geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a compound of two

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

is constructed by two overlapping

, usually implied as regular tetrahedra.

Stellated octahedron

There is only one uniform polyhedral compound, the

stellated octahedron The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depi ...

, which has

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhed ...

, order 48. It has a regular

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...

core, and shares the same 8 vertices with the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...

. If the edge crossings were treated as their own vertices, the compound would have identical surface topology to the

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...

; were face crossings also considered edges of their own the shape would effectively become a nonconvex

triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube. It can be seen as an octahedron with triangul ...

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 In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...

, creating compounds of two tetragonal or two rhombic disphenoids, with a

bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does ...

or rhombic fusil cores. This is first in a set of uniform compound of two antiprisms. * A facetting of a

trigonal trapezohedron In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or '' isohedral rh ...

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 reflection In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

s of each other.

Other compounds

If two regular tetrahedra are given the same orientation on the 3-fold axis, a different compound is made, with D_3h, ,2symmetry, order 12. : Compound two tetrahedra twisted

Other orientations can be chosen as 2 tetrahedra within the

compound of five tetrahedra The compound of five tetrahedra is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876. It can be seen as a faceting of a regular ...

and

compound of ten tetrahedra The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876. It can be seen as a ...

the latter of which can be seen as a

hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...

mic pyramid: : Compound_tetrahedra_2_of_5

References

* Cundy, H. and Rollett, A. "Five Tetrahedra in a Dodecahedron". §3.10.8 in ''

Mathematical Models A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

'', 3rd ed. Stradbroke, England: Tarquin Pub., pp. 139-141, 1989.

External links

* {{mathworld , urlname = Tetrahedron2-Compound , title = Compound of two tetrahedra
Compounds of Polyhedra

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...

model

Polyhedral compounds

Stellated octahedron

Lower symmetry constructions

Other compounds

See also

References

External links