Tetrakis hexahedron

Tetrakis hexahedron
(Click here for rotating model)
Type	Catalan solid
Coxeter diagram
Conway notation	kC
Face type	V4.6.6 isosceles triangle
Faces	24
Edges	36
Vertices	14
Vertices by type	6{4}+8{6}
Symmetry group	Oh, B3, [4,3], (*432)
Rotation group	O, [4,3]+, (432)
Dihedral angle	143°07′48″ arccos(−⁠4/5⁠)
Properties	convex, face-transitive
Truncated octahedron (dual polyhedron)	Net

Dual compound of truncated octahedron and tetrakis hexahedron. The woodcut on the left is from Perspectiva Corporum Regularium (1568) by Wenzel Jamnitzer.

Die and crystal model

Drawing and crystal model of variant with tetrahedral symmetry called hexakis tetrahedron

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.

As a Kleetope

Non-convex tetrakis hexahedron with equilateral triangle faces

The tetrakis hexahedron can be considered 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).

Denoting 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 $\arctan {\tfrac {1}{2}}\approx 26.565^{\circ }$ (sequence A073000 in the OEIS). 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 (OEIS: A204188). 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 ⁠${\tfrac {3a^{3}}{2}}.$⁠

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.

Cartesian coordinates

Cartesian coordinates for the 14 vertices of a tetrakis hexahedron centered at the origin, are the points $${\bigl (}{\pm {\tfrac {3}{2}}},0,0{\bigr )},\ {\bigl (}0,\pm {\tfrac {3}{2}},0{\bigr )},\ {\bigl (}0,0,\pm {\tfrac {3}{2}}{\bigr )},\ {\bigl (}{\pm 1},\pm 1,\pm 1{\bigr )}.$$

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 $\arccos {\tfrac {1}{9}}\approx 83.62^{\circ }$ and the two smaller ones equal $\arccos {\tfrac {2}{3}}\approx 48.19^{\circ }$.

Applications

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 T_d, [3,3] (*332) tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from six 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.

See also

• Disdyakis triacontahedron
• Disdyakis dodecahedron
• Kisrhombille tiling
• Compound of three octahedra
• Deltoidal icositetrahedron, another 24-face Catalan solid.

References

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Tetrakis hexahedron)

External links

• Weisstein, Eric W., "Tetrakis hexahedron" ("Catalan solid") at MathWorld.
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model
  • Conway Notation for Polyhedra Try: "dtO" or "kC"
• Tetrakis Hexahedron – Interactive Polyhedron model
• The Uniform Polyhedra