User:Tomruen/List of Hanner polytopes

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.

Construction
The Hanner polytopes are constructed recursively by the following rules: 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.
 * 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.

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

Lists
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.

Key:
 * C$n$=n-cube, coordinates, (±1,±1,±1...±1), 2n vertices
 * C=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
The binary construction is named x because any value, 1-cube, or 1-orthoplex produce a line segment, { }.

Polygons
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.

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

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

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

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

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

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

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

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

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

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

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

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

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

16-polytopes
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
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

 * Hanner Polytopes