# Truncated cube

Truncated cube
Rank3
TypeUniform
SpaceSpherical
Bowers style acronymTic
Coxeter diagramx4x3o ()
Elements
Vertex figureIsosceles triangle, edge lengths 1, 2+2, 2+2
Faces8 triangles, 6 octagons
Edges12+24
Vertices24
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{7+4\sqrt2}}{2} ≈ 1.77882}$
Volume${\displaystyle 7\frac{3+2\sqrt2}{3} ≈ 13.59966}$
Dihedral angles8–3: ${\displaystyle \arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439^\circ}$
8–8: 90°
Central density1
Euler characteristic2
Flag count144
Number of pieces14
Level of complexity1
Related polytopes
ArmyTic
RegimentTic
DualTriakis octahedron
ConjugateQuasitruncated hexahedron
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

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

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

• ${\displaystyle \left(±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).}$

## Representations

A truncated cube has the following Coxeter diagrams:

## Semi-uniform variant

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 ${\displaystyle \frac{\sqrt{3a^2+4b^2+4ab\sqrt2}}{2}}$ and its volume is given by ${\displaystyle a^3+6ab^2+(9a^2b+5b^3)\frac{\sqrt2}{3}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(±\frac{a+b\sqrt2}{2},\,±\frac{a+b\sqrt2}{2},\,±\frac{a}{2}\right).}$

## Related polyhedra

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
Truncated cube tic t{4,3} x4x3o
Cuboctahedron co r{4,3} o4x3o
Truncated octahedron toe t{3,4} o4x3x
Octahedron oct {3,4} o4o3x
Small rhombicuboctahedron sirco rr{4,3} x4o3x
Great rhombicuboctahedron girco tr{4,3} x4x3x
Snub cube snic sr{4,3} s4s3s