Polyhedraa

Polyhedraa are a hypothetical generalization of polyhedra conceived to explore richer local face arrangements on closed three-dimensional surfaces. A polyhedraan is a finite, connected, compact surface in three-dimensional Euclidean space made of polygonal faces, with edges and vertices arranged so that around every vertex the incident faces follow a repeating sequence of three face types in a fixed cyclic order, for example A-B-C-A-B-C. Each face is drawn from a chosen set of polygons, and the numbers of sides can vary between faces. The overall structure is described by a parameter that encodes the frequency of each face type and the vertex visit sequence, leading to families with different symmetry and curvature properties.

Construction and varieties. Polyhedraa can be generated by systematic operations on known polyhedra, such as alternating truncations, stellations, or face-type substitutions applied along edges. By controlling how many times each face type appears and how they meet at vertices, researchers obtain convex and non-convex realizations, as well as highly symmetric or more irregular examples. Some studies focus on uniform polyhedraa, where all vertices see the same cyclic sequence and all faces belong to a restricted set of polygons.

Properties and relationships. Like ordinary polyhedra, polyhedraa have vertices, edges, and faces that satisfy topological constraints; Euler characteristic is used to classify spherical realizations (typically V−E+F=2). Their study intersects tiling theory, computational geometry, and architectural modeling, offering a framework to examine how local face-type rules influence global shape and symmetry.

Etymology and context. The term polyhedraa combines polyhedron with a plural-like suffix to indicate a family of related solids. It remains a niche concept used primarily in theoretical discussions and speculative design rather than standard geometric practice. See also polyhedra, polyhedrons, tilings.