Truncated tetrahedron

The truncated tetrahedron, or tut, is one of the 13 Archimedean solids, and the only one with tetrahedral symmetry. It consists of 4 triangles and 4 hexagons. Each vertex joins one triangle and two hexagons. As the name suggests, it can be obtained by truncation of the tetrahedron.

Vertex coordinates
A truncated tetrahedron of edge length 1 has vertex coordinates given by all permutations and even sign changes of:
 * $$\left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right).$$

Representations
A truncated tetrahedron has the following Coxeter diagrams:


 * x3x3o (full symmetry)
 * s4o3x (as triangle-alternated small rhombicuboctahedron)
 * xux3oox&#xt (A2 axial, triangle-first)
 * xuxo oxux&#xt (A1×A1 axial, edge-first)

Semi-uniform variant
The truncated tetrahedron has a semi-uniform variant of the form x3y3o that maintains its full symmetry. This variant has 4 triangles of size y and 4 ditrigons as faces.

With edges of length a (between two ditrigons) and b (between a ditrigon and a triangle), its circumradius is given by $$\sqrt{\frac{3a^2+4b^2+4ab}{8}}$$ and its volume is given by $$(a^3+6a^2b+12ab^2+4b^3)\frac{\sqrt2}{12}$$.

It has coordinates given by all permutations and even sign changes of:


 * $$\left((a+2b)\frac{\sqrt2}{4},\,a\frac{\sqrt2}{4},\,a\frac{\sqrt2}{4}\right).$$

These semi-uniform truncated tetrahedra occur as vertex figures of two uniform polychora, the small ditetrahedronary hexacosihecatonicosachoron and ditetrahedronary dishecatonicosachoron.

Related polyhedra
It is possible to augment one of the hexagonal faces of the truncated tetrahedron with a triangular cupola to form the augmented truncated tetrahedron.

A number of uniform polyhedron compounds are composed of truncated tetrahedra:


 * Truncated stella octangula (2)
 * Truncated chiricosahedron (5)
 * Truncated icosicosahedron (10)