Tetrakis hexahedron

Tetrakis hexahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymTekah
Coxeter diagramo4m3m ()
Conway notationkC
Elements
Faces24 isosceles triangles
Edges12+24
Vertices6+8
Vertex figure6 squares, 8 hexagons
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\frac {4}{5}}\right)\approx 143.13010^{\circ }}$
Central density1
Number of external pieces24
Level of complexity3
Related polytopes
ArmyTekah
RegimentTekah
DualTruncated octahedron
ConjugateNone
Abstract & topological properties
Flag count144
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
Flag orbits3
ConvexYes
NatureTame

The tetrakis hexahedron is one of the 13 Catalan solids. It has 24 isosceles triangles as faces, with 6 order-4 and 8 order-6 vertices. It is the dual of the uniform truncated octahedron.

It can also be obtained as the convex hull of a cube and an octahedron, where the edges of the octahedron are ${\displaystyle {\frac {3{\sqrt {2}}}{4}}\approx 1.06066}$ times the length of those of the cube. Using an octahedron that is any number less than ${\displaystyle {\sqrt {2}}\approx 1.41421}$ times the edge length of the cube (including if the two have the same edge length) gives a fully symmetric variant of this polyhedron. The lower limit is ${\displaystyle {\frac {\sqrt {2}}{2}}}$ times that of the cube, where the octahedron's vertices will coincide with the cube's face centers.

Each face of this polyhedron is an isosceles triangle with base side length ${\displaystyle {\frac {4}{3}}\approx 1.33333}$ times those of the side edges. These triangles have apex angle ${\displaystyle \arccos \left({\frac {1}{9}}\right)\approx 83.62063^{\circ }}$ and base angles ${\displaystyle \arccos \left({\frac {2}{3}}\right)\approx 48.18969^{\circ }}$.

Vertex coordinates

A tetrakis hexahedron with dual edge length 1 has vertex coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {3{\sqrt {2}}}{4}},\,\pm {\frac {3{\sqrt {2}}}{4}},\,\pm {\frac {3{\sqrt {2}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {9{\sqrt {2}}}{8}},\,0,\,0\right)}$.

Variations

In addition to its fully-symmetric variants, the tetrakis hexahedron has variants that remain isotopic under tetrahedral symmetry. This variant can be called the disdyakis hexahedron and uses scalene triangles for faces.