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User:Tomruen/List of Hanner polytopes

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In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956.[1]

Construction

[edit]

The Hanner polytopes are constructed recursively by the following rules:[2]

  • A line segment is a one-dimensional Hanner polytope
  • The Cartesian product of every two Hanner polytopes is another Hanner polytope, whose dimension is the sum of the dimensions of the two given polytopes
  • The dual of a Hanner polytope is another Hanner polytope of the same dimension.

They are exactly the polytopes that can be constructed using only these rules: that is, every Hanner polytope can be formed from line segments by a sequence of product and dual operations.[2]

Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by Cartesian products and direct sums, the dual of the Cartesian products. This direct sum operation combines two polytopes by placing them in two linearly independent subspaces of a larger space and then constructing the convex hull of their union.

Counts

[edit]

Binary cases are complete up to n=5, and then new cases are added with cases more that doubling each new dimension.

n Count Binary
1 1 1
2 1 1
3 2 2
4 4 4
5 8 8
6 18 16
7 40 32
8 94 64
9 224 128
10 548 256
11 1,356 512
12 3,418 1,024
13 8,692 2,048
14 22,352 4,096
15 57,932 8,192
16 151,312 16,384
17 397,628 32,768

Lists

[edit]

This article lists solutions up to dimension 7. There are 1, 1, 2, 4, 8, 18, and 40 Hanner polytopes in dimensions 1 to 7, respectively.

They exist in dual pairs and are listed below as n-polytopes in n-dimensions.[3]

Key:

Cn=n-cube, coordinates, (±1,±1,±1...±1), 2n vertices
CΔ
n
=dual polytope=n-orthoplex, coordinates as permutations of (±1,0,0...,0), 2n vertices.
bip P := P ⊕ { } denotes a fusil, adding two vertices in an added dimension
prism P := P × { } refers to a prism construction, doubling the vertices in an added dimension

Coordinates are assigned left to right in sets by the original polytope and the extending polytope, each set separated by a semicolon rather than comma. uniform polytopes here only require a single coordinate type.

The binary construction reads right to left, with 1 for prism, 0 for fusil, x is either, and xx is either, so ...000xx is an n-orthoplex, and ...111xx is an n-cube. There are powers of two binary expressions possible after the x's, while starting at 6D, some solutions can't be expressed this way.

For example, the binary construction 10010xx is interpreted right-to-left, with oxx as an octahedron, {3,4}, then 1 implying a prism, {3,4}×{}, next 00 (square) as a di-fusil, {3,4}×{}+{4}, and final 1 as a prism, ({3,4}×{}+{4})×{}. It can be called a octahedral-prism,square di-fusil prism.

For 7,8,9 dimensions the counts are 94, 224, 548, but are unlisted. The binary cases would be 64, 128, and 256, leaving 30, 96, and 292 special cases.

Hanner polytopes with ringed Coxeter–Dynkin diagram are (vertex-transitive) uniform polytopes. Their facet-transitive duals can be named by replacing rings with vertical lines through the nodes.

Line segment

[edit]

The binary construction is named x because any value, 1-cube, or 1-orthoplex produce a line segment, { }.

1D
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Coxeter f0
1 x=0 or 1 CΔ
1
or C1
line segment d{ }
= { }

2 (±1)

Polygons

[edit]

1-orthoplex

1-cube

There is only one Hanner polygon, a square, which can be in two orientations. The 2-cube construction has 4 vertices (±1; ±1). The dual 2-orthoplex construction vertices are listed at ([±1,0]), with the brackets to imply the bracket coordinates need to be permuted, here as (±1; 0), (0; ±1).

The binary construction is named xx because any values produce a square.

2D
# Construction Polytope f-vector Coordinates
binary Name Name Schläfli Coxeter f0 f1
1 xx=00 or 01
or 10 or 11
CΔ
2

= C2
square {4} = { }+{ }
= { }×{ }
=
=
4 4 ([±1,0]) = (±1; 0), (0; ±1)
(±1; ±1)

Polyhedra

[edit]

Octahedron
{3,4}

Cube
{4,3}

There are two Hanner polyhedra, the regular cube and octahedron.

