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.

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.

Examples
D4h, [4,2], order 16 D4, [4], order 8 D3d, [2+,6], order 12
Compound of two disphenoids.png
Compound of two tetragonal disphenoids in square prism
ß{2,4} or CDel node h3.pngCDel 2x.pngCDel node h3.pngCDel 4.pngCDel node.png
Digonal disphenoid compound.png
Compound of two digonal disphenoids
Compound of two triangular pyramids.png
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.

Compound two tetrahedra twisted.png

Other orientations can be chosen as 2 tetrahedra within the compound of five tetrahedra and compound of ten tetrahedral the latter of which can be seen as a hexagrammic pyramid:

Compound tetrahedra 2 of 5.pngCompound of tetrahedra 2 of 10.png

References[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]