Truncated tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.
| Truncated tetrahedron | |
|---|---|
(Click here for rotating model) | |
| Type | Archimedean solid Uniform polyhedron |
| Elements | F = 8, E = 18, V = 12 (χ = 2) |
| Faces by sides | 4{3}+4{6} |
| Conway notation | tT |
| Schläfli symbols | t{3,3} = h2{4,3} |
| t0,1{3,3} | |
| Wythoff symbol | 2 3 | 3 |
| Coxeter diagram | = |
| Symmetry group | Td, A3, [3,3], (*332), order 24 |
| Rotation group | T, [3,3]+, (332), order 12 |
| Dihedral angle | 3-6: 109°28′16″ 6-6: 70°31′44″ |
| References | U02, C16, W6 |
| Properties | Semiregular convex |
Colored faces |
3.6.6 (Vertex figure) |
Triakis tetrahedron (dual polyhedron) |
Net |
A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron.
A truncated tetrahedron is the Goldberg polyhedron GIII(1,1), containing triangular and hexagonal faces.
A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.