## Tilings by Jaap Scherphuis

https://www.jaapsch.net/tilings/#pentagon

The 15 types of pentagonal tilings discovered so far, the 15th (convex monohedral) pentagonal tiling, discovered in 2015

## Terminology

Tiling or tessellation
A dissection of the infinite flat plane into shapes of a finite area.
Tile
One of the shapes that forms a tiling.
Isometry
A distance-preserving mapping of the plane. There are four types: translations, rotations, reflections, and glide reflections.
Symmetry of a tiling
An isometry that maps the tile boundaries onto tile boundaries. In other words this is some transformation that leaves the tiling looking the same as before.
Periodic tiling
A tiling that has two independent translation symmetries, i.e. a tiling that repeats itself along two different axes like a wallpaper pattern.
Primitive unit or Unit Parallelogram
A section of the tiling (usually a parallelogram or a set of neighbouring tiles) that generates the whole tiling using only translations, and which is as small as possible.
Fundamental unit
A section of the tiling (usually a set of neighbouring tiles) that generates the whole tiling using the tiling symmetries (not just the translations), and which is as small as possible.
Monohedral tiling
A tiling where all the tiles are congruent to each other, i.e. all have the same size and shape (though they are allowed to be mirror images).
Isohedral tiling
A monohedral tiling where for any two tiles there is a symmetry of the tiling that maps one tile to the other.
k-Isohedral tiling (k is a positive integer)
A monohedral tiling where the tiles form k classes such that for any two in the same class there is a symmetry of the tiling that maps one tile to the other, and for any pair of tiles in different classes no such symmetry exists. Note that 1-isohedral is the same as isohedral. In the applet, each class of tile has its own colour.
Edge-to-edge tiling
A tiling of polygons such that no corner of one tile touches the side of another.