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  O_{h}, B_{3}, [4,3], (*432) 
Rotation group  O, [4,3]^{+}, (432) 
Dihedral angle  143°07′48″ arccos(−4/5) 
Properties  convex, facetransitive 
Truncated octahedron (dual polyhedron) 
Net 
In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron and kiscube^{[1]}) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.
It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron.
Contents
Orthogonal projections
The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one midedge.
Projective symmetry 
[2]  [4]  [6] 

Tetrakis hexahedron 

Truncated octahedron 
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 24cell viewed under a vertexfirst 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 SL_{4}(R). Its Tits boundary has the structure of a spherical building whose apartments are 2dimensional 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 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, midedges, and a central point.
The edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles. These edges can also be seen as a compound of 3 orthogonal square hosohedrons.
stereographic projection  Perspective  

[4]  [3]  [2]  Skew 
Dimensions
If we denote the edge length of the base cube by a, the height of each pyramid summit above the cube is a/4. The inclination of each triangular face of the pyramid versus the cube face is arctan(1/2), approximately 26.565° (sequence A073000 in the OEIS). One edge of the isosceles triangles has length a, the other two have length 3a/4, which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of √5a/4 in the triangle ( A204188). Its area is √5a/8, and the internal angles are arccos(2/3) (approximately 48.1897°) and the complementary 180° − 2 arccos(2/3) (approximately 83.6206°).
The volume of the pyramid is a^{3}/12; so the total volume of the six pyramids and the cube in the hexahedron is 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.
Cubic pyramid
It is very similar to the net for a cubic pyramid, as the net for a square based is a square with triangles attached to each edge, the net for a cubic pyramid is a cube with square pyramids attached to each face.
Related polyhedra and tilings
Uniform octahedral polyhedra  

Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2) 

{4,3}  t{4,3}  r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
= 

Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
*n32 symmetry mutation of truncated tilings: n.6.6  

Sym. *n42 [n,3] 
Spherical  Euclid.  Compact  Parac.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 
[12i,3]  [9i,3]  [6i,3]  
Truncated figures 

Config.  2.6.6  3.6.6  4.6.6  5.6.6  6.6.6  7.6.6  8.6.6  ∞.6.6  12i.6.6  9i.6.6  6i.6.6  
nkis figures 

Config.  V2.6.6  V3.6.6  V4.6.6  V5.6.6  V6.6.6  V7.6.6  V8.6.6  V∞.6.6  V12i.6.6  V9i.6.6  V6i.6.6 
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.
*n32 symmetry mutations of omnitruncated tilings: 4.6.2n  

Sym. *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3] 
*∞32 [∞,3] 
[12i,3] 
[9i,3] 
[6i,3] 
[3i,3] 

Figures  
Config.  4.6.4  4.6.6  4.6.8  4.6.10  4.6.12  4.6.14  4.6.16  4.6.∞  4.6.24i  4.6.18i  4.6.12i  4.6.6i 
Duals  
Config.  V4.6.4  V4.6.6  V4.6.8  V4.6.10  V4.6.12  V4.6.14  V4.6.16  V4.6.∞  V4.6.24i  V4.6.18i  V4.6.12i  V4.6.6i 
See also
 Disdyakis triacontahedron
 Disdyakis dodecahedron
 Kisrhombille tiling
 Compound of three octahedra
 Deltoidal icositetrahedron, another 24face Catalan solid.
References
 ^ Conway, Symmetries of Things, p.284
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 9780521543255, MR 730208 (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron)
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Tetrakis hexahedron)
External links
 Eric W. Weisstein, 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