sidepaired

Sidepaired, or side-paired, describes a polyhedron or polygon together with a specified side-pairing: a set of isometries that identify its faces (sides) in pairs. In the planar case, a polygon’s edges are paired by isometries, while in higher dimensions the pairing involves the (n−1)-dimensional faces. The pairing determines how points on identified faces are glued to one another.

A side-paired polyhedron P gives rise to a quotient space P/~ by identifying each point on a side with its image under the corresponding pairing isometry. When the side-pairing maps generate a discrete group acting properly discontinuously on the ambient space, P serves as a fundamental domain for that group, and the quotient space often forms a manifold or orbispace. The resulting space inherits geometric and topological properties from both P and the glue maps.

Side-pairings are central to the constructive approach in geometry and topology for producing manifolds and orbifolds, particularly in hyperbolic geometry and tessellation theory. They also provide concrete methods for presenting groups via generators and relations: the isometries used in the pairings yield a group presentation describing the symmetries of the glued space. Classic examples include polygonal schemes that create closed surfaces of prescribed genus in two dimensions and hyperbolic manifolds in higher dimensions.

Terminology varies; the object may be called a side-paired polyhedron, a polyhedron with a side-pairing, or simply a side pairing. The study of side-pairings intersects combinatorial topology, geometric group theory, and the theory of tessellations.