Guilded Purls

Wednesday, January 11, 2006

In case anyone cares....

.... here's an explanation of the hyperbolic plane -along with Dave's opining re: the people who have time to crochet them. I guess HE'S not getting one for Christmas ;)

A hyperbolic plane is something that looks like a plane but with a
specific (different) way of measuring distances.

If you use the trains in Chicago, you soon decide that Jackson Park
is just exactly as far from O'Hare as it is from Midway airport.
On a map, Midway looks much closer, but the only way to get there
on the trains is to go to the loop first, and from there it's
equidistant to either airport. (I'm approximating here.) The point
is that it makes perfect sense,in some circumstances, to have a
familiar object (Chicago, say) on which there may be a perfectly
reasonable kind of distance defined ("as the crow flies", say), and
yet for other specific purposes it makes sense to have a different
way to measure distances between the same points ("train distance").

The hyperbolic plane is one such animal. There are different ways
to describe it. One way is to envision an ordinary disk. Those are
the points, and you may think you already know how to measure
distances between those points (with a ruler, say). But it becomes
the hyperbolic disk when I specify a new way to measure distances.
The particular distance mechanism ("metric") which makes it a
hyperbolic plane involves a variation on the following theme:

You know those little rollers that surveyors use to measure distances?
(They walk along, pushing this unicycle thingy which counts the
number of revolutions of its wheel; then distance from A to B is
(# of revolutions)x(circumference of wheel). OK?) That's the
ordinary crow-flying Euclidean distance. To get the hyperbolic
distance between two points you have to envision one of these
walker things that magically shrinks as you approach the boundary
of the disk. The consequence of this is that when you try to measure
the distance between points near the edge of the hyperbolic disk,
you'll have to keep that little wheel spinning many many more times,
and you thus conclude that the distance is much longer than it appears.
In fact (once all the formulas are set up) it is by this metric
infinitely far from the center of the hyperbolic disk to the boundary.

People with too much time on their hands crochet these things.
That is, you take a bunch of concentric circles but you make their
radii grow too fast as you move from the center outward. (The connection
with the math above is that you're trying to help the inhabitants of
hyperbolic space figure out distances by saying "just measure the
(ordinary) length of string you need to get from point A to point B".)
I'm sure you know, fabric-wise, what will happen if you try to make a doily
this way: there will be lots of puckering around the edges because the
fabric has nowhere to go. So realistically there's a limit to how much
of the hyperbolic plane you can stitch together. But you end up
with something that looks like the lay of the land in a pleasant
valley surrounded by gnarly hills: motion near the center is easy
but if you have to travel near places far from the center, even points that
look close together on the map are really far apart for a pedestrian
because you have to go up and down so many hills. The map distance,
(the "Euclidean" distance) is small but the pedestrian distance
(the "hyperbolic" distance) is large.

The concept is of fundamental importance in physics (relativity theory).
Within mathematics it is famous historically because it gave a way
to demonstrate the independence of Euclid's Parallel Postulate (i.e.
that the PP cannot be derived from the other, more natural-sounding axioms).
I'm sure -- without even looking for it -- that there are good entries
in wikipedia and mathworld on hyperbolic geometry.



  • I unfortunately do not have a picture and could not find one online, but the woman who made it is a math professor and she said that she uses it for a model in her classes because it is more sturdy than ones made out of paper. So, in her case, and in her defence it was not a pointless crocheting project but a useful hands on classroom model (now on the other hand i want to make one and I can honestly say that I have no use for it at all and just think it would be a fun way to use up yarn). ;-)

    By Blogger Kristin, at 4:09 PM  

  • Here's a link to a site on crocheting it and some nice pictures too.

    By Blogger Kristin, at 5:18 PM  

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