Tetrakis hexahedron  

(Click here for rotating model) 

Type  Catalan solid 
Coxeter diagram  
Face type  isosceles triangle 
Faces  24 
Edges  36 
Vertices  14 
Vertices by type  6{4}+8{6} 
Face configuration  V4.6.6 
Symmetry group  O_{h}, BC_{3}, [4,3], (*432) 
Rotation group  O, [4,3]^{+}, (432) 
Dihedral angle  143° 7' 48" 
Properties  convex, facetransitive 
Truncated octahedron (dual polyhedron) 
Net 
In geometry, a tetrakis hexahedron (also known as a tetrahexahedron) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube.
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.
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 five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B_{2} and A_{2} Coxeter planes.
Centered by  Vertex  Edge 46 
Edge 66 
Face Square 
Face Hexagon 

Truncated octahedron 

Hexakis hexahedron 

Projective symmetry 
[2]  [2]  [2]  [4]  [6] 
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.
Symmetry
With T_{d}, [3,3] (*332) tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahdral symmetry. This polyhedron can be constructed from 6 great circles on a sphere.
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 degrees (sequence A073000 in 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 √5 a/4 in the triangle ( A204188). Its area is √5a/8, and the internal angles are arccos(2/3) (approximately 48.1897 degrees) and the complementary 1802arccos(2/3) (approximately 83.6206 degrees).
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.
Related polyhedra and tilings
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{4,3} 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} 
Symmetry *n42 [n,3] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  

*232 [2,3] D_{3h} 
*332 [3,3] T_{d} 
*432 [4,3] O_{h} 
*532 [5,3] I_{h} 
*632 [6,3] P6m 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 

Order  12  24  48  120  ∞  
Truncated figures 
2.6.6 
3.6.6 
4.6.6 
5.6.6 
6.6.6 
7.6.6 
8.6.6 
∞.6.6 
Coxeter Schläfli 
t{3,2} 
t{3,3} 
t{3,4} 
t{3,5} 
t{3,6} 
t{3,7} 
t{3,8} 
t{3,∞} 
Uniform dual figures  
nkis figures 
V2.6.6 
V3.6.6 
V4.6.6 
V5.6.6 
V6.6.6 
V7.6.6 
V8.6.6 
V∞.6.6 
Coxeter 
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
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.
Symmetry *n32 [n,3] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  

*232 [2,3] D_{3h} 
*332 [3,3] T_{d} 
*432 [4,3] O_{h} 
*532 [5,3] I_{h} 
*632 [6,3] P6m 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 

Coxeter Schläfli 
tr{2,3} 
tr{3,3} 
tr{4,3} 
tr{5,3} 
tr{6,3} 
tr{7,3} 
tr{8,3} 
tr{∞,3} 
Omnitruncated figure 

Vertex figure  4.6.4  4.6.6  4.6.8  4.6.10  4.6.12  4.6.14  4.6.16  4.6.∞ 
Dual figures  
Coxeter  
Omnitruncated duals 

Face configuration 
V4.6.4  V4.6.6  V4.6.8  V4.6.10  V4.6.12  V4.6.14  V4.6.16  V4.6.∞ 
See also
References
 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
