scalenohedra

Scalenohedron is a polyhedron all of whose faces are scalene triangles—triangles whose three side lengths are pairwise different. The term combines “scalene” with the suffix for a solid, and scalenohedra may be either convex or non-convex. In many contexts they belong to the broader class of simplicial polyhedra, since every face is a triangle and the surface is topologically a sphere.

In a scalenohedron, each face is a triangle with three distinct edge lengths, so no face is equilateral or isosceles. The polyhedron is determined by a triangulation of the sphere; combinatorially it has F triangular faces, E edges, and V vertices, with 3F = 2E and V − E + F = 2.

Construction and examples can be understood by starting from any convex triangulation of the sphere and applying a small generic perturbation to vertex positions. With probability 1, such a perturbation destroys any equal-length edges in faces, yielding a scalenohedron. There are infinitely many distinct combinatorial types of scalenohedra.

Crystallography usage: the term also appears in mineralogy, where a scalenohedron denotes a crystal form bounded by scalene triangular faces. Calcite is frequently observed in scalenohedral forms, producing crystals with elongated, irregular triangular faces.

See also: Triangulated polyhedron, Simplicial polyhedron, Crystallography crystal form.