User:Tomruen/List of hyperbolic symmetry groups

Hyperbolic plane
A first few hyperbolic groups, ordered by their Euler characteristic are:

Hyperbolic groups from regular polygons
Every regular polyhedron/tiling {p,2q} represents a regular polygon reflective domain, orbifold *qp. Higher symmetry groups can be constructed by:
 * 1) Adding a order-p gyration point in the center as p*q. (order ×p)
 * 2) If p has divisor r, (p/r)*qr.  (order ×p/r)
 * 3) An order-p reflection point in the center creates right triangle domains, *(2q).p.2.  (order ×2p)
 * 4) If p is even,
 * 5) you can make an isoceles triangle domain *2q.2q.(p/2), (order ×p)
 * 6) as well as triangle *p.p.q  (order ×p)
 * 7) and also by an alternate set of p/2 central mirrors, as kite-shaped fundamental domains *2.q.2.(p/2).  (order ×p)

Mutations of orbifolds
Orbifolds with the same structure can be mutated between different symmetry classes, including across curvature domains from spherical, to Euclidean to Hyperbolic. This table shows mutation classes. This table is not complete for possible hyperbolic orbifolds.

Example, comparing 22* symmetry of the plane to 23* symmetry of the hyperbolic plane:
 * 23star orbifold example.png