Tetrakis hexahedron

Tetrakis hexahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Tekah
Coxeter diagram	o4m3m ()
Elements
Faces	24 isosceles triangles
Edges	12+24
Vertices	6+8
Vertex figure	6 squares, 8 hexagons
Measures (edge length 1)
Dihedral angle	$\arccos\left(-\frac45\right) \approx 143.13010^\circ$
Central density	1
Number of external pieces	24
Level of complexity	3
Related polytopes
Army	Tekah
Regiment	Tekah
Dual	Truncated octahedron
Conjugate	None
Abstract & topological properties
Flag count	144
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

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 $\frac{3\sqrt2}{4} \approx 1.06066$ times the length of those of the cube. Using an octahedron that is any number less than $\sqrt2 \approx 1.41421$ 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 $\frac{\sqrt2}{2}$ 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 $\frac43 \approx 1.33333$ times those of the side edges. These triangles have apex angle $\arccos\left(\frac19\right) \approx 83.62063^\circ$ and base angles $\arccos\left(\frac23\right) \approx 48.18969^\circ$.

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.

External links

• Klitzing, Richard. "tekah".

• Wikipedia Contributors. "Tetrakis hexahedron".
• McCooey, David. "Tetrakis Hexahedron"

• Quickfur. "The Tetrakis Hexahedron".