Introduction to Tessellation Symmetry

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In a tessellation, there is repeating. The repeating makes a pattern. The part that repeats is called a "tile". Even if it's shaped like a whale or a horse, we still call it a "tile". That's because loooong ago, tessellations were only made with tiles. They were tiles like the ones you've seen on floors and walls, but laid out in more colorful, fancy patterns.

On this page you see a few of the many ways a "tile" can repeat. On the next few pages you'll see each of those ways, with examples.

Symmetry is a complex mathematical subject, but this website tries to keep that science simple. There are several excellent websites such as which teach you the complicated version of the math for tessellation. Teachers, see "links" on our menu to more of those websites. I also recommend these 2 books (click here to read about them).

Look at the cat picture on this page. The cats all have the same shape, right? They repeat by just copy-catting and then ssssssssliding to a new place. That's the easiest kind of tessellation symmetry. You can call it "Slide" or "Glide" or "Translate", because these three words have the same meaning in tessellation.

Now look at the whales. In the top row of whales, they all face left. In the middle row, they all face right...but they're the same shape, just facing the other way. You can call this way of repeating "Mirror" or "Flip" or "Reflect". These three words mean the same thing, in tessellation.

Now look at the picture of stingrays. The stingrays gather around a few center-points. They repeat by copying and then turning. That's a third kind of tessellation symmetry. You can call that "turn" or "spin" or "rotate".

So...that's three ways to repeat a "tile" to make a tessellation. Are there only three ways? Nope. There are at least 17.

For flat stuff...what your math teacher would call "2D planes" and your department store would call "wallpaper"...we think there are only 17 possible patterns. Wow...we know of only 17 kinds of symmetry patterns for flat areas! We call these "the 17 symmetry patterns", or just "the 17 wallpaper groups".

Do you want to see a list of those 17 symmetry patterns...Ummm... I mean, "the 17 wallpaper groups" ? OK, Click here to see the Pólya illustration in the Escher section of our website. When you're done, come back to this page and read more about symmetry.

cat tessellation showing translation/slide/glide symmetry
Slide / Glide / Translate

whale tessellation showing a flip mirror reflection
Mirror / Flip / Reflect

Turn / Spin / Rotate