Hexagonal duoprism

Hexagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Hiddip
Coxeter diagram	x6o x6o ()
Elements
Cells	12 hexagonal prisms
Faces	36 squares, 12 hexagons
Edges	72
Vertices	36
Vertex figure	Tetragonal disphenoid, edge lengths √3 (base) and √2 (sides)
Measures (edge length 1)
Circumradius	$\sqrt2 ≈ 1.41421$
Inradius	$\frac{\sqrt3}{2} ≈ 0.86603$
Hypervolume	$\frac{27}{4} = 6.75$
Dichoral angles	Hip–6–hip: 120°
	Hip–4–hip: 90°
Central density	1
Number of external pieces	12
Level of complexity	3
Related polytopes
Army	Hiddip
Regiment	Hiddip
Dual	Hexagonal duotegum
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	G2≀S2, order 288
Convex	Yes
Nature	Tame

The hexagonal duoprism or hiddip, also known as the hexagonal-hexagonal duoprism, the 6 duoprism or the 6-6 duoprism, is a noble uniform duoprism that consists of 12 hexagonal prisms, with 4 joining at each vertex. It is also the 12-5 gyrochoron. It is the first in an infinite family of isogonal hexagonal dihedral swirlchora and also the first in an infinite family of isochoric hexagonal hosohedral swirlchora.

This polychoron can be alternated into a triangular duoantiprism, although it cannot be made uniform.

A unit hexagonal duoprism can be vertex-inscribed into the antifrustary distetracontoctachoron and ditetrahedronary dishecatonicosachoron.

Gallery

• 

  Wireframe, cell, net

Vertex coordinates

Coordinates for the vertices of a hexagonal duoprism of edge length 1, centered at the origin, are given by:

• $\left(0,\,±1,\,0,\,±1\right),$
• $\left(0,\,±1,\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(±\frac{\sqrt3}{2},\,±\frac12,\,0,\,±1\right),$
• $\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12\right).$

Representations

A hexagonal duoprism has the following Coxeter diagrams:

• x6o x6o (full symmetry)
• x3x x6o (one hexagon seen as ditrigon)
• x3x x3x (both hexagons seen as ditrigons, triangular duoprismatic symmetry)
• xux xxx6ooo&#xt (hexagonal axial)
• xux xxx3xxx&#xt (ditrigonal axial)

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Klitzing, Richard. "hiddip".

• Wikipedia Contributors. "6-6 duoprism".