3D
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Coxeter f0 f1 f2
1 0xx CΔ
3
octahedron { }+{ }+{ } = 3{ } = {3,4} = (= ) 6 12 8 ([±1,0,0]) = (±1; 0; 0), (0; ±1; 0), (0; 0; ±1)
2 1xx C3 cube { }×{ }×{ } = { }3 = {4,3} = 8 12 6 (±1,±1,±1)

4-polytopes

[edit]

16-cell
{3,3,4}
([±1,0,0,0])

Tesseract
{4,3,3}
(±1,±1,±1,±1)
File:Cubic fusil-ortho.png
Cubic difusil
{4,3} + { }
(±1,±1,±1; 0), (0,0,0; ±1)

Octahedral prism
{3,4}×{ }
([±1,0,0]; ±1)

There are 4 Hanner polytopes in 4-dimensions, all from 22 binary constructions.

4D
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Coxeter Vertices f0 f1 f2 f3
1 00xx CΔ
4
16-cell { }+{ }+{ }+{ } = 4{ } = {3,3,4} = 8 8 24 32 16 ([±1,0,0,0])
2 11xx C4 4-cube { }×{ }×{ }×{ } = { }4 = {4,3,3} = 16 16 32 24 8 (±1,±1,±1,±1)
3 01xx bip C3 cubical fusil {4,3}+{ } = dt{2,3,4} 8 + 2 10 28 30 12 (±1,±1,±1; 0) (0,0,0; ±1)
4 10xx prism CΔ
3
octahedral prism {3,4}×{ } = t{2,3,4} 6×2 12 30 28 10 ([±1,0,0]; ±1)

5-polytopes

[edit]

5-orthoplex
{3,3,3,4}

5-cube
{4,3,3,3}

There are 8 Hanner polytopes in 5-dimensions, all from 23 binary constructions.

5D
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Coxeter Vertices f0 f1 f2 f3 f4
1 000xx CΔ
5
5-orthoplex 5{ } = {3,3,3,4} = 10 10 40 80 80 32 ([±1,0,0,0,0])
2 111xx C5 5-cube { }5 = {4,3,3,3} 32 32 80 80 40 10 (±1,±1,±1,±1,±1)
3 001xx bip bip C3 cube,square di-fusil {4,3}+{4} 8 + 4 12 48 86 72 24 (±1,±1,±1; 0,0) (0,0,0; ±1,±1)
4 110xx prism prism CΔ
3
octahedron,square di-prism {3,4}×{4} 6×4 24 72 86 48 12 ([±1,0,0]; ±1,±1)
5 010xx bip prism CΔ
3
octahedral prism fusil {3,4}×{ }+{ } 6×2 + 2 14 54 88 66 20 ([±1,0,0]; ±1; 0) (0,0,0; 0; ±1)
6 101xx prism bip C3 cubic fusil prism ({4,3}+{ })×{ } (8 + 2)×2 20 66 88 54 14 (±1,±1,±1; 0; ±1) (0,0,0; ±1; ±1)
7 100xx prism CΔ
4
16-cell prism {3,3,4}×{ } 8×2 16 56 88 64 18 ([±1,0,0,0]; ±1)
8 011xx bip C4 tesseractic fusil {4,3,3}+{ } 16 + 2 18 64 88 56 16 (±1,±1,±1,±1; 0) (0,0,0,0; ±1)

6-polytopes

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6-orthoplex
{3,3,3,3,4}

6-cube
{4,3,3,3,3}

There are 18 Hanner polytopes in 6-dimensions, 16 from 24 binary constructions, and 2 requiring di-prisms or di-fusils.

