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A tiling (or sometimes called as a "tessellation") is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. They can also be generalized into higher dimensions, just like polytopes.

Euclidean Tilings[]

Below are quick definitions of the 3 possible euclidean one-polygon 2-dimensional tilings.

Triangular Tiling[]

Triangular tiling is a plane filled with triangles. It has the Schläfli symbol {3, 6} and is the dual of the hexagonal tiling.

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Square Tiling[]

Square tiling is a plane filled with squares. It has the Schläfli symbol {4, 4}.

Hexagonal Tiling[]

Hexagonal tiling is one of the three regular tilings of the plane and consists of an infinite number of hexagons. It has a Schläfli symbol of {6, 3}, meaning that it has three hexagons located around each vertex.

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Spherical Tilings[]

Regular spherical tilings are just the Platonic Solids.

Hyperbolic Tilings[]

There are an infinite amount of regular Hyperbolic Tilings, because these are the tilings that are not euclidean or spherical.

Order-5 square tiling[]

The Order-5 square tiling has a Schläfli symbol of {4, 5}. It can be thought of as a "hyperbolic cube".

Order-3 heptagonal tiling[]

The Order-3 heptagonal tiling has a Schäfli symbol of {7, 3}. It can be thought of as a "hyperbolic dodecahedron".

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