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)
Tetrakis hexahedron Net
Net
Dual compound of truncated octahedron and tetrakis hexahedron. The woodcut on the left is from Perspectiva Corporum Regularium (1568) by Wenzel Jamnitzer.
Drawing and crystal model of variant with tetrahedral symmetry called hexakis tetrahedron [1]

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

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 and the two smaller ones equal .

Orthogonal projections

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

Orthogonal projections
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 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.

Truncated
octahedron
Disdyakis
hexahedron
Deltoidal
dodecahedron
Rhombic
hexahedron
Tetrahedron

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 The inclination of each triangular face of the pyramid versus the cube face is (sequence A073000 in the OEIS). One edge of the isosceles triangles has length a, the other two have length which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of in the triangle (OEISA204188). Its area is and the internal angles are and the complementary

The volume of the pyramid is so the total volume of the six pyramids and the cube in the hexahedron is

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 3D net for a 4D 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{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}

=

=

=
=
or
=
or
=





Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
*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
n-kis
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 mutation 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

References

  1. ^ 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).
  2. ^ Conway, Symmetries of Things, p.284
  3. ^ Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem", Milan Journal of Mathematics, 78 (2): 643–682, doi:10.1007/s00032-010-0124-5, MR 2781856