6D
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Coxeter Vertices f0 f1 f2 f3 f4 f5
1 0000xx CΔ
6
6-orthoplex {3,3,3,3,4} 12 12 60 160 240 192 64 (±1,0,0,0,0,0)
2 1111xx C6 6-cube {4,3,3,3,3} 64 64 192 240 160 60 12 (±1,±1,±1,±1,±1,±1)
3 0001xx bip bip bip C3 octahedron,cube di-fusil {4,3}+{3,4} 8 + 6 14 72 182 244 168 48 (±1,±1,±1; 0,0,0) (0,0,0; [±1,0,0])
4 1110xx prism prism prism CΔ
3
octahedron,cube di-prism {4,3}×{3,4} 8×6 48 168 244 182 72 14 (±1,±1,±1; [±1,0,0])
5 0010xx bip bip prism CΔ
3
octahedral-prism,square di-fusil {3,4}×{ }+{4} 6×2 + 4 16 82 196 242 152 40 ([±1,0,0]; ±1; 0,0) (0,0,0; 0; ±1,±1)
6 1101xx prism prism bip C3 cubic-fusil,square di-prism ({4,3}+{ })×{4} 8 + 2×4 40 152 242 196 82 16 (±1,±1,±1; 0; ±1,±1) (0,0,0; ±1; ±1,±1)
7 0100xx bip prism CΔ
4
16-cell prism fusil {3,3,4}×{ }+{ } 8×2 + 2 18 88 200 240 146 36 ([±1,0,0,0]; ±1; 0) (0,0,0,0; 0; ±1)
8 1011xx prism bip C4 tesseract fusil prism ({4,3,3}+{ })×{ } 16 + 2×2 36 146 240 200 88 18 (±1,±1,±1,±1; 0; ±1) (0,0,0,0; ±1; ±1)
9 0011xx bip bip C4 tesseract,square di-fusil {4,3,3}+{4} 16 + 4 20 100 216 232 128 32 (±1,±1,±1,±1; 0,0) (0,0,0,0; ±1,±1)
10 1100xx prism prism CΔ
4
16-cell,square di-prism {3,3,4}×{4} 8×4 32 128 232 216 100 20 ([±1,0,0,0]; ±1; ±1)
11 1000xx prism CΔ
5
5-orthoplex prism {3,3,3,4}×{ } 10×2 20 90 200 240 144 34 ([±1,0,0,0,0]; ±1)
12 0111xx bip C5 5-cube fusil {4,3,3,3}+{ } 32 + 2 34 144 240 200 90 20 (±1,±1,±1,±1,±1; 0) (0,0,0,0,0; ±1)
13 0101xx bip prism bip C3 cubic fusil prism fusil ({4,3}+{ })×{ }+{ } (8 + 2)×2 + 2 22 106 220 230 122 28 (±1,±1,±1; 0; ±1; 0) (0,0,0; ±1; ±1; 0) (0,0,0; 0; 0; ±1)
14 1010xx prism bip prism CΔ
3
octahedral prism fusil prism ({3,4}×{ }+{ })×{ } (6×2 + 2)×2 28 122 230 220 106 22 ([±1,0,0]; ±1; 0; ±1) (0,0,0; 0; ±1; ±1)
15 1001xx prism bip bip C3 cube,square di-fusil prism ({4,3}+{4})×{ } (8 + 4)×2 24 108 220 230 120 26 (±1,±1,±1; 0,0; ±1) (0,0,0; ±1,±1; ±1)
16 0110xx bip prism prism CΔ
3
octahedron,square di-prism fusil {3,4}×{4}+{ } 6×4 + 2 26 120 230 220 108 24 ([±1,0,0]; ±1,±1; 0) (0,0,0; 0,0; ±1)
17 -- C3 ⊕ C3 cube,cube di-fusil {4,3}+{4,3} 8 + 8 16 88 204 240 144 36 (±1,±1,±1; 0,0,0) (0,0,0; ±1,±1,±1)
18 -- CΔ
3
× CΔ
3
octahedron,octahedron di-prism {3,4}×{3,4} 6×6 36 144 240 204 88 16 ([±1,0,0]; [±1,0,0])

7-polytopes

[edit]

7-orthoplex
{3,3,3,3,3,4}

7-cube
{4,3,3,3,3,3}

There are 40 Hanner polytopes in 7-dimensions, 32 from 25 binary constructions, and 8 requiring di-prisms or di-fusils.

