Truncated tetrahedron

Truncated tetrahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tut
Coxeter diagram	x3x3o ()
Stewart notation	T3
Elements
Faces	4 triangles, 4 hexagons
Edges	6+12
Vertices	12
Vertex figure	Isosceles triangle, edge lengths 1, √3, √3 ;
Measures (edge length 1)
Circumradius
Volume
Dihedral angles	6–3:
	6–6:
Central density	1
Number of pieces	8
Level of complexity	3
Related polytopes
Army	Tut
Regiment	Tut
Dual	Triakis tetrahedron
Conjugate	None
Abstract properties
Flag count	72
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A3, order 24
Convex	Yes
Nature	Tame

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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:

o3o3o truncations
Name	OBSA	Schläfli symbol	CD diagram
Tetrahedron	tet	{3,3}	x3o3o
Truncated tetrahedron	tut	t{3,3}	x3x3o
Tetratetrahedron = Octahedron	oct	r{3,3}	o3x3o
Truncated tetrahedron	tut	t{3,3}	o3x3x
Tetrahedron	tet	{3,3}	o3o3x
Small rhombitetratetrahedron = Cuboctahedron	co	rr{3,3}	x3o3x
Great rhombitetratetrahedron = Truncated octahedron	toe	tr{3,3}	x3x3x
Snub tetrahedron = Icosahedron	ike	sr{3,3}	s3s3s