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!
 
If you're tired of counting with number symbols (1 2 3), check out this slideshow video I made:
 
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Most of the time, in math class, we use the flat Cartesian plane.  This is represented by the x- and y-axes we draw on our graph paper. Sometimes in our class (when we want to be really fancy) we use another axis--the z-axis--to make our space three-dimensional.

All that is fine--after all, our paper is flat, and the real world is three-dimensional.  But sometimes the real world gets boring, and we want to draw shapes on different surfaces.  What happens to our shape when we draw it on a balloon? Or what if we wanted to draw a straight line on a donut? These are questions that got mathematicians thinking about other spaces besides flat and 3D spaces.

HYPERBOLIC PLANES

Hyperbolic planes are fun, wiggly surfaces we can imagine instead of the boring old flat planes.  Hyperbolic planes are cool because if we were to walk on one, from any point, we could always choose to walk up hill, down hill, or flat.  In other words, we could climb an endless hill (if we wanted to get a lot of exercise) or we could slide down an endless descent (which sounds like a lot more fun).  And if we were so inclined, we could take the scenic route and stroll along a flat path with the hills and valleys on either side of us.

LET'S MAKE ONE

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Okay, it might be kind of hard to imagine what this kind of world would look like.  You can make a really simple hyperbolic plane by cutting two circles out of paper.

Cut a pizza-shaped wedge out of one circle and cut a slit in the other circle up to the middle.  Then tape the pizza wedge into the slit--it seems like something you shouldn't be able to do, and the paper won't like it.  You're forcing your circle to have too much surface and it will get all wavy and weird.  Try changing the size of your pizza wedge to change how wiggly your plane is.

NOW WE'RE READY TO MAKE A COOLER ONE

Okay, let's try the coolest, easiest, and least useful crocheting project ever.  This is the very first thing I ever crocheted.  Don't worry about what kind of yarn or hook you use--it'll turn out awesome no matter what.  To learn how to start the yarn on your hook, read here.
  • Chain 6 and join into a loop with a slip stitch.
  • Chain 3
  • Double Crochet twice in the first stitch of your loop. (Yes, put two stitches right in the same stitch!)
  • Continue to double crochet around in a spiral, putting two double crochets in each stitch.
  • Watch your creation get wavy!
  • When you think your hyperbolic plane is big enough, make a row of single crochets all the way around.
  • Finish by pulling a loop of yarn through the next stitch and then pulling your yarn end all the way through to make a knot.
  • Hide your yarn ends by weaving them into the hyperbolic plane.
  • Voila!
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These make great toys for small children, cats, and mathematicians.  Try making more and varying how many double crochets you put in each stitch.  You can also change the gauge of your yarn or hook for a different wiggle in your plane.

 
Here is a game to play with a group of about ten people.  Many of you may have played it before:
  • Stand in a tight circle and have each person reach into the middle with his or her left hand and grab someone else's left hand.  If you have an odd person out, she should hold someone else's right hand with her left.
  • Then have everyone reach into the middle of the circle with his or her right hand and grab the right hand of a different person (i.e. not the person whose right hand they're holding).
  • Now, without letting go of hands, try to untangle the knot to the simplest form you can manage.  You might have to climb over or under other people in the knot to get untangled.  See what happens!
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Knot theory is a real branch of mathematics where we look at squiggles with the ends connected together and try to figure out what shapes they become if we simplify them by uncrossing lines and untwisting loops.  The simplest knot isn't a knot at all: it's a circle, and it's called the unknot.  You can think of twisting up a rubber band until it's a mess of tangles.  No matter what you do, it will still untwist back into a circular shape.

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We can also have interlocking unknots, called loops.  Think of these like the magician's magic rings, except instead of magically separating into separate unknots, they stay linked together.


A third common type of knot is the trefoil knot.  Tie an overhand knot in a piece of yarn and then tie the two ends of the yarn together.  Pull out the loops of the knot and lay it out until it looks like this:

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Beautiful, isn't it?

Now, try your human knot game again, but this time see if you can figure out what to do at the start to purposefully get one of these three types of knots in the end.  

If you want to see a video with more knot theory games, check out Vi Hart's video about doodling in math class.