*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

## LET'S MAKE ONE

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

- 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!