## Euclidean Patterns in Non-Euclidean Tilings |

The tilings enumerated within the EPINET database are all tilings of the hyperbolic plane by finite bounded tiles. It is also possible to tile the hyperbolic plane by infinite unbounded tiles. The edge–skeletons of these tilings have infinitely many components that may be geodesic lines or infinite trees. If the tiling has a symmetry that is compatible with a TPMS, it can be projected onto the surface to give multi–component networks that may be interwoven, interpenetrating networks, generalised rod packings, or a union of finite polyhedral edge–skeletons.

The following pages display some example tree and line packings in the hyperbolic plane that map onto the P, D, and G, surfaces. For more details see the papers:

- S.T. Hyde and C. Oguey, “Hyperbolic 2D forests and euclidean entangled thickets”, Eur. Phys. J. B, 16, 613–630 (2000). [EDP sciences link]
- S.T. Hyde and S.J. Ramsden, “Polycontinuous morphologies and interwoven helical networks”, Europhys. Lett., 50, 135–141, (2000). [EDP sciences link]

A tiling by tiles with infinitely many sides. The edge–skeleton is a forest of regular 3–coordinated trees with edge–length arccosh(3).

A tiling with unbounded tiles that have no vertices. The edge–skeleton is an arrangement of hyperbolic lines that correspond to the Voronoi partition of the trees in the above forest.

A tiling by tiles with infinitely many sides. The edge–skeleton is a forest of regular 3–coordinated trees with edge–length arccosh(5).

A tiling with unbounded tiles that have no vertices. The edge–skeleton is an arrangement of hyperbolic lines that correspond to the Voronoi partition of the trees in the above forest.

A tiling by tiles with infinitely many sides. The edge–skeleton is a forest of regular 3–coordinated trees with edge–length arccosh(15).

A tiling by tiles with infinitely many sides. The edge–skeleton is a forest of regular 4–coordinated trees with edge–length arccosh(5).