From the point of view of symmetry, there are exactly 17 ways to create a repeating two-dimensional design. This web page is the introduction to a catalog of such wallpaper designs, with each mathematically different design represented using pattern blocks.
Creating such a catalog seemed like a good idea, given that on the one hand, pattern blocks are ubiquitous in schools, and on the other hand, symmetry is an important part of mathematics. There are no doubt other places on the Web where one can learn about wallpaper groups. This catalog is aimed at math teachers, in the hope of providing a context for learning and teaching about symmetry, with an emphasis on the geometric and visual (rather than on the algebra of group theory).
I analyzed one wallpaper pattern in depth in these blog posts: Part 1 | Part 2.
Mathematical terminology and background
You can enjoy this catalog without knowing a lot of math, so you can skip this section. On the other hand, the more you know, the more you'll get out of it.
A figure is symmetric if it is its own image in an isometry.
An isometry of the plane is a one-to-one geometric transformation which preserves distance — one of these four: translation, rotation, reflection, or glide reflection. Read more about these on my Geometric Transformations page. The first three are Common Core-sanctioned, and appear in the curriculum starting in 8th grade. Read about the fourth here: Glide Reflection.
An excellent resource to learn about symmetry is Peter Stevens' Handbook of Regular Patterns (MIT Press). It features crystal-clear explanations and design examples from cultures all over the world. I cannot recommend it highly enough.
How to Use this Catalog
For each design, I share a pattern block figure, the name of the symmetry group it illustrates (using crystallographers' notation), and a list of its symmetries. In the lists, I use the word "mirror" to refer to the reflection lines.
I encourage you to actively engage with the images:
- Am I correct? If you see a mistake, let me know!
- All designs have translation symmetry. Find two non-parallel minimal translations that preserve the design.
- If there is rotational symmetry, find each center of rotation.
- If there is reflection symmetry, find each line of symmetry.
- If there is glide reflection symmetry, find the reflection and translation that are the components for each.
Things to think about as you study the images:
- How are the centers of rotation arranged with respect to each other? to the reflection lines?
- What happens at the intersection of reflection lines?
- Can you find a group of blocks that could serve as a tile for this pattern?
Finally: is there a better representation of this wallpaper group using pattern blocks? One that is more economical (uses fewer different blocks in each constituent part)? One that is more aesthetically pleasing? Such designs must exist. Send me your creations! I will post the best ones here and credit you. Thanks in advance!
Katherine Paur sent me quite a few designs, which I've added to the catalog. She labeled each in a different notation (from Conway's The Symmetry of Things). She placed a small cube at each center of rotational symmetry, labeling those as either kaleidoscopes (centers of rotation at the intersection of mirrors) or gyrations (not on mirrors.)
And now, dive into the catalog!