Compound of four tetrahedra explained

bgcolor=#e7dcc3 colspan=2	Compound of 4 digonal antiprisms
align=center colspan=2
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.

Uniform compounds

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

\pi/ 4 . It has dihedral symmetry, D_8h, 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

\pi/ 4 . 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:

Four tetrahedra that are not spread equally in

\pi/ 4 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

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:

With edge-length as a unit, it has a surface area equal to S = (1291)/(210 ⋅ \sqrt 3) ≈ 3 . 5493 .

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

References

• John . Skilling . Uniform Compounds of Uniform Polyhedra . Mathematical Proceedings of the Cambridge Philosophical Society . 79 . 3 . 453–454 . 1976 . 10.1017/S0305004100052440 . 123279687 . 0397554 . 0322.50007 .
• Webb . Robert . Stella Models . Symmetry: Culture and Science . 13 . 3-4 . 391-399 . 2002 . (Figure 6.a "Compounds")
• Web site: Weisstein . Eric W. . Tetrahedron 4-Compound . MathWorld . Wolfram Alpha .

See also

• Compound of three tetrahedra
• Compound of five tetrahedra
• Compound of six tetrahedra

External links

• Tetrahedron 4-Compound (nonuniform) with adjustable angles at GeoGebra