7D
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Vertices f0 f1 f2 f3 f4 f5 f6
1 00000xx CΔ
7
7-orthoplex {3,3,3,3,3,4} 14 14 84 280 560 672 448 128 (±1,0,0,0,0,0,0)
2 11111xx C7 7-cube {4,3,3,3,3,3} 128 128 448 672 560 280 84 14 (±1,±1,±1,±1,±1,±1,±1)
3 00001xx bip bip bip bip C3 cube,16-cell di-fusil {4,3}+{3,3,4} 8 + 8 16 100 326 608 656 384 96 (±1,±1,±1; 0,0,0,0) (0,0,0; [±1,0,0,0]}
4 11110xx prism prism prism prism CΔ
3
octahedron,tesseract di-prism {3,4}×{4,3,3} 6×16 96 384 656 608 326 100 16 ([±1,0,0]; ±1,±1,±1,±1)
5 00010xx bip bip bip prism CΔ
3
octahedral-prism,octahedron di-fusil {3,4}×{ }+{3,4} 6×2 + 6 18 114 360 634 636 344 80 ([±1,0,0]; ±1; 0,0,0) (0,0,0; 0; [±1,0,0])
6 11101xx prism prism prism bip C3 cubic-bipyramid,cube di-prism ({4,3}+{ })×{4,3} (8 + 2)×8 80 344 636 634 360 114 18 (±1,±1,±1; 0; ±1,±1,±1) (0,0,0; ±1; ±1,±1,±1)
7 00011xx bip bip bip C4 tesseract,octahedron di-fusil {4,3,3}+{3,4} 16 + 6 22 140 416 904 656 592 64 (±1,±1,±1,±1; 0,0,0) 0,0,0,0; [±1,0,0])
8 11100xx prism prism prism CΔ
4
16-cell,cube di-prism {3,3,4}×{4,3} 8×8 64 592 656 904 416 140 22 ([±1,0,0,0]; ±1,±1,±1]
9 00100xx bip bip prism CΔ
4
16-cell-prism,square di-fusil ({3,3,4}×{ })+{4} (8×2) + 4 20 124 376 640 626 328 72 ([±1,0,0,0]; ±1; 0,0) (0,0,0,0; 0; ±1,±1)
10 11011xx prism prism bip C4 tesseractic-bipyramid,square di-prism ({4,3,3}+{ })×{4} (16 + 2)×4 72 328 626 640 376 124 20 (±1,±1,±1,±1; 0; ±1,±1) (0,0,0,0; ±1; ±1,±1)
11 00101xx bip bip prism bip C3 square-bipyramid-prism,square di-fusil ({4,3}+{ })×{ }+{4} (8 + 2)×2 + 4 24 150 432 670 582 272 56 (±1,±1,±1; 0; ±1; 0,0) (0,0,0; ±1; ±1; 0,0) (0,0,0; 0; 0; ±1,±1)
12 11010xx prism prism bip prism CΔ
3
octahedral-prism-bipyramid,square di-prism ({3,4}×{ }+{ })×{4} (6×2 + 2)×4 56 272 582 670 432 150 24 ([±1,0,0]; ±1; 0, ±1,±1) (0,0,0; 0; ±1; ±1,±1)
13 00110xx bip bip prism prism CΔ
3
(octahedron,square di-prism),square di-fusil {3,4}×{4}+{4} 6×4 + 4 28 172 470 680 548 240 48 ([±1,0,0]; ±1,±1; 0,0) (0,0,0; 0,0; ±1,±1)
