Picture
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

Picture
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!
Picture
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.