Compound of four tetrahedra
Introduction Compound of 4 digonal antiprismsframelessTypeUniform compoundIndexUC23(n=4, p=2, q=1)Polyhedra4 digonal antiprisms (tetrahedra)Faces16 trianglesEdges32Vertices16Symmetry group''D''8h, order 32Subgroup restricting to one stella octangula''O''h, order 48''D''4h, order 163D model of a compound of four tetrahedra or digonal antiprisms.In geometry, a '''compound of four tetrahedra''' can be constructed by four tetrahedra in a number of different symmetry positions.
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 . 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 . 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:+ Perspectives200px200pxTop viewSide viewFour tetrahedra that are not spread equally in  angles over ''C''2 can still hold uniform symmetry when allowed rotational freedom. In this case, these tetrahedra share a symmetric arrangement over the common axis of symmetry ''C''2 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 . 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 .This compound is self-dual, meaning its dual polyhedron is the same compound polyhedron.
References ** (Figure 6.a "Compounds")*
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''*: