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In geometry, a tetrahedron (pl.: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra .
Polytetrahedron is a term used for three distinct types of objects, all based on the tetrahedron: . A uniform convex 4-polytope made up of 600 tetrahedral cells.It is more commonly known as a 600-cell or hexacosichoron.
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters: [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
If two regular tetrahedra are given the same orientation on the 3-fold axis, a different compound is made, with D 3h, [3,2] symmetry, order 12.. Other orientations can be chosen as 2 tetrahedra within the compound of five tetrahedra and compound of ten tetrahedra the latter of which can be seen as a hexagrammic pyramid:
A Boerdijk helical sphere packing has each sphere centered at a vertex of the Coxeter helix. Each sphere is in contact with 6 neighboring spheres. The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and Arie Hendrick Boerdijk [], is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices.
A regular tetrahedron, an example of a solid with full tetrahedral symmetry. A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
There are three polyhedral groups: The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron.It is isomorphic to A 4.. The conjugacy classes of T are:
Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (tetrahedrons and octahedrons in the case of the rectified 5-cell). The Coxeter symbol for the rectified 5-cell is 0 21 .
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