# Infinite-order square tiling

Last updated
Infinite-order square tiling Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 4
Schläfli symbol {4,}
Wythoff symbol 4 2
Coxeter diagram         Symmetry group [,4], (*42)
Dual Order-4 apeirogonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

## Uniform colorings

There is a half symmetry form,    , seen with alternating colors: ## Symmetry

This tiling represents the mirror lines of *∞∞∞∞ symmetry. The dual to this tiling defines the fundamental domains of (*2) orbifold symmetry. This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

## Related Research Articles In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,5}. In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}. In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1{4,5} or r{4,5}. In geometry, the truncated order-5 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,5}. In geometry, the snub pentapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,5}, constructed from two regular pentagons and three equilateral triangles around every vertex. In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}. In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,7}. In geometry, the order-7 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,7}. In geometry, the rhombitetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{4,7}. It can be seen as constructed as a rectified tetraheptagonal tiling, r{7,4}, as well as an expanded order-4 heptagonal tiling or expanded order-7 square tiling. In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons and three equilateral triangles around every vertex. In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}. In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling. In geometry, the snub octaoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,8}. In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}. In geometry, the snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}. In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection. In geometry, the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}. In 2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling. It has Schläfli symbol of {5,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection. In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232 orbifold notation, and can be seen as a half symmetry of *443 and *662, and quarter symmetry of *642. In 2-dimensional hyperbolic geometry, the infinite-order hexagonal tiling is a regular tiling. It has Schläfli symbol of {6,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

• John H. Conway; Heidi Burgiel; Chaim Goodman-Strass (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations". The Symmetries of Things. ISBN   978-1-56881-220-5.
• H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN   0-486-40919-8. LCCN   99035678.