Polyhedroja

Polyhedroja is a term used in geometric literature to denote a family of polyhedra defined by assembling regular polygonal faces around vertices according to a specified configuration. Unlike strictly regular or Archimedean solids, polyhedroja allow mixed face types, such as triangles, squares, and hexagons, while maintaining a regular vertex figure in which the same cyclic order of faces meets at every vertex.

Construction proceeds by selecting a finite set of face types and a vertex configuration, then arranging faces so that at each vertex the same sequence of faces occurs. The solid is completed by adjoining the faces consistently around all vertices. Convex polyhedroja require angle-sum constraints at each vertex; non-convex or star-faced variants may relax these constraints, yielding a broader family. Examples include configurations such as 3-4-3-4 or 3-3-4-4 around each vertex, which, when extended to all vertices, produce symmetric finite solids.

In mathematics, polyhedroja occupy a position between regular and semi-regular solids and the more general classes of uniform and Johnson polyhedra. They are studied for their combinatorial structure, symmetry groups, and potential uses in tiling the sphere or the plane. In computer graphics and architectural design, polyhedroja can offer compact templates for meshes and structural forms with balanced face types.

Etymology and usage: The term polyhedroja is a modern coinage used in some discussions of polyhedral geometry, and there is no single canonical definition. Different authors may adopt varying constraints on vertex configurations or face types, leading to a family of related but non-identical concepts. See also: polyhedron, Archimedean solid, Johnson solid, tiling.