# Hexagrammic antiprism

Hexagrammic antiprism Rank3
TypeUniform
SpaceSpherical
Notation
Coxeter diagram     Elements
Components2 octahedra
Faces4 triangles (as 2 hexagrams)
12 triangles
Edges12+12
Vertices12
Vertex figureSquare, 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 density2
Number of external pieces38
Level of complexity11
Related polytopes
ArmySemi-uniform Hip
Regiment*
DualHexagrammic antitegum
ConjugateNone
Convex coreOrder-6-truncated hexagonal tegum
Abstract & topological properties
Flag count96
OrientableYes
Properties
SymmetryG2×A1, order 24
ConvexNo
NatureTame

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:

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