Compound of two tetrahedra

In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.

Pair of two dual tetrahedra

Stellated octahedron

Further information: stellated octahedron

There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral symmetry, order 48. It has a regular octahedron core, and shares the same 8 vertices with the cube.

If the edge crossings were treated as their own vertices, the compound would have identical surface topology to the rhombic dodecahedron; were face crossings also considered edges of their own the shape would effectively become a nonconvex triakis octahedron.

A tetrahedron and its dual tetrahedron

The intersection of both solids is the octahedron, and their convex hull is the cube.

Orthographic projections from the different symmetry axes

If the edge crossings were vertices, the mapping on a sphere would be the same as that of a rhombic dodecahedron.

Lower symmetry constructions

There are lower symmetry variations on this compound, based on lower symmetry forms of the tetrahedron.

• A facetting of a rectangular cuboid, creating compounds of two tetragonal or two rhombic disphenoids, with a bipyramid or rhombic fusil cores. This is first in a set of uniform compound of two antiprisms.
• A facetting of a trigonal trapezohedron creates a compound of two right triangular pyramids with a triangular antiprism core. This is first in a set of compounds of two pyramids positioned as point reflections of each other.

Examples

D4h, [4,2], order 16	C4v, [4], order 8	D3d, [2+,6], order 12
Compound of two tetragonal disphenoids in square prism ß{2,4} or	Compound of two digonal disphenoids	Compound of two right triangular pyramids in triangular trapezohedron

Other compounds

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:

See also

• Compound of cube and octahedron
• Compound of dodecahedron and icosahedron
• Compound of small stellated dodecahedron and great dodecahedron
• Compound of great stellated dodecahedron and great icosahedron

References

• Cundy, H. and Rollett, A. "Five Tetrahedra in a Dodecahedron". §3.10.8 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 139–141, 1989.

External links

• Weisstein, Eric W. "Compound of two tetrahedra". MathWorld.
• Compounds of Polyhedra VRML model: [1]