14 11001xx prism prism bip bip C3 (cube,square di-fusil),square di-prism ({4,3}+{4})×{4} (8 + 4)×4 48 240 548 680 470 172 28 (±1,±1,±1; 0,0; ±1,±1) (0,0,0; ±1,±1; ±1,±1)
15 00111xx bip bip C5 5-cube,square di-fusil {4,3,3,3}+{4} 32 + 4 36 212 528 680 490 200 40 (±1,±1,±1,±1,±1; 0,0) (0,0,0,0,0; ±1,±1)
16 11000xx prism prism CΔ
5
5-orthoplex,square di-prism {3,3,3,4}×{4} 10×4 40 200 490 680 528 212 36 ([±1,0,0,0,0]; ±1,±1)
17 01000xx bip prism CΔ
5
5-orthoplex-prism fusil {3,3,3,4}×{ }+{ } 10×2 + 2 22 130 380 640 624 322 68 ([±1,0,0,0,0]; ±1; 0) (0,0,0,0,0; 0; ±1)
18 10111xx prism bip C5 5-cube-bipyramid prism ({4,3,3,3}+{ })×{ } (32 + 2)×2 68 322 624 640 380 130 22 (±1,±1,±1,±1; 0; ±1) (0,0,0,0,0; ±1; ±1)
19 01001xx bip prism bip bip C3 cube,square di-fusil prism fusil ({4,3}+{4})×{ }+{ } (8 + 4)×2 + 2 26 156 436 670 580 266 52 (±1,±1,±1; 0,0; ±1; 0) (0,0,0; ±1,±1; ±1; 0) (0,0,0; 0,0; 0; ±1)
20 10110xx prism bip prism prism CΔ
3
octahedron-square di-prism fusil prism ({3,4}×{4}+{ })×{ } (6×4 + 2)×2 52 266 580 670 436 156 26 ([±1,0,0]; ±1,±1; 0; ±1) (0,0,0; 0,0; ±1; ±1)
21 01010xx bip prism bip prism CΔ
3
octahedron prism fusil prism fusil ({3,4}×{ }+{ })×{ }+{ } (6×2 + 2)×2 + 2 30 178 474 680 546 234 44 ([±1,0,0]; ±1; 0; ±1; 0) (0,0,0; 0; ±1; ±1; 0) (0,0,0; 0; 0; ±1)
22 10101xx prism bip prism bip C3 cube fusil prism fusil prism (({4,3}+{ })×{ }+{ })×{ } ((8 + 2)×2 + 2)×2 44 234 546 680 474 178 30 (±1,±1,±1; 0; ±1; 0; ±1) (0,0,0; ±1; ±1; 0; ±1) (0,0,0; 0; 0; ±1; ±1)
23 01011xx bip prism bip C4 tesseract fusil prism fusil ({4,3,3}+{ })×{ }+{ } (16 + 2)×2 + 2 38 216 528 684 496 192 32 (±1,±1,±1,±1; 0; ±1; 0) (0,0,0,0; ±1; ±1; 0) (0,0,0,0; 0; 0; ±1)
24 10100xx prism bip prism CΔ
4
16-cell prism fusil prism ({3,3,4}×{ }+{ })×{ } (8×2 + 2)×2 36 192 496 684 528 216 38 ([±1,0,0,0]; ±1; 0; ±1) (0,0,0,0; 0; ±1; ±1)
25 01100xx bip prism prism CΔ
4
16-cell-square di-prism fusil {3,3,4}×{4}+{ } 8×4 + 2 34 192 488 680 532 220 40 ([±1,0,0,0]; ±1,±1; 0) (0,0,0,0; 0,0; ±1)
26 10011xx prism bip bip C4 tesseract,square di-fusil prism ({4,3,3}+{4})×{ } (16 + 4)×2 40 220 532 680 488 192 34 (±1,±1,±1,±1; 0,0; ±1) (0,0,0,0; ±1,±1; ±1)
