Truncated cube

Truncated cube
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tic
Coxeter diagram	x4x3o ()
Elements
Faces	8 triangles, 6 octagons
Edges	12+24
Vertices	24
Vertex figure	Isosceles triangle, edge lengths 1, √2+√2, √2+√2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{7+4\sqrt2}}{2} ≈ 1.77882$
Volume	$7\frac{3+2\sqrt2}{3} ≈ 13.59966$
Dihedral angles	8–3: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439^\circ$
	8–8: 90°
Central density	1
Number of external pieces	14
Level of complexity	3
Related polytopes
Army	Tic
Regiment	Tic
Dual	Triakis octahedron
Conjugate	Quasitruncated hexahedron
Abstract & topological properties
Flag count	144
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

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:

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

Representations

A truncated cube has the following Coxeter diagrams:

• x4x3o (full symmetry)
• xwwx4xoox&#xt (BC2 axial, octagon-first)
• xwwxoo3ooxwwx&#xt (A2 axial, triangle-first)
• wx3oo3xw&#zx (A3 subsymmetry, as hull of 2 small rhombitetratetrahedra)
• wx xw4xo&#zx (BC2×A1 symmetry)
• wwx wxw xww&#zx (A1×A1×A1 symmetry)
• oxwUwxo xwwxwwx&#xt (A1×A1 axial)

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

It has coordinates given by all permutations of:

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

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

External links

• Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#11).

• Klitzing, Richard. "tic".

• Quickfur. "The Truncated Cube".

• Wikipedia Contributors. "Truncated cube".
• McCooey, David. "Truncated cube"