Compound of four tetrahedra

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Compound of 4 digonal antiprisms
Type Uniform compound
Index

UC23
(n=4, p=2, q=1)

Polyhedra 4 digonal antiprisms (tetrahedra)
Faces 16 triangles
Edges 32
Vertices 16
Symmetry group D8h, order 32
Subgroup restricting to one stella octangula Oh, order 48
D4h, order 16

In geometry, a compound of four tetrahedra can be constructed by four tetrahedra in a number of different symmetry positions.

3D model of a compound of four tetrahedra or digonal antiprisms.

Uniform compounds

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A uniform compound of four tetrahedra can be constructed by rotating tetrahedra along an axis of symmetry C2 (that is the middle of an edge) in multiples of  . It has dihedral symmetry, D8h, and the same vertex arrangement as the convex octagonal prism.

This compound can also be seen as two compounds of stella octangulae fit evenly on the same C2 plane of symmetry, with one pair of tetrahedra shifted  . It is a special case of a p/q-gonal prismatic compound of antiprisms, where in this case the component p/q = 2 is a digonal antiprism, or tetrahedron.

Below are two perspective viewpoints of the uniform compound of four tetrahedra, with each color representing one regular tetrahedron:

Perspectives
   
Top view Side view

Four tetrahedra that are not spread equally in   angles over C2 can still hold uniform symmetry when allowed rotational freedom. In this case, these tetrahedra share a symmetric arrangement over the common axis of symmetry C2 that is rotated by equal and opposite angles. This compound is indexed as UC22, with parameters p/q = 2 and n = 4 as well.

Other compounds

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A nonuniform compound can be generated by rotating tetrahedra about lines extending from the center of each face and through the centroid (as altitudes), with varying degrees of rotation.

A model for this compound polyhedron was first published by Robert Webb, using his program Stella, in 2004, following studies of polyhedron models:

 
Compound of four tetrahedra rotated by  . This particular model was built by Robert Webb using Stella, a computer software for generating polyhedra.

With edge-length as a unit, it has a surface area equal to

 .

This compound is self-dual, meaning its dual polyhedron is the same compound polyhedron.

References

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  • Skilling, John (1976). "Uniform Compounds of Uniform Polyhedra". Mathematical Proceedings of the Cambridge Philosophical Society. 79 (3): 453–454. doi:10.1017/S0305004100052440. MR 0397554. S2CID 123279687. Zbl 0322.50007.
  • Webb, Robert (2002). "Stella Models". Symmetry: Culture and Science. 13 (3–4): 391–399. (Figure 6.a "Compounds")

See also

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