# Truncated tetrahedron

(Redirected from Tut)
Truncated tetrahedron
Rank3
TypeUniform
Notation
Bowers style acronymTut
Coxeter diagramx3x3o ()
Conway notationtT
Stewart notationT3
Elements
Faces4 triangles, 4 hexagons
Edges6+12
Vertices12
Vertex figureIsosceles triangle, edge lengths 1, 3, 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {22}}{4}}\approx 1.17260}$
Volume${\displaystyle {\frac {23{\sqrt {2}}}{12}}\approx 2.71057}$
Dihedral angles6–3: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
6–6: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density1
Number of external pieces8
Level of complexity3
Related polytopes
ArmyTut
RegimentTut
DualTriakis tetrahedron
Petrie dualPetrial truncated tetrahedron
ConjugateNone
Abstract & topological properties
Flag count72
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA3, order 24
Flag orbits3
ConvexYes
NatureTame

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:

• ${\displaystyle \left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)}$.

Scaling these coordinates by a factor of ${\displaystyle 2{\sqrt {2}}}$ results in integer coordinates, making the truncated tetrahedron an integral polytope.

## Representations

A truncated tetrahedron has the following Coxeter diagrams:

## 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 ${\displaystyle {\sqrt {\frac {3a^{2}+4b^{2}+4ab}{8}}}}$ and its volume is given by ${\displaystyle (a^{3}+6a^{2}b+12ab^{2}+4b^{3}){\frac {\sqrt {2}}{12}}}$.

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

• ${\displaystyle \left((a+2b){\frac {\sqrt {2}}{4}},\,a{\frac {\sqrt {2}}{4}},\,a{\frac {\sqrt {2}}{4}}\right)}$.

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: