One of my fellow ScienceBloggers, Andrew Bleiman from Zooilogix, sent me an amusing link. If you've done things like study topology, then you'll know about non-euclidean spaces. Non-euclidean spaces are often very strange, and with the exception of a few simple cases (like the surface of a sphere), getting a handle on just what a non-euclidean space looks like can be extremely difficult.
One of the simple to define but hard to understand examples is called a hyperbolic space. The simplest definition of a hyperbolic space is a space
where if you take open spheres of increasing radius around a point, the amount of space in those open spheres increases exponentially.
If you think of a sheet of paper, if you take a point, and you draw progressively larger circles around the point, the size of the circles increases
with the square of the radius: for a circle with radius R, the amount of space inside the circle is proportional to R2. If you did it in three dimensions, the amount of space in the spheres would be proportional to R3. But it's always a fixed exponent.
In a hyperbolic space, you've got a constant N, which defines the "dimensionality" of the space - and the open spheres around it enclose a
quantity of space proportional to NR. The larger the open circle around
a point, the higher the exponent.
What Andrew sent me is a link about how you can create models of hyperbolic
spaces using simple crochet. And then you can get a sense of just how a hyperbolic space works by playing with the thing you crocheted!
It's absolutely brilliant. Once you see it, it's totally obvious
that this is a great model of a hyperbolic space, and just about anyone
can make it, and then experiment with it to get an actual tactile sense
of how it works!
It just happens that right near where I live, there's a great yarn shop whose owners my wife and I have become friends with. So if you're interested in trying this out, you should go to their shop, Flying Fingers, and buy yourself some yarn and crochet hooks, and crochet yourself some hyperbolic surfaces! And tell Elise and Kevin that I sent you!
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About a year ago, I visited the Institute for Figuring website, to read about the Crocheted Hyperbolic Coral Reef. I crocheted a few simple hyperbolic planes to acquaint myself with the technique, and then used it to create sea lettuce and nudibranchs from yarn. You can see photos of a few of these at my Pez Luna crafts blog (URL linked through my screen name-my posts always get held up when I embed links). The July 31 2007 entry has a photo of a yarn Aplysia californica, for which I used hyperbolic crochet to create the parapodia.
Works great for crocheted jellyfish too!
There are other interesting math + fiber art websites and articles linked at the IFF...check it out!
I highly recommend using Sugar and Cream Cotton yarn. Then, when you are done, they work very well for dishclothes/scrubbers. (After all, at some point you're going to have to figure out something useful to do with all those non-euclidean models lying around your house.)
Hmm, an interesting piece of artwrk, but I can't get my brain around it... seemss... wrong...
R'LYEH! R'LYEH!! CTHULU FTAGN!
The problem with crocheting a hyperbolic surface is that it gets boring after a while... In that case, try crocheting a SierpiÅski triangle scarf, there are a number of patterns out on the web. I made my wife's out of Koigu KPPPM since I didn't intend for it to become a dishcloth ;-)
Or, if you're really bored, you can try crocheting a Lorenz manifold...
A buddy of mine took a different tack, and made a rather large model of the hyperbolic plane using a triangulation: Hyperbolic Immersion.
The idea: you can think of geometries as being obtained by gluing together equilateral triangles. If you glue them so that 5 or fewer triangles come together at every vertex, you obtain a spherical geometry. If you glue them so that 6 triangles come together at every vertex, you obtain a flat geometry. More than 6 gives a hyperbolic geometry; that sculpture was made by bolting together 7 triangles at each vertex.
Tremella grows like this.
That formula for hyperbolic volume looks all wrong to me, Mark; where does it come from? I think the volume of a hyperbolic n-ball of radius r is proportional to integral0r sinhn-1 t dt, so asymptotically it looks like C exp (n-1)r. (This is in the standard hyperbolic geometry with curvature 1 everywhere.)
So do some kinds of leaf lettuces. And then there's the lovely fractals of Romanesco cauliflowers...
I really like hyperbolic geometry. One interesting thing to imagine is a city built in a hyperbolic space. Since angles are scale-dependent, it'd be impossible for the inhabitants to make an accurate scale model of the city.
I went to Cornell and took David Henderson's Experiencing Geometry class. He had models that his wife, Daina Taimina had crocheted, and they were great educational tool. My understanding of how things work on a hyperbolic plane was helped immensely by these models.
I've crocheted a couple myself now, and they're neat -- always a good conversation starter, too.
The problem with crocheting a hyperbolic surface is that it gets boring after a while... In that case, try crocheting a SierpiÅski triangle scarf,
Hence the nudibranchs and cnidarians.... Also, there are a number of mathematically-inclined individuals in my family, so they appreciate the hyperbolic planes and spheres. ;-)
Would love to see a photo of the Sierpinski triangle scarf; Koigu yarns are luscious! *drools*
The Flying Fingers shop looks awesome, and I love that they have a Yarn Bus!
After this you can put on your Klein bottle hat... :)
http://www.kleinbottle.com/klein_bottle_hats.htm