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)
Vertex figureIsosceles triangle, edge lengths 1, 2+2, 2+2
Truncated cube vertfig.png
Faces8 triangles, 6 octagons
Measures (edge length 1)
Dihedral angles8–3:
 8–8: 90°
Central density1
Euler characteristic2
Flag count144
Number of pieces14
Level of complexity1
Related polytopes
DualTriakis octahedron
ConjugateQuasitruncated hexahedron
Topological properties
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".