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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