Polyhedrot

Polyhedrot is a term used to describe a family of geometric structures that generalize conventional polyhedra by allowing rotational degrees of freedom at the seams between adjacent polygonal facets. In this framework, a polyhedrot consists of a finite set of polygonal faces connected along edges, with hinges at the edges that permit rotation of neighboring faces relative to each other. The combinatorial skeleton is that of a standard polyhedron, but the geometry is parameterized by a vector of hinge angles.

A polyhedrot of a given combinatorial type is specified by a planar graph together with a hinge-angle specification. If all hinge angles are fixed at 0 or 180 degrees, the object reduces to a conventional polyhedron. If hinge angles vary within feasible ranges, the structure becomes a flexible polyhedral mechanism. Allowing faces to be nonplanar extends the idea to curved facets, creating a broader class of deformations.

Properties of polyhedrots include rigidity or flexibility, depending on the hinge network and facet shapes. The symmetry group of a polyhedrot is generally a subgroup of the ambient isometry group, and the volume can change as the hinges rotate. Realizations are often examined as linkage networks or deployable structures and have connections to origami-inspired folding, tensegrity-inspired assemblies, and kirigami concepts.

Examples are typically described as hypothetical variants such as the tetrahedrot or cube-rot family, used mainly to illustrate how polyhedral combinatorics interact with hinge-based deformations. Polyhedrot is not a standard term in mainstream mathematics; it appears in speculative or educational discussions as a way to explore polyhedra under controlled deformations.

See also: polyhedron, flexible polyhedron, hinge, linkage, origami, tensegrity.