Tetrakis hexahedron

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Tetrakis hexahedron
Tetrakishexahedron.jpg
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Conway notation kC
Face type V4.6.6
DU08 facets.png

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° 7' 48"
 \arccos ( -\frac{4}{5} )
Properties convex, face-transitive
Truncated octahedron.png
Truncated octahedron
(dual polyhedron)
Tetrakis hexahedron Net
Net

In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, kiscube[1] or the jetty snowflake) 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.

Orthogonal projections[edit]

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
Dual cube t12 e66.png Dual cube t12 B2.png Dual cube t12.png
Truncated
octahedron
Cube t12 e66.png 3-cube t12 B2.svg 3-cube t12.svg

Uses[edit]

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[edit]

With Td, [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. 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.

Tetrahedral reflection domains.png Disdyakis cube.png Tetrakis hexahedron tetrahedral.png Tetrakis hexahedron cubic.png

Seen in stereographic projection 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:

Tetrakis hexahedron stereographic D4.png Tetrakis hexahedron stereographic D3.png Tetrakis hexahedron stereographic D2.png
[4] [3] [2]

Dimensions[edit]

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 (OEISA204188). Its area is √5a/8, and the internal angles are arccos(2/3) (approximately 48.1897 degrees) and the complementary 180-2arccos(2/3) (approximately 83.6206 degrees).

The volume of the pyramid is a3/12; so the total volume of the six pyramids and the cube in the hexahedron is 3a3/2.

Kleetope[edit]

It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube.

Pyramid augmented cube.png

Cubic pyramid[edit]

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[edit]

Spherical tetrakis hexahedron
Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+, (432) [3+,4], (3*2)
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t12.svg Uniform polyhedron-43-t2.svg Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-43-h01.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
{4,3} t{4,3} r{4,3} t{3,4} {3,4} rr{4,3} tr{4,3} sr{4,3} s{3,4}
Duals to uniform polyhedra
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Dodecahedron.svg
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V35
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact hyperb. 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
Hexagonal dihedron.png Uniform tiling 332-t12.png Uniform tiling 432-t12.png Uniform tiling 532-t12.png Uniform tiling 63-t12.png H2 tiling 237-6.png H2 tiling 238-6.png H2 tiling 23i-6.png H2 tiling 23j12-6.png H2 tiling 23j9-6.png H2 tiling 23j-6.png
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
Hexagonal Hosohedron.svg Spherical triakis tetrahedron.png Spherical tetrakis hexahedron.png Spherical pentakis dodecahedron.png Uniform tiling 63-t2.png Order3 heptakis heptagonal til.png H2checkers 334.png H2checkers 33i.png
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 \ge 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 Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png H2 tiling 23j12-7.png H2 tiling 23j9-7.png H2 tiling 23j6-7.png H2 tiling 23j3-7.png
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 Spherical hexagonal bipyramid.png Spherical tetrakis hexahedron.png Spherical disdyakis dodecahedron.png Spherical disdyakis triacontahedron.png Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg H2checkers 237.png H2checkers 238.png H2checkers 23i.png H2 checkers 23j12.png H2 checkers 23j9.png H2 checkers 23j6.png H2 checkers 23j3.png
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[edit]

References[edit]

  1. ^ Conway, Symmetries of things, p.284

External links[edit]