27 01101xx bip prism prism bip C3 cubic-bipyramid,square di-prism fusil ({4,3}+{ })×{4}+{ } (8 + 2)×4 + 2 42 232 546 680 474 180 32 (±1,±1,±1; 0; ±1,±1; 0) (0,0,0; ±1; ±1,±1; 0) (0,0,0; 0; 0,0; ±1)
28 10010xx prism bip bip prism CΔ
3
octahedral-prism,square di-fusil prism ({3,4}×{ }+{4})×{ } (6×2 + 4)×2 32 180 474 680 546 232 42 ([±1,0,0]; ±1; 0,0; ±1) (,0,0; 0; ±1,±1; ±1)
29 01110xx bip prism prism prism CΔ
3
octahedron,square di-prism fusil {3,4}×{4,3}+{ } 6×8 + 2 50 264 580 670 436 158 28 ([±1,0,0]; ±1,±1,±1; 0) (0,0,0; 0,0,0; ±1)
30 10001xx prism bip bip bip C3 cube,octahedron di-fusil prism ({4,3}+{3,4})×{ } (8 + 6)×2 28 158 436 670 580 264 50 (±1,±1,±1; 0,0,0; ±1) (0,0,0; [±1,0,0]; ±1)
31 01111xx bip C6 6-cube fusil {4,3,3,3,3}+{ } 64 + 2 66 320 624 640 380 132 24 (±1,±1,±1,±1,±1,±1; 0) (0,0,0,0,0,0; ±1)
32 10000xx prism CΔ
6
6-orthoplex prism {3,3,3,3,4}×{ } 12×2 24 132 380 640 624 320 66 ([±1,0,0,0,0,0]; ±1)
33 -- bip (C3⊕C3) cube,cube di-fusil fusil {4,3}+{4,3}+{ } 8 + 8 + 2 18 120 380 648 624 324 72 (±1,±1,±1; 0,0,0; 0) (0,0,0; ±1,±1,±1; 0) (0,0,0; 0,0,0; ±1)
34 -- prism (CΔ
3
×CΔ
3
)
octahedron,octahedron di-prism prism {3,4}×{3,4}×{ } 6×6×2 72 324 624 648 380 120 18 ([±1,0,0]; [±1,0,0]; ±1)
35 -- prism (C3⊕C3) cube,cube di-fusil prism ({4,3}+{4,3})×{ } (8 + 8)×2 32 192 496 684 528 216 38 (±1,±1,±1; 0,0,0; ±1) (0,0,0; ±1,±1,±1; ±1)
36 -- bip (CΔ
3
×CΔ
3
)
octahedron,octahedron di-prism fusil {3,4}×{3,4}+{ } 6×6 + 2 38 216 528 684 496 192 32 ([±1,0,0]; [±1,0,0]; 0) (0,0,0; 0,0,0; ±1)
37 -- C4⊕C3 tesseract,cube di-fusil {4,3,3}+{4,3} 16 + 8 24 172 478 680 544 240 48 (±1,±1,±1,±1; 0,0,0) (0,0,0,0; ±1,±1,±1)
38 -- CΔ
4
×CΔ
3
16-cell,octahedron di-prism {3,3,4}×{3,4} 8×6 48 240 544 680 478 172 24 ([±1,0,0,0]; [±1,0,0])
39 -- (prism CΔ
3
)⊕C3
octahedral-prism,cube di-fusil {3,4}×{ }+{4,3} 6×2 + 8 20 138 418 666 596 288 60 (0,0,0; 0; ±1,±1,±1) ([±1,0,0]; ±1; 0,0,0)
40 -- (bip C3)×CΔ
3
cubical-fusil,octahedron di-prism ({4,3}+{ })×{3,4} (8 + 2)×6 60 288 596 666 418 138 20 (±1,±1,±1; 0; [±1,0,0]) (0,0,0; ±1; [±1,0,0])

