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Transformations and SymmetryWallpaper Groups

In the previous sections we have now seen two different kinds of symmetry, that correspond to two different transformations: rotations and reflections. But there is also a symmetry for the third kind of rigid transformation: .

Translational symmetry does not work for isolated objects like flowers or butterflies, but it does for regular patterns that extend into every direction:

Hexagonal honyecomb

Ceramic wall tiling

In addition to reflectional, rotational and translational symmetry, there even is a fourth kind: glide reflections. This is a combination of a reflection and a translation in the same direction as the axis of reflection.

A pattern can have more than one type of symmetry. And just like for squares, we can find the symmetry group of a pattern, which contains all its different symmetries.

These groups don’t tell you much about how the pattern looks like (e.g. its colours and shapes), just how it is repeated. Multiple different patterns can have the same symmetry group – as long are arranged and repeated in the same way.

These two patterns have the same symmetries, even though they look very different. But symmetries are not about colours, or superficial shapes.

These two patterns also have the same symmetries – even though they look more similar to the corresponding patterns on the left, than to each other.

It turns out that, while there are infinitely many possible patterns, they all have one of just 17 different symmetry groups. These are called the Wallpaper Groups. Every wallpaper group is defined by a combination of translations, rotations, reflections and glide reflections. Can you see the centers of rotation and the axes of reflection in these examples?

Type P1
Just translations

Type P2
Rotations of order 2

Type P3
Four rotations of order 2 (180°)

Type P4
Rotations of order 3 (120°)

Type P6
Rotations of order 6 (60°), order 3, and order 2

Type PM
Parallel axes of reflection

Type PMM
Perpendicular reflections and rotations of order 2

Type P4M
Rotations of order 2 and order 4 (90°), reflections, glide reflections

Type P6M
Rotations of order 2 and order 6, reflections, glide reflections

Type P3M1
Rotations of order 3, reflections in three directions, glide reflections

Type P31M
Rotations of order 3, reflections in three directions, glide reflections

Type P4G
Rotations of order 2 and order 4, reflections, glide reflections

Type CMM
Perpendicular reflections and rotations of order 2

Type PMG
Reflections, glide reflections, and rotations of order 2

Type PG
Just parallel glide reflections

Type CM
Reflections and glide reflections

Type PGG
Perpendicular glide reflections, and rotations of order 2

Unfortunately there is no simple reason why there are 17 of these groups. Proving it requires much more advanced mathematics…

Instead, you can try drawing your own repeated patterns for each of the 17 wallpaper groups:

Examples of other students’ drawings

The wallpaper groups were all about flat, two-dimensional patterns. We can do something similar for three-dimensional patterns: these are called crystallographic groups, and there are 219 of them!

In addition to translations, reflections, rotations, and glide reflections, these groups include symmetries like glide planes and screw axes (think about the motion when unscrewing a bottle).

Boron-Nitride has its molecules arranged in this crystal lattice, which has a 3-dimensional symmetry group.