Crocheted Hyperbolic Planes

Instructions for Crocheting a Hyperbolic Plane

For your first attempt at hyperbolic crochet I recommend a standard weight (“worsted weight”) yarn and a size G or H crochet hook.  In theory a hyperbolic plane can be made with any weight of yarn and the appropriately sized crochet hook, but very bulky or very fine yarns can be harder to handle.

If you have never crocheted before, these instructions will probably be insufficient! Ask your grandmother, OR, if she is not handy, there are good written and video instructions online. I’m partial to the instructions at Lion Brand Yarn; go to the website and click on “Learning Center,” then select “Learn to Crochet.”

Your yarn can be acrylic, cotton, or wool. I recommend starting with an inexpensive yarn, perhaps an acrylic which has a little bit of elasticity, when you first attempt this.

1. Make a slip knot about 6 inches from the end of your yarn.(Your slip knot should have an opening about the size of a pea, or slightly larger.)

2. Insert the hook through the hole in the slip knot, catch the yarn in the hook, and pull through the hole. (Be careful to catch the yarn that’s attached to the ball, not the short “tail” connected to the slip knot. The tail will never be used for making stitches, but you can use it as a handle to hold onto the work as you get going!)

3. Keeping the hook inside the new loop created in step 2, catch the yarn in the hook again and again pull through.

4. Do this about 4 more times and you will have a chain of about 6 links.

5. Leaving your hook in the final loop, put the two ends of the chain together to form a circle. Do this by inserting the hook into the first loop in the chain, closest to the knot. Again grab the yarn with your hook and pull through. (Once more, be careful to catch the yarn that’s attached to the ball, not the short tail connected to the slip knot.)

6. Continuing to keep your hook in the loop you just made, put your hook into the large open middle of the circle you just created. Catch the yarn in your hook and pull through the circle. You should now have two loops on your crochet hook.

7. Catch the yarn on your hook once more (but this time don’t put the hook through the circle.) Pull the yarn through both loops. Now you have a (new) single loop your hook again. By following steps 6 and 7 you have just made a “single crochet” stitch.

8. Repeat steps 6 and 7 about six to eight (6-8) times, working your way around the edge of the circle .

9. Now work your way around the outside edge of the entire circle a second time, but this time insert your hook under the new stitches at the edge of the circle and pull the yarn through. At this stage it can be hard to identify the right places to insert your hook, but this will become clearer as you gain practice. Forge ahead even if you think you have made a mistake!

10. Once you have completed your second time around the entire circle, continue crocheting in the same direction you have been going. From now until you finish your hyperbolic plane you will be continuing to crochet in a spiral path, and the spiral will get larger and larger. Each time you make a new stitch, you will start by inserting your hook in the “hole” under the V-shaped stitches on the edge of the plane. When you do this, your hook will be perpendicular to the plane and directly under the V’s on the edge of the plane.

11. Once you have completed step 9 you are ready to start increasing. Do this by making a second “single crochet” stitch in every other hole. Doing two single crochet stitches in the same hole is known as “increasing.”

12. Continue until you are content with the size and shape of your plane. Make a knot at the end and sew the loose ends of the yarn (the one at the center and the one at the edge) back into the crocheted plane to hide them.

Once you have finished your hyperbolic plane, you can try experimenting with different procedures. What happens if you increase (double up) in every hole? What happens if you wait to increase by making three single crochet stitches, then increasing only at the fourth hole? You can also try other crochet stitches, such as the half-double stitch, as long as you use the same stitch consistently throughout. And try different materials, such as cotton yarn to make a washcloth. Have fun!

Thanks to Diana Taimina, who first had the insight to crochet a hyperbolic plane, and to Rachel Gogan for her hyperbolic dishcloth workshop at MIT’s Fiber Camp 2011. See Daina Taimina’s book, Crocheting Adventures With Hyperbolic Planes.

© Sarah Kuhn, November 2011