(r,s)-Polytopes in Higher Dimensions

Here is a list of some higher dimensional r-simple and s-simplicial d-polytopes for r and s greater than 1 and r+s=d. This list is not complete or systematic in any way, it just contains examples that I came across or constructed some time ago.

The N, M, and K-series can be found in Gruenbaum's book on Convex Polytopes, some polytopes arose from the E-construction, and the 16 polytopes in the ND-series are from a generalisation of the E-construction.

The type of the polytope is given in the form (r,s), meining that the polytope is r-simple and s-simplicial. This notation agrees with the one in my thesis.

Dimension Five | Dimension Six | Dimension Seven | Dimension Eight

Dimension 5

E₁(5-simplex) : 12 vertices, 45 facets, type (3, 2) [poly]
E₂(5-simplex) : 12 vertices, 20 facets, type (3, 2) [poly]
K₅₂ : 15 vertices, 12 facets, type (3, 2) [poly]
E₂(stacked 5-simplex) : 17 vertices, 30 facets, type (3, 2) [poly]
K₅₃ : 20 vertices, 12 facets, type (3, 2) [poly]
ND₁ : 22 vertices, 35 facets, type (2, 3) [poly]
N₅ : 26 vertices, 16 facets, type (3, 2) [poly]
E₁(E₁(5-simplex)) : 27 vertices, 45 facets, type (3, 2) [poly]
E₃(K₅₂) : 28 vertices, 46 facets, type (3, 2) [poly]
ND₂ : 28 vertices, 43 facets, type (2, 3) [poly]
ND₃ : 34 vertices, 53 facets, type (2, 3) [poly]
ND₄ : 40 vertices, 62 facets, type (2, 3) [poly]
E₂(5-cube) : 42 vertices, 80 facets, type (3, 2) [poly]
ND₅ : 46 vertices, 71 facets, type (2, 3) [poly]
E₂(tr(5-cube)) : 47 vertices, 90 facets, type (3, 2) [poly]
ND₆ : 52 vertices, 80 facets, type (2, 3) [poly]
ND₇ : 58 vertices, 89 facets, type (2, 3) [poly]
ND₈ : 64 vertices, 98 facets, type (2, 3) [poly]
ND₉ : 70 vertices, 107 facets, type (2, 3) [poly]
E₃(E₃(E₃(5-simplex))) : 72 vertices, 150 facets, type (3, 2) [poly]
ND₁₀ : 76 vertices, 116 facets, type (2, 3) [poly]
M₅ : 80 vertices, 42 facets, type (3, 2) [poly]
ND₁₁ : 82 vertices, 125 facets, type (2, 3) [poly]
ND₁₂ : 88 vertices, 134 facets, type (2, 3) [poly]
ND₁₃ : 94 vertices, 143 facets, type (2, 3) [poly]
ND₁₄ : 100 vertices, 152 facets, type (2, 3) [poly]
ND₁₅ : 106 vertices, 161 facets, type (2, 3) [poly]
ND₁₆ : 112 vertices, 170 facets, type (2, 3) [poly]

Dimension 6

K₆₂ : 21 vertices, 14 facets, type (4, 2) [poly]
K₆₃ : 35 vertices, 14 facets, type (4, 2) [poly]
N₆ : 44 vertices, 32 facets, type (3, 3) [poly]
E₂(6-cube) : 76 vertices, 240 facets, type (2, 4) [poly]
M₆ : 240 vertices, 76 facets, type (4, 2) [poly]

Dimension 7

E₃(7-simplex) : 16 vertices, 70 facets, type (2, 5) [poly]
K₇₂ : 28 vertices, 16 facets, type (5, 2) [poly]
E₃(K₇₂) : 44 vertices, 350 facets, type (2, 5) [poly]
K₇₃ : 56 vertices, 16 facets, type (5, 2) [poly]
K₇₄ : 70 vertices, 16 facets, type (5, 2) [poly]
N₇ : 78 vertices, 64 facets, type (3, 4) [poly]
M₇ : 672 vertices, 142 facets, type (5, 2) [poly]

Dimension 8

E₄(8-simplex) : 18 vertices, 126 facets, type (2, 6) [poly]
K₈₂ : 36 vertices, 18 facets, type (6, 2) [poly]
K₈₃ : 84 vertices, 18 facets, type (6, 2) [poly]
K₈₄ : 126 vertices, 18 facets, type (6, 2) [poly]
N₈^dual : 128 vertices, 144 facets, type (5, 3) [poly]
N₈ : 144 vertices, 128 facets, type (3, 5) [poly]
M₈ : 1792 vertices, 272 facets, type (6, 2) [poly]