Order-6 square tiling


In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with six squares around every vertex. This symmetry by orbifold notation is called with 4 order-3 mirror intersections. In Coxeter notation can be represented as , removing two of three mirrors in the 642 symmetry| symmetry. The *3333 symmetry can be doubled to 663 symmetry by adding a mirror bisecting the fundamental domain.
This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction t1. A second 6-color symmetry can be constructed from a hexagonal symmetry domain.
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Example artwork

Around 1956, M.C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings Circle Limit I–IV demonstrate this concept. The last one Circle Limit IV , tiles repeating angels and devils by symmetry on a hyperbolic plane in a Poincaré disk projection.
The artwork seen below has an approximate hyperbolic mirror overlay added to show the square symmetry domains of the order-6 square tiling. If you look closely, you can see one of four angels and devils around each square are drawn as back sides. Without this variation, the art would have a 4-fold gyration point at the center of each square, giving, symmetry.

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure.
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol, and Coxeter diagram, progressing to infinity.