disphenoidal

Disphenoidal refers to a tetrahedron whose four faces are congruent triangles. It is also known as an equifacial tetrahedron or isosceles tetrahedron. A convenient equivalent condition is that opposite edges are equal in length: if the vertices are labeled A, B, C, D, then AB = CD, AC = BD, and AD = BC. With opposite edges paired this way, every face has the same side lengths, so all four triangular faces are congruent.

This symmetry implies several geometric features. The three pairs of opposite edges are equal, and the three dihedral angles along opposite edges are equal. A disphenoidal tetrahedron possesses a high degree of symmetry: its rotational symmetry group is typically D2d, with three two-fold axes through the midpoints of opposite edges and corresponding mirror planes.

Construction and models often use three edge-length parameters a, b, c to represent the lengths of the opposite-edge pairs: AB = CD = a, AC = BD = b, and AD = BC = c. Any tetrahedron with these edge equalities is disphenoidal, and conversely a disphenoid has this edge pattern. A simple way to picture it is by taking two congruent triangles in parallel planes and connecting corresponding vertices so that the resulting tetrahedron has four congruent faces.

In applications, disphenoids appear in geometry as a benchmark for symmetry in tetrahedra and in crystallography to describe a tetragonal disphenoidal crystal habit. They are a natural generalization of isosceles triangles into three dimensions and serve as a classic example of equifacial polyhedra.

See also: Isosceles tetrahedron; Disphenoid.