Hexagonal duoantiprism

Hexagonal duoantiprism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Hiddap
Coxeter diagram	s12o2s12o ()
Elements
Cells	72 tetragonal disphenoids, 24 hexagonal antiprisms
Faces	288 isosceles triangles, 24 hexagons
Edges	144+144
Vertices	72
Vertex figure	Gyrobifastigium
Measures (based on hexagons of edge length 1)
Edge lengths	Lacing (144): $\sqrt3-1 ≈ 0.73205$
	Edges of hexagons (144): 1
Circumradius	$\sqrt2 ≈ 1.41421$
Central density	1
Related polytopes
Army	Hiddap
Regiment	Hiddap
Dual	Hexagonal duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(12)≀S2)/2, order 576
Convex	Yes
Nature	Tame

The hexagonal duoantiprism or hiddap, also known as the hexagonal-hexagonal duoantiprism, the 6 duoantiprism or the 6-6 duoantiprism, is a convex isogonal polychoron that consists of 24 hexagonal antiprisms and 72 tetragonal disphenoids. 4 hexagonal antiprisms and 4 tetragonal disphenoids join at each vertex. It can be obtained through the process of alternating the dodecagonal duoprism. However, it cannot be made uniform, and has two edge lengths. It is the second in an infinite family of isogonal hexagonal dihedral swirlchora.

The ratio between the longest and shortest edges is 1:$\frac{1+\sqrt3}{2}$ ≈ 1:1.36603.

Vertex coordinates

The vertices of a hexagonal duoantiprism based on hexagons of edge length 1, centered at the origin, are given by:

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