Tessellations, Symmetry, and the Wallpaper Groups

This is a picture of a carpet piece beneath my piano bench. The design is an example of a tessellation—a "tiling" of a two-dimensional surface in a pattern that can be repeated forever with no gaps or overlaps.

I notice that I can imagine extending the pattern indefinitely.
I notice that I can translate (slide) the design so that it looks as if it has not moved.
I notice that I can simplify the design into a familiar pattern of squares and octagons.
I notice a lot of symmetry in the design!

I wonder how many different lines of reflection there are.
I wonder how many centers of rotation there are?
I notice that some centers make the image match after rotating quarter-turns.
I notice that other centers make the image match only when rotating by half-turns.

I notice that some of the lines and centers of symmetry look the same as others.
I wonder how many different lines and centers of symmetry there are.
I wonder what counts as a different line or center.

I wonder if there are other tessellations that have other kinds of symmetry.
I wonder if there are patterns to how the symmetries of a tessellation fit together.

Symmetries of tessellations fit together in intricate yet predictable patterns. Mathematicians have discovered that there are exactly 17 of these patterns. These are called the wallpaper groups. The wallpaper group of the symmetry patterns on my carpet is named p4mm (p4m for short) or *442. Can you find different tessellating designs that have the same pattern of symmetries?

This Wikipedia page has many beautiful examples of each wallpaper group near the bottom of the page. The examples include drawings and photographs of pavings and mosaics, designs on textiles and pottery, and computer generated images. They span cultures across the globe and hundreds of years of time. The article also has technical information (near the top) for those who are interested. For example, it will tell you why the wallpaper group for my carpet square is named p4mm.

If you are exploring these ideas in your classroom, you and your students may find it easier to begin with frieze groups. (Check here for additional information, or try your own google search.) These are one-dimensional versions of tessellations; that is, the repeating pattern takes place back and forth along a single line. You often see frieze patterns along the borders of walls and ceilings. It can be fun to have students find examples of frieze groups or wallpaper groups in their homes and towns. It is even more fun for them to create their own!