faceregular

Faceregular is a geometric term used to describe a polyhedron in which every face is a regular polygon. A face is regular if it is both equilateral and equiangular, so a faceregular polyhedron may have faces of one or several different regular polygon types, but each face individually meets the regular-polygon criteria. The term is sometimes used interchangeably with “regular-faced polyhedron.”

The concept encompasses a wide range of solids. Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and icosahedron—are by definition faceregular, since all their faces are congruent regular polygons and their vertices are highly symmetric. Archimedean solids also fall into this category because all faces are regular polygons (though not all faces are identical) and the solids exhibit uniform vertex configurations. Johnson solids, a larger family of convex polyhedra with regular polygonal faces but without full vertex-transitivity, are typically described as faceregular as well. In contrast, polyhedra with non-regular faces, such as those with trapezoids or irregular polygons, do not meet the criterion.

In mathematical contexts, faceregular polyhedra are studied for their combinatorial and geometric properties, including face configurations, edge relations, and symmetry. The term is more common in descriptive geometry and historical literature; many modern references prefer the phrase regular-faced polyhedron or simply note that all faces are regular polygons. Faceregular polyhedra also extend to higher-dimensional polytopes, where the analogue is that every facet is a regular polygonal figure. See also regular polyhedra, Archimedean solids, and Johnson solids.