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Compound of two tetrahedra

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Pair of two dual tetrahedra

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

Stellated octahedron[edit]

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[edit]

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

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[edit]

If two regular tetrahedra are given the same orientation on the 3-fold axis, a different compound is made, with D3h, [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[edit]


  • 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[edit]

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