I made an infographic depicting every distinct possible symmetry of 2D periodic groups.  There are 17 distinct symmetries, known as the Wallpaper groups.  Here is a preview.

“Every 2D Periodic Symmetry”. You can find the full image on my flickr. Vector graphics available on request.

As an origamist, I’m very interested in symmetry groups because knowing the mathematical possibilities can open the way to novel mathematical art. I’ve often referenced the relevant Wikipedia articles on different symmetry groups, but Wikipedia doesn’t really convey the intuitive understanding that I would like to have as an artist.

So I’ve previously made infographics about the octahedral and icosahedral symmetry groups, which are frequently used in modular origami.  Now I’ve made one for the Wallpaper groups, which are often used in origami tessellations.

There are some tricky aspects about the Wallpaper groups that made it impossible to follow the same design of my other infographics.  My previous infographics depict the symmetry groups in a sort of hierarchy, with the most symmetric group on top, and all its subgroups below.  But there is no “most symmetric” wallpaper group.  You can always increase the symmetry of a wallpaper group by doubling the frequency of its repetitions.  For example, if you have a pattern that repeats every 10 cm, then you can make it more symmetric by having it repeat every 5 cm instead.

So to make this infographic, I ignored all subgroup relationships that change the frequency of repetitions.  (In technical terms, I’m taking the quotient of each wallpaper group with its translation group.)  I’ve also inverted the hierarchy, so the least symmetric group is on top, and more symmetric groups are on bottom.  To depict each Wallpaper group, I show an example pattern, and then I show a single unit cell with all its symmetries.  The latter diagram is basically borrowed from Wikipedia.

I find that depicting the wallpaper groups in a hierarchy helps me understand them better, and even memorize them.  Before, I only knew them as an unstructured list of length 17.

I’m currently working on a series of origami tessellations to depict each of the 17 Wallpaper groups.