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 even permutations 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 (betwwen a ditrigon and a triangle), its circumradius is given by $$\sqrt{\frac{3a^2+4b^2+4ab}{8}}$$ and its volume is given by $$(a^3+6a^2b+12ab^2+4b^3)\frac{\sqrt2}{12}$$.

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)