Truncated cube

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Truncated cube
Truncated hexahedron.png
Bowers style acronymTic
Coxeter diagramx4x3o (CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png)
Faces8 triangles, 6 octagons
Vertex figureIsosceles triangle, edge lengths 1, 2+2, 2+2
Truncated cube vertfig.png
Measures (edge length 1)
Dihedral angles8–3:
 8–8: 90°
Central density1
Number of external pieces14
Level of complexity3
Related polytopes
DualTriakis octahedron
ConjugateQuasitruncated hexahedron
Abstract & topological properties
Flag count144
Euler characteristic2
SymmetryB3, order 48

The truncated cube, the truncated hexahedron, or tic, is one of the 13 Archimedean solids. It consists of 8 triangles and 6 octagons. Each vertex joins one triangle and two octagons. As the name suggests, it can be obtained by truncation of the cube.

Vertex coordinates[edit | edit source]

A truncated cube of edge length 1 has vertex coordinates given by all permutations of:

Representations[edit | edit source]

A truncated cube has the following Coxeter diagrams:

Semi-uniform variant[edit | edit source]

The truncated cube has a semi-uniform variant of the form x4y3o that maintains its full symmetry. This variant has 8 triangles of size y and 6 ditetragons as faces.

With edges of length a (between two ditetragons) and b (between a ditetragon and a triangle), its circumradius is given by and its volume is given by .

It has coordinates given by all permutations of:

Related polyhedra[edit | edit source]

A truncated cube can be augmented by attaching a square cupola to one of its octagonal faces, forming the augmented truncated cube. If a second square cupola is attached to the opposite octagonal face, the result is the biaugmented truncated cube.

The truncated rhombihedron is a uniform polyhedron compound composed of 5 truncated cubes.

o4o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cube cube {4,3} x4o3o
Uniform polyhedron-43-t0.png
Truncated cube tic t{4,3} x4x3o
Uniform polyhedron-43-t01.png
Cuboctahedron co r{4,3} o4x3o
Uniform polyhedron-43-t1.png
Truncated octahedron toe t{3,4} o4x3x
Uniform polyhedron-43-t12.png
Octahedron oct {3,4} o4o3x
Uniform polyhedron-43-t2.png
Small rhombicuboctahedron sirco rr{4,3} x4o3x
Uniform polyhedron-43-t02.png
Great rhombicuboctahedron girco tr{4,3} x4x3x
Uniform polyhedron-43-t012.png
Snub cube snic sr{4,3} s4s3s
Uniform polyhedron-43-s012.png

External links[edit | edit source]

  • Klitzing, Richard. "tic".