8-polytopes

[edit]

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

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

There are 94 Hanner polytopes in 8-dimensions, 64 from 26 binary constructions, and 30 requiring di-prisms or di-fusils.

8D (incomplete)
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Vertices f0 f1 f2 f3 f4 f5 f6 f7
1 000000xx CΔ
8
8-orthoplex {3,3,3,3,3,3,4} 16 16 112 448 1120 1792 1792 1024 256 (±1,0,0,0,0,0,0,0)
2 111111xx C8 8-cube {4,3,3,3,3,3,3} 256 256 1024 1792 1792 1120 448 112 16 (±1,±1,±1,±1,±1,±1,±1,±1)

9-polytopes

[edit]

9-orthoplex
{3,3,3,3,3,3,3,4}

9-cube
{4,3,3,3,3,3,3,3}

There are 224 Hanner polytopes in 9-dimensions, 128 from 27 binary constructions, and 96 requiring di-prisms or di-fusils.

9D (incomplete)
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Vertices f0 f1 f2 f3 f4 f5 f6 f7 f8
1 0000000xx CΔ
9
9-orthoplex {3,3,3,3,3,3,3,4} 18 18 144 672 2016 4032 5376 4608 2304 512 (±1,0,0,0,0,0,0,0,0)
2 1111111xx C9 9-cube {4,3,3,3,3,3,3,3} 512 512 2304 4608 5376 4032 2016 672 144 18 (±1,±1,±1,±1,±1,±1,±1,±1,±1)

10-polytopes

[edit]

10-orthoplex
{3,3,3,3,3,3,3,4}

10-cube
{4,3,3,3,3,3,3,3}

There are 548 Hanner polytopes in 10-dimensions, 256 from 28 binary constructions, and 292 requiring di-prisms or di-fusils.

10D (incomplete)
# Binary construction Polytope f-vector Coordinates
binary Name Name Schläfli Vertices f0 f1 f2 f3 f4 f5 f6 f7 f8 f9
1 00000000xx CΔ
10
10-orthoplex {3,3,3,3,3,3,3,4} 20 20 180 960 3360 8064 13440 15360 11520 5120 1024 (±1,0,0,0,0,0,0,0,0,0)
2 11111111xx C10 10-cube {4,3,3,3,3,3,3,3} 1024 1024 5120 11520 15360 13440 8064 3360 960 180 20 (±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)

11-polytopes

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There are 1356 Hanner polytopes in 11-dimensions, 512 from 29 binary constructions, and 884 requiring di-prisms or di-fusils.

12-polytopes

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There are 3418 Hanner polytopes in 12-dimensions, 1024 from 210 binary constructions, and 2394 requiring di-prisms or di-fusils.

13-polytopes

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There are 8692 Hanner polytopes in 13-dimensions, 2048 from 211 binary constructions, and 6644 requiring di-prisms or di-fusils.

14-polytopes

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There are 22352 Hanner polytopes in 14-dimensions, 4096 from 212 binary constructions, and 18256 requiring di-prisms or di-fusils.

15-polytopes

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There are 57932 Hanner polytopes in 15-dimensions, 8192 from 213 binary constructions, and 49740 requiring di-prisms or di-fusils.

16-polytopes

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There are 151,312 Hanner polytopes in 16-dimensions, 16,384 from 214 binary constructions, and 134,928 requiring di-prisms or di-fusils.

17-polytopes

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There are 397,628 Hanner polytopes in 17-dimensions, 32,768 from 215 binary constructions, and 364,860 requiring di-prisms or di-fusils.

Refernces

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  1. ^ Hanner, Olof (1956), "Intersections of translates of convex bodies", Mathematica Scandinavica, 4: 65–87, MR 0082696.
  2. ^ a b Freij, Ragnar (2012), Topics in algorithmic, enumerative and geometric combinatorics (PDF), Ph.D. thesis, Department of Mathematical Sciences, Chalmers Institute of Technology.
  3. ^ Raman Sanyal, Axel Werner Gunter, M. Ziegler, On Kalai’s conjectures concerning centrally symmetric polytopes, Discrete Comput Geom (2009) 41: 183–198, DOI 10.1007/s00454-008-9104-8 [1] pp. 190, 196