Hexagrammic antiprism

Hexagrammic antiprism
Rank	3
Type	Uniform
Space	Spherical
Notation
Coxeter diagram
Elements
Components	2 octahedra
Faces	4 triangles (as 2 hexagrams) 12 triangles
Edges	12+12
Vertices	12
Vertex figure	Square, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} \approx 0.70711$
Volume	$\frac{2\sqrt2}{3} \approx 0.94281$
Dihedral angle	$\arccos\left(-\frac13\right) \approx 109.47122^\circ$
Height	$\frac{\sqrt6}{3} \approx 0.81650$
Central density	2
Number of external pieces	38
Level of complexity	11
Related polytopes
Army	Semi-uniform Hip
Regiment	*
Dual	Hexagrammic antitegum
Conjugate	None
Convex core	Order-6-truncated hexagonal tegum
Abstract & topological properties
Flag count	96
Orientable	Yes
Properties
Symmetry	G2×A1, order 24
Convex	No
Nature	Tame

The hexagrammic antiprism, compound of two triangular antiprisms, or compound of two octahedra, is a prismatic uniform polyhedron. It consists of 12 triangles and 2 hexagrams. Each vertex joins one hexagram and three triangles. As the name suggests, it is an antiprism based on a hexagram.

Its quotient prismatic equivalent is the digonal-triangular duoantiprism, which is four-dimensional.

A less-symmetric variant of the hexagrammic antiprism with golden hexagrams as bases occurs as a combocell type of every baby monster snub.

Vertex coordinates

A hexagrammic antiprism of edge length 1 has vertex coordinates given by:

• $\left(\pm\frac12,\,\pm\frac{\sqrt3}{6},\,\pm\frac{\sqrt6}{6}\right),$
• $\left(0,\,\pm\frac{\sqrt3}{3},\,\pm\frac{\sqrt6}{6}\right).$