Tetrakis hexahedron

In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube ) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.

It can be called a disdyakis hexahedron or hexakis tetrahedron as the dual of an omnitruncated tetrahedron, and as the barycentric subdivision of a tetrahedron.

Cartesian coordinates
Cartesian coordinates for the 14 vertices of a tetrakis hexahedron centered at the origin, are the points $$ \bigl( {\pm\tfrac32}, 0, 0 \bigr),\ \bigl( 0, \pm\tfrac32, 0 \bigr),\ \bigl( 0, 0, \pm\tfrac32 \bigr),\ \bigl({\pm1}, \pm1, \pm1 \bigr). $$

The length of the shorter edges of this tetrakis hexahedron equals 3/2 and that of the longer edges equals 2. The faces are acute isosceles triangles. The larger angle of these equals $$\arccos\tfrac19 \approx 83.62^{\circ}$$ and the two smaller ones equal $$\arccos\tfrac23 \approx 48.19^{\circ}$$.

Orthogonal projections
The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge.

Uses
Naturally occurring (crystal) formations of tetrahexahedra are observed in copper and fluorite systems.

Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers.

A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron, with the rhombic faces divided into two triangles.

The tetrakis hexahedron appears as one of the simplest examples in building theory. Consider the Riemannian symmetric space associated to the group SL4(R). Its Tits boundary has the structure of a spherical building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices (chambers) can be obtained by taking the radial projection of a tetrakis hexahedron.

Symmetry
With Td, [3,3] (*332) tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 great circles on a sphere. It can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point.

The edges of the spherical tetrakis hexahedron belong to six great circles, which correspond to mirror planes in tetrahedral symmetry. They can be grouped into three pairs of orthogonal circles (which typically intersect on one coordinate axis each). In the images below these square hosohedra are colored red, green and blue.

Dimensions
If we denote the edge length of the base cube by $a$, the height of each pyramid summit above the cube is $\tfrac{a}{4}.$ The inclination of each triangular face of the pyramid versus the cube face is $$\arctan\tfrac{1}{2} \approx 26.565^\circ$$. One edge of the isosceles triangles has length $a$, the other two have length $\tfrac{3a}{4},$ which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of $$\tfrac{\sqrt 5 a}{4}$$ in the triangle. Its area is $$\tfrac{\sqrt 5 a^2}{8},$$ and the internal angles are $$\arccos\tfrac{2}{3} \approx 48.1897^\circ$$ and the complementary $$180^\circ - 2\arccos\tfrac{2}{3} \approx 83.6206^\circ.$$

The volume of the pyramid is $\tfrac{a^3}{12};$ so the total volume of the six pyramids and the cube in the hexahedron is $\tfrac{3a^3}{2}.$

Kleetope
It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube. A non-convex form of this shape, with equilateral triangle faces, has the same surface geometry as the regular octahedron, and a paper octahedron model can be re-folded into this shape. This form of the tetrakis hexahedron was illustrated by Leonardo da Vinci in Luca Pacioli's Divina proportione (1509).

This non-convex form of the tetrakis hexahedron can be folded along the square faces of the inner cube as a net for a four-dimensional cubic pyramid.

Related polyhedra and tilings
It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7.

With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.

Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.