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.
Construction The truncated tetrahedron can be constructed from a regular tetrahedron by cutting all of its vertices off, a process known as truncation. The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices. With edge length 1, the Cartesian coordinates of the 12 vertices are the permutations of
( ± 3 2 4 , ± 2 4 , ± 2 4 ) {\displaystyle {\bigl (}{\pm {\tfrac {3{\sqrt {2}}}{4}}},\pm {\tfrac {\sqrt {2}}{4}},\pm {\tfrac {\sqrt {2}}{4}}{\bigr )}}
that have an even number of minus signs.
Properties Given the edge length a {\displaystyle a} . The surface area of a truncated tetrahedron A {\displaystyle A} is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume V {\displaystyle V} is:
A = 7 3 a 2 ≈ 12.124 a 2 , V = 23 12 2 a 3 ≈ 2.711 a 3 . {\displaystyle {\begin{aligned}A&=7{\sqrt {3}}a^{2}&&\approx 12.124a^{2},\\V&={\tfrac {23}{12}}{\sqrt {2}}a^{3}&&\approx 2.711a^{3}.\end{aligned}}}
The dihedral angle of a truncated tetrahedron between triangle-to-hexagon is approximately 109.47°, and that between adjacent hexagonal faces is approximately 70.53°. The densest packing of the truncated tetrahedron is believed to be Φ = 207 208 {\textstyle \Phi ={\frac {207}{208}}} , as reported by two independent groups using Monte Carlo methods by Damasceno, Engel & Glotzer (2012) and Jiao & Torquato (2011). Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independence of the findings make it unlikely that an even denser packing is to be found. If the truncation of the corners is slightly smaller than that of a truncated tetrahedron, this new shape can be used to fill space completely.
The truncated tetrahedron is an Archimedean solid, meaning it is a vertex-transitive and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. The truncated tetrahedron has the same three-dimensional group symmetry as the regular tetrahedron, the tetrahedral symmetry T h {\displaystyle \mathrm {T} _{\mathrm {h} }} . The polygonal faces that meet for every vertex are one equilateral triangle and two regular hexagons, and the vertex figure is denoted as 3 ⋅ 6 2 {\displaystyle 3\cdot 6^{2}} . Its dual polyhedron is triakis tetrahedron, a Catalan solid, shares the same symmetry as the truncated tetrahedron.
Related polyhedra
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