**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.

main page | the 17 plane symmetry groups | links | references | historic tiling | tilings arranged by symmetry group | Color tiling | Aperiodic Tiling and Penrose Tiles

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