Compound of six tetrahedra (prismatic symmetry)

Compound of six tetrahedra (prismatic symmetry)
Rank3
TypeUniform
Elements
Components6 tetrahedra
Faces24 triangles
Edges12+24
Vertices24
Vertex figureEquilateral triangle, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{4}}\approx 0.61237}$
Volume${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Dihedral angle${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }}$
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density6
Related polytopes
ArmySemi-uniform Twip, edge lengths ${\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$ (base), ${\displaystyle {\frac {\sqrt {2}}{2}}}$ (sides)
Regiment*
DualCompound of six tetrahedra
ConjugateCompound of six tetrahedra
Abstract & topological properties
Flag count144
OrientableYes
Properties
SymmetryI2(12)×A1, order 48
ConvexNo
NatureTame

The compound of six tetrahedra with prismatic symmetry is a prismatic uniform polyhedron compound. It consists of 24 triangles, with three at each vertex. It is an antiprism based on the compound of six digons.

Its quotient prismatic equivalent is the digonal hexateroorthowedge alterprism, which is eight-dimensional.

Vertex coordinates

Coordinates for the vertices of a compound of six tetrahedra of edge length 1 centered at the origin are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {\sqrt {2}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {2}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{4}},\,\pm {\frac {\sqrt {3}}{4}},\,\pm {\frac {\sqrt {2}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{4}},\,\pm {\frac {1}{4}},\,\pm {\frac {\sqrt {2}}{4}}\right).}$

Variations

This compound has variants where the bases are non-regular compounds of six digons. In these cases the compound has only hexagonal antiprismatic symmetry and the convex hull is a dihexagonal alterprism.