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. 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 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. Area and volume The area A and the volume V of a truncated tetrahedron of edge length a are: Densest packing The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = , as reported by two independent groups using Monte Carlo methods. Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independency of the findings make it unlikely that an even denser packing is to be found. In fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space. Cartesian coordinates Cartesian coordinates for the 12 vertices of a truncated tetrahedron centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs: (+3,+1,+1), (+1,+3,+1), (+1,+1,+3) (−3,−1,+1), (−1,−3,+1), (−1,−1,+3) (−3,+1,−1), (−1,+3,−1), (−1,+1,−3) (+3,−1,−1), (+1,−3,−1), (+1,−1,−3) Another simple construction exists in 4-space as cells of the truncated 16-cell, with vertices as coordinate permutation of: (0,0,1,2) Orthogonal projection Spherical tiling The truncated tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. Friauf polyhedron A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D2d symmetry) is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a