Saturday, March 28, 2009

Non-Euclidean Geometry meets Crochet

Ever heard of Hyperbolic Geometry? Apparently it is the geometric opposite of the sphere. On a sphere, the surface curves in on itself and is closed. On a hyperbolic plane the surface curves away from itself at every point. In other words....
In Euclidean geometry, as a circle gets larger, the length of the circumference increases linearly. But with a hyperbolic plane, we're talking EXPONENTIAL increases. Got that?
Oddly enough, the concept of a hyperbolic plane can be physically modeled using crochet.

Yes, you can impress your family and friends by making your own hyperbolic surface! All you do is increase the number of stitches in each successive row or round of a crochet project. That's it. Here's how to make a 'pseudosphere' (like the one I made pictured above):

1. CH 2. SC 6 in the second chain from the hook (6 SC total).
2. SC 2 in each stitch all the way around (12 SC total).
*BTW, I like to crochet in a spiral and not worry about ending rounds and starting another. So just keep going with step 3 - don't worry about making a SL ST to the last stitch in the round and all that stuff.
3. SC 2 in each stitch all the way around (24 SC total).
4. SC 2 in each stitch all the way around (48 SC total).
5. SC 2 in each stitch all the way around (96 SC total).
I think I stopped after step 5 but feel free to keep going!

You'll get a kinesthetic feel for what 'expanding exponentially' REALLY means. Early rounds go quickly, but each successive round takes longer and longer (and also eats up exponentially increasing amounts of yarn.)

If you want to learn a little more about the mathematics involved, check out this article or just google 'hyperbolic crochet.'


Deneen said...

Very cool!

Teal Marie Chimblo Fyrberg said...

Love it! Going to make this for a baby shower present. Thank you for posting!

Chanell D Gautreaux said...

This is awesome! I have to get some cotton and start making these!

Dr. Pam said...

My sis is a mathematics professor and loves crochet! I'm going to make her one of these to use as a model in her classroom. Way cool!

Cindy said...

This looks really cool. I might need to figure out how to crochet just to make one :)

Posh Pooch Designs said...

Love it and love the colors! new Foloower here.

Unknown said...

Hello. I found this pattern through Ravelry and just had to tell you that I loved it. I am very new to crochet, but found your exponentially-expanding scrubber to be great practice. Thanks for the inspiration.
Annapolis, MD

BreeAnna said...

LOVING THIS! Thank you so very much!!

Rachel said...

...what do pure math, coral reefs and crochet have in common...? Hyperbolic planes of course...!!!
We stumbled across this in our quest to learn more about fractals and Escher...fantastic...!!! Thanks so much.