4) | 68040 | 15120 | |||||||

61 | t_{0,3,4,5} | Pentisteriruncinated 8-simplex | (0,0,0,1,2,3,3,3,4) | 50400 | 10080 | |||||||

62 | t_{1,3,4,5} | Bisteriruncicantellated 8-simplex | (0,0,0,1,2,3,3,4,4) | 75600 | 15120 | |||||||

63 | t_{2,3,4,5} | Triruncicantitruncated 8-simplex | (0,0,0,1,2,3,4,4,4) | 40320 | 10080 | |||||||

64 | t_{0,1,2,6} | Hexicantitruncated 8-simplex | (0,0,1,1,1,1,2,3,4) | 52920 | 7560 | |||||||

65 | t_{0,1,3,6} | Hexiruncitruncated 8-simplex | (0,0,1,1,1,2,2,3,4) | 113400 | 15120 | |||||||

66 | t_{0,2,3,6} | Hexiruncicantellated 8-simplex | (0,0,1,1,1,2,3,3,4) | 98280 | 15120 | |||||||

67 | t_{1,2,3,6} | Bipenticantitruncated 8-simplex | (0,0,1,1,1,2,3,4,4) | 90720 | 15120 | |||||||

68 | t_{0,1,4,6} | Hexisteritruncated 8-simplex | (0,0,1,1,2,2,2,3,4) | 105840 | 15120 | |||||||

69 | t_{0,2,4,6} | Hexistericantellated 8-simplex | (0,0,1,1,2,2,3,3,4) | 158760 | 22680 | |||||||

70 | t_{1,2,4,6} | Bipentiruncitruncated 8-simplex | (0,0,1,1,2,2,3,4,4) | 136080 | 22680 | |||||||

71 | t_{0,3,4,6} | Hexisteriruncinated 8-simplex | (0,0,1,1,2,3,3,3,4) | 90720 | 15120 | |||||||

72 | t_{1,3,4,6} | Bipentiruncicantellated 8-simplex | (0,0,1,1,2,3,3,4,4) | 136080 | 22680 | |||||||

73 | t_{0,1,5,6} | Hexipentitruncated 8-simplex | (0,0,1,2,2,2,2,3,4) | 41580 | 7560 | |||||||

74 | t_{0,2,5,6} | Hexipenticantellated 8-simplex | (0,0,1,2,2,2,3,3,4) | 98280 | 15120 | |||||||

75 | t_{1,2,5,6} | Bipentisteritruncated 8-simplex | (0,0,1,2,2,2,3,4,4) | 75600 | 15120 | |||||||

76 | t_{0,3,5,6} | Hexipentiruncinated 8-simplex | (0,0,1,2,2,3,3,3,4) | 98280 | 15120 | |||||||

77 | t_{0,4,5,6} | Hexipentistericated 8-simplex | (0,0,1,2,3,3,3,3,4) | 41580 | 7560 | |||||||

78 | t_{0,1,2,7} | Hepticantitruncated 8-simplex | (0,1,1,1,1,1,2,3,4) | 18144 | 3024 | |||||||

79 |