# wallpaper group

A wallpaper group, also known as a **crystallographic group**, is a distinct way
to tile the plane that repeats indefinitely
in two dimensions; that is, a collection of two-dimensional symmetric patterns
on a plane surface, containing two nonparallel translations (see periodic
tiling). There are only 17 kinds of these patterns, known as isometries,
each uniquely identified by its translation and rotation symmetries, as
discovered in the late nineteenth century by E. S. Fedorov
and, independently, by the German A. M. Schoenflies and the Englishman William
Barlow. Thirteen of the isometries include some kind of rotational symmetry,
while four do not; twelve show rectangular symmetries, while five involve
hexagonal symmetries. Every two-dimensional repetitive pattern in wallpaper,
textiles, brickwork, or the arrangement of atoms in a plane of a crystal
is just a minor variations on one of these 17 patterns.