Many of you may be familiar with M. C. Escher's famous tessellations.  They are beautiful works of art and fascinating mathematical objects at the same time.  For a really fantastic example, check out this fractal tessellation.  The idea of a tessellation is to figure out different ways to tile a surface.  We all know from our bathroom floors that square tiles can cover a surface, and we know from brick walls that rectangular tiles work as well.  But what about other shapes?
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I decided to make some tessellations of my own.  First, I tried making one with triangles.  That worked out well, and I got a cool pattern. I noticed that I could also make the tessellation a different way:

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I think I like this second tessellation even better because the triangles make a diamond shape when they're lined up this way.  It's no Escher, but aesthetics counts in all of mathematics.  We always try to find the most beautiful way to solve a problem.

Next, I tried making a tessellation with a different shape.  I was thinking about honeycombs when I made this one:

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The hexagon tessellation is a beautiful, isn't it?  I particularly like the fact that as we build it out on all sides, we can make the whole tiled surface have the shape of a hexagon.  We could have done the same thing with the second triangle tessellation (making the whole surface triangular.  Do you see i.  That's another thing that makes it preferable to the first triangle tessellation I made.

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We can make tessellations with more than one shape--in fact sometimes we have to!  I wanted to tessellate stars, but I couldn't fit them together without putting other shapes in between.

I needed to add rhombuses and pentagons.  Now think about those two shapes... Could we have made a tessellation with just rhombuses (diamonds) or pentagons?  Here's a hint: it works for one shape but not the other!  Why would that be?

I like my geometric tessellations, but I wanted to make something more like the tessellations Escher did.  He often tessellated much more complex shapes like birds or lizards.  So to start my new creation, I took my first triangle tessellation and turned it on its side. (It may not be as pretty as my second one, but it does have a use.)  I wiggled the edges of the triangles and played around with the shape until it looked like this:
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You can make your own tessellation this way.  Just start with any geometric tessellation and move the edges around until they look the way you want.  Remember that what you do to one edge has to be mirrored on another--that's the trickiest part.  Your shapes don't have to look like animals--they can be abstract or look like flowers or whatever you want!