# Hexagrammic antiprism

Hexagrammic antiprism
Rank3
TypeUniform
Notation
Coxeter diagram
Elements
Components2 octahedra
Faces12 triangles, 4 triangles as 2 hexagrams
Edges12+12
Vertices12
Vertex figureSquare, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Volume${\displaystyle {\frac {2{\sqrt {2}}}{3}}\approx 0.94281}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density2
Number of external pieces38
Level of complexity11
Related polytopes
ArmySemi-uniform Hip, edge lengths ${\displaystyle {\frac {\sqrt {3}}{3}}}$ (base), ${\displaystyle {\frac {\sqrt {6}}{3}}}$ (sides)
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.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{6}},\,\pm {\frac {\sqrt {6}}{6}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {\sqrt {3}}{3}},\,\pm {\frac {\sqrt {6}}{6}}\right).}$

## Variations

This compound has variants where the bases are non-regular compounds of two triangles. In these cases the compound has only triangular antiprismatic symmetry and the convex hull is a ditrigonal alterprism.

The case of this variant with golden hexagrams as bases occurs as a combocell type of every baby monster snub.