Compound of four tetrahedra

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

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:

Perspectives

Top view	Side view

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:

Compound of four tetrahedra rotated by $\pi /3$. 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

$S={\frac {1291}{210\cdot {\sqrt {3}}}}\approx 3.5493$.

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

References

• 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")

• 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

  Rotating model on YouTube