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  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, tetrakis cube, and kiscube^{[2]}) 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.^{[3]}
Cartesian coordinates
Cartesian coordinates for the 14 vertices of a tetrakis hexahedron centered at the origin, are the points [math]\displaystyle{ \bigl( {\pm\tfrac32}, 0, 0 \bigr),\ \bigl( 0, \pm\tfrac32, 0 \bigr),\ \bigl( 0, 0, \pm\tfrac32 \bigr),\ \bigl({\pm1}, \pm1, \pm1 \bigr). }[/math]
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 [math]\displaystyle{ \arccos\tfrac19 \approx 83.62^{\circ} }[/math] and the two smaller ones equal [math]\displaystyle{ \arccos\tfrac23 \approx 48.19^{\circ} }[/math].
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.
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Truncated octahedron 
Disdyakis hexahedron 
Deltoidal dodecahedron 
Rhombic hexahedron 
Tetrahedron 
Spherical polyhedron  

(see rotating model)  Orthographic projections from 2, 3 and 4fold axes 
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.
Stereographic projections  

2fold  3fold  4fold  
Dimensions
If we denote the edge length of the base cube by a, the height of each pyramid summit above the cube is [math]\displaystyle{ \tfrac{a}{4}. }[/math] The inclination of each triangular face of the pyramid versus the cube face is [math]\displaystyle{ \arctan\tfrac{1}{2} \approx 26.565^\circ }[/math] (sequence A073000 in the OEIS). One edge of the isosceles triangles has length a, the other two have length [math]\displaystyle{ \tfrac{3a}{4}, }[/math] which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of [math]\displaystyle{ \tfrac{\sqrt 5 a}{4} }[/math] in the triangle (OEIS: A204188). Its area is [math]\displaystyle{ \tfrac{\sqrt 5 a^2}{8}, }[/math] and the internal angles are [math]\displaystyle{ \arccos\tfrac{2}{3} \approx 48.1897^\circ }[/math] and the complementary [math]\displaystyle{ 180^\circ  2\arccos\tfrac{2}{3} \approx 83.6206^\circ. }[/math]
The volume of the pyramid is [math]\displaystyle{ \tfrac{a^3}{12}; }[/math] so the total volume of the six pyramids and the cube in the hexahedron is [math]\displaystyle{ \tfrac{3a^3}{2}. }[/math]
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 nonconvex form of this shape, with equilateral triangle faces, has the same surface geometry as the regular octahedron, and a paper octahedron model can be refolded into this shape.<ref>{{citation
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This nonconvex form of the tetrakis hexahedron can be folded along the square faces of the inner cube as a net for a fourdimensional 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.
See also
 Disdyakis triacontahedron
 Disdyakis dodecahedron
 Kisrhombille tiling
 Compound of three octahedra
 Deltoidal icositetrahedron, another 24face Catalan solid.
References
 ↑ Hexakistetraeder in German, see e.g. Meyers page and Brockhaus page. The same drawing appears in Brockhaus and Efron as преломленный пирамидальный тетраэдр (refracted pyramidal tetrahedron).
 ↑ Conway, Symmetries of Things, p.284
 ↑ Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of Bernoulli: the fortyeight faces of a mathematical gem", Milan Journal of Mathematics 78 (2): 643–682, doi:10.1007/s0003201001245
 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, doi:10.1017/CBO9780511569371, ISBN 9780521543255 (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron)
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, 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
Original source: https://en.wikipedia.org/wiki/Tetrakis hexahedron.
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