Tessellation is an arrangement of shapes that has the appearance of repeating an image seamlessly forever. This post is my attempt to present a basic understanding of tessellation.
Some shapes fit together perfectly, with no gaps between them: these shapes, then, have the ability to tessellate without any alteration. In the picture above, the squares, triangles, and hexagons are tessellating, but the starburst shape, although it is showing off some radial symmetry, it is not tessellating.
I think that it’s easy to see how some shapes fit neatly together. The part about tessellation that’s not easy to see is when shapes aren’t so neatly defined. We all have patterned clothing that show tessellations, but since the repeating shapes are not laid out in neat little squares the repeating can be hard to see. This is the fun of this kind of design, creating that illusion that there are no lines governing where the repeat happens.
Lets say you’ve drawn a few hedgehogs and you really must cover a whole surface with these images. It won’t repeat neatly so you make some modifications to turn it into a shape that tessellates. The easiest and least elegant way to do this would be to draw a big square all around it and then tile up the squares together. This will have the look of a wall made up of individual tiles, which could be fine, but if the design is for fabric, or wrapping paper or wall paper, there are options beyond the tiled look.
Here I’ve drawn a rectangle so tightly around the drawings that some of the parts of the image are hanging out the sides. I want to be careful that I leave white spaces so that whatever is hanging out can fit in, like a puzzle piece, on the other side of the rectangle, then I make copies and see how it fits together. This is easier to see than explain, so take a look what I’ve done below.
The hanging out parts have been added back in. Now it’s ready to become a tessellating image. Here’s what it looks like finished. Hedgehogs forever.
The reason I’m doing this post right now is because this week there’s a hashtag on twitter, #tiles, which people are posting many tessellating patterns. I’ve tried to embed a couple of links to some photos, but it’s not clear to me that the photos will show up. If you want to see more or know more about this take a look at http://summermathphotochallenge.weebly.com/about.html ( it’s in English, French and Spanish!)
— Juan Fernando López (@jflopez2011) June 16, 2015
— Katherine Bryant (@MathSciEditor) June 16, 2015
There is much more to know about this kind of repeat pattern, but, hopefully, this post will have explained some of the basics. And now look around: you are surrounded by tessellations!