Polyhedrats

Polyhedrats are a class of polyhedral solids defined as the finite union of convex polyhedra arranged to meet face-to-face along a chosen set of congruent polygonal faces, forming a single connected solid whose exterior boundary is a polyhedral surface. Each constituent polyhedron is called a cell; the shared faces become internal, and only the outer faces constitute the boundary. The term emphasizes the assembly process rather than a single canonical shape.

Construction and notation: A polyhedrat is described combinatorially by a gluing graph G, whose vertices correspond to the cells and whose edges indicate shared faces. Each glued face comes from a polygonal type F, such as a p-gon, that matches with the corresponding face on the adjacent cell. A compact shorthand P(p,q,r) may be used to denote a family where p-gonal faces are used, q faces meet at each vertex, and r layers or rounds of gluing occur around each central cell. In the simple case r=1 and G consists of a single glue, the polyhedrat reduces to a conventional polyhedron with Schläfli-type data (p,q).

Properties: Polyhedrats can be convex or non-convex. The symmetry group generally decreases relative to the original cells; vertex-figures may vary across the surface, and the Euler characteristic is computed from the resulting incidence data. The concept overlaps with polyhedral complexes and tilings on spheres or other manifolds, and the exterior surface remains a polygonal tessellation.

Examples and related ideas: If only one cell is used, a polyhedrat is a standard polyhedron. Gluing two congruent cells along a common face yields a simple two-cell polyhedrat, illustrating the assembly concept. Polyhedrats relate to polyhedral complexes, uniform tilings, and broader study in polyhedral combinatorics.

See also: Polyhedron, Polyhedral complex, Uniform polyhedron, Geometric assembly.