Order-4 apeirogonal tiling
| Order-4 apeirogonal tiling | |
|---|---|
Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | ∞4 |
| Schläfli symbol | {∞,4} r{∞,∞} t(∞,∞,∞) t0,1,2,3(∞,∞,∞,∞) |
| Wythoff symbol | 4 | ∞ 2 2 | ∞ ∞ ∞ ∞ | ∞ |
| Coxeter diagram | |
| Symmetry group | [∞,4], (*∞42) [∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞) (*∞∞∞∞) |
| Dual | Infinite-order square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive edge-transitive |
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It covers the hyperbolic plane, which is a non-Euclidean surface with constant negative curvature, with a repeating pattern of congruent shapes that fill the plane completely without gaps or overlaps.
This tiling is made from apeirogons, which are polygons with infinitely many sides. In this pattern, four apeirogons meet at each vertex. It can be understood as the hyperbolic analogue of the square tiling of the Euclidean plane, where four squares meet at each vertex. Its Schläfli symbol is {∞,4}, meaning that each face has infinitely many sides and four faces meet at every vertex.