# Compound of four octahedra (prismatic symmetry)

Compound of four octahedra (prismatic symmetry)
Rank3
TypeUniform
Elements
Components4 octahedra
Faces24 triangles, 8 triangles as 2 tetratriangles
Edges24+24
Vertices24
Vertex figureSquare, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Volume${\displaystyle {\frac {4{\sqrt {2}}}{3}}\approx 1.88562}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density4
Related polytopes
ArmySemi-uniform Twip, edge lengths ${\displaystyle {\frac {3{\sqrt {2}}-{\sqrt {6}}}{6}}}$ (base), ${\displaystyle {\frac {\sqrt {6}}{3}}}$ (sides)
Regiment*
DualCompound of four cubes
ConjugateCompound of four octahedra
Abstract & topological properties
Flag count192
OrientableYes
Properties
SymmetryI2(12)×A1, order 48
ConvexNo
NatureTame

The tetratriangular antiprism or compound of four octahedra with prismatic symmetry is a prismatic uniform polyhedron compound. It consists of 2 tetratriangles and 24 triangles. Each vertex joins one tetratriangle and three triangles. As the name suggests, it is an antiprism based on a tetratriangle.

Its quotient prismatic equivalent is the triangular tetrahedroorthowedge alterprism, which is six-dimensional.

## Vertex coordinates

Coordinates for the vertices of a tetratriangular antiprism of edge length 1 centered at the origin are given by:

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

## Variations

This compound has variants where the bases are non-regular compounds of four triangles (seen as two-hexagram compounds). In these cases the compound has only hexagonal prismatic symmetry and the convex hull is a dihexagonal prism.