# Tiling Definitions

Fundamental Region-- In a periodic tiling, a smallest piece that generates the tiling when all symmetries are applied.  A fundamental region can have many shapes.

Isohedral tiling-- a tiling in which all tiles form one transitivity class. An example. A non-isohedral example.  More generally, a tiling is n-isohedral if there are n transitivity classes.

Isometry--In the plane, as elsewhere, a function which preserves distances and maps the plane onto itself.  There are four types of isometries in the plane.  Isometries are also known as symmetries.

Isogonal tiling-- a tiling for which every vertex is mapped to every other vertex by at least one symmetry.

Lattice-- In a periodic tiling, if you take any point in the pattern and apply to that point all the translations (but not any of the other 3 types of isometries), what results is a lattice for the pattern.  An example.  For periodic tilings, there are 5 types of lattices

Monohedral tiling--a tiling in which all tiles are congruent.  An example.

Periodic tiling-- one which admits at least two translations in non-parallel directions.  An example.

Normal tiling-- every tile is a disk, topologically speaking. Every intersection of two tiles is in a connected set, and the tiles are uniformly bounded (above and below) by a radius of some size. A tiling with disconnected intersections.  A tiling with tiles unbounded above and below.

Primitive Cell-- in a periodic tiling, a type of region formed by connecting lattice points.  Such a region is a primitive cell if the region generates the entire pattern by translations, and if no smaller part of the region generates.

Regular tiling--one for which the group acts transitively on flags. A flag is a tile along with an edge of that tile and a vertex of that edge.  There are only 3 regular tilings, made of equilateral triangles, squares, and regular hexagons.  Regular pentagons don't tile the plane.

Symmetric tiling--a tiling which can be mapped to itself at least one symmetry besides the identity. An example.  An example of a tiling with only one reflection.

Transitivity class-- in a symmetric tiling, the transitivity class of a tile is the collection of all tiles in the tiling that are mapped to the given tile by an isometry of the tiling.  See Isohedral tiling.

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