disphenoids

Disphenoids, also known as equifacial tetrahedra, are tetrahedra in which all four faces are congruent triangles. This uniformity of faces is the defining feature and leads to a distinctive equality of opposite edges: AB = CD, AC = BD, and AD = BC. As a consequence, the four triangular faces are all congruent, so the disphenoid is a special case of an isosceles tetrahedron.

The symmetry of a disphenoid is high for a tetrahedron. It typically has three mirror planes and three twofold rotation axes, giving it a D2d-type symmetry. The opposite edges being equal also implies a symmetric arrangement of the dihedral angles along corresponding edges.

A standard way to realize a disphenoid is to choose four points arranged so that the pairs of opposite edges are equal in length; geometrically this can be achieved by placing the vertices in a configuration that is symmetric with respect to three orthogonal planes. In this setup the four faces are congruent triangles, and any pair of opposite edges has the same length.

All tetrahedra admit a circumscribed sphere through their four vertices, and disphenoids possess this natural circumsphere just like other tetrahedra. The regularity of their faces and the equality of opposite edges make disphenoids useful in studies of tetrahedral symmetry and in certain geometric constructions where equifacial properties are desirable.

See also: equifacial tetrahedron, isosceles tetrahedron, tetrahedron.