Patterns are all around us. We can observe them in our homes, schools, where we work, in our art and in our architecture.

A repeating pattern without gaps or overlaps is known as a tessellation. In a tessellation there is a base tile that is repeated over a plane. There are 17 known symmetry groups that form what is referred to as wallpaper patterns. These symmetry groups refer to two-dimensional repetitive patterns and have been catorgoried by their symmentries: the types of rotations, reflections, translations, and glide-reflections.

  • Rotations are clockwise and can rotate by half-turns, 120 degree turns, 90 degree turns, and 60 degree turns.

  • A reflection is a flip along an axis:horizontal, vertical, or at some angle.

  • A translation is a move in which everything is moved by the same amount in the same direction.

  • Glide reflections are composed of a reflection across an axis and a translation along the axis.

While it may seem surprising, there are only 17 known patterns for tilings. Mathematician Evgraf Fedorov provided a proof in 1891, and George Plya provided another for this fact in 1924.

A chart created by Dorothy Washburn and Donald Crowe can help you determine which of the 17 patterns your tiling is part of:

Source: Symmetries of Culture by Donald W. Crowe

Groups start with either p or c (primitive cell or centered cell).
The number n represents the type of rotation.

m refers to mirror
g refers to glide
1 refers to translation

So for example Symmetry group p1 is made up of only translations. The picture remains unchanged no matter how many translations you apply. This group has the simplest pattern and is easy to see:
p1.png p1_v2.png

Symmetry group 2: p2 is made up of both translations and rotations. 180 degree rotations are referred to as half-turns.

Symmetry group 3 (pm) has reflections and translations. There are 2 types of parallel reflections axes. These reflections are referred to as bilateral symmetries.

Symmetry group 4 (pg) contains glide reflections and translations. Glide reflections are sometimes difficult to find when you are looking at the pattern. The direction of a glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation.

Symmetry group 5 (cm) contains reflections and glide-reflections with parallel axes and translations.

Symmetry group 6 (pmm)contains reflections whose axes are perpendicular. The rotations are half-turns.

Symmetry group 7 (pmg) has reflections and glide reflections, as well as translations. The red lines are the axes of reflection, green lines are the axes of glide reflections, and black dots are the fixed points of the half-turns.

Symmetry group 8 (pgg) contains glide-reflections, half turn rotations, and translations.

Symmetry group 9 (cmm) contains reflections and 180 degree rotations.

Symmetry Group 10 (p4) contains rotations and translations. This group contains 90 degree rotations, while the previous groups do not. The black dots in figure 11 represent the centers for the half turns and the blue squares represent the centers of the 90 degree turns.

Symmetry Group 11 (p4m) contains rotations, translations, and reflections. The rotation centers lie on the reflection axes. There are also glide-reflections in this group. This group contains 90 degree turns, half turns, and reflection axes at 45 degree angle.

Symmetry Group 12 (p4g) This group contains reflections, glide reflections, and rotations. The axes of reflection are perpendicular at 90 and 180 degree angles. The lattice is square.

Symmetry Group 13 (p3) This group contains rotations and translations. The lattice is a hexagon. The rotations centers are found at the vertices and centers of the triangles, and are 120 degree rotations.

Symmetry group 14 (p31m) This group contains reflections, rotations, and glide reflections. The reflections are at 60 degree inclines to each other and the rotations are 120 degree turns. The lattice is a hexagon. The reflection axes are parallel and make equilateral triangles. The axes of glide reflections are halfway between the reflection axes.

Symmetry Group 15 (p3m1) This group is similar to p31m. The difference is that all of the centers of rotation lie on the reflection axes. The lattice is a hexagon.

Symmetry Group 16 (p6) This group contains rotations and translations. The rotations are at 60 degrees, 180 degrees, and half turns. The lattice is a hexagon. There are no reflections in this group.

Symmetry group 17 (p6m) This is the most complicated group of them all. It contains reflections, rotations, translations, and glide reflections. The rotations are at 120 degrees, 60 degrees, and 180 degrees. The lattice is a hexagon. The axes of glide reflections are halfway between the parallel reflection axes. They pass through the centers of the half turns.

There are three regular polygons that satisfy the requirement of covering a plane without gaps or overlaps. These regular polygons are:
  • Triangles
  • Squares
  • Hexagons
There is a way to symbolize the tilings using numbers. Each number in the sequence denotes the polygon at each vertex. 333333 is the symbol for the tessellation of triangles
4444 is the symbol for squares
666 is the symbol for hexagons

A complete set of tiling symmetries requires taking these shapes and adding rotations and reflections. There are 17 known tiling symmetries.