Tag Archives: topology

The difference between a hole and a handle

I have learnt topology in a very haphazard fashion. So sometimes when I observe something for the first time, my mind is blown. Topology is beautiful. Today’s lesson was in the difference between a hole and a handle.

different-non-separating-cycles-on-surface

So here’s the example, on the left are two examples of non-separating cycles on an orientable surface of genus 2.  If we cut along them we get two surfaces that are topologically equivalent (right).  To get back to the original surface, we can fill in the holes with disks (red), identify the two disks and then delete the disk.  Call the two disks A and B.  Since the surface on the right is orientable, when the disks are glued into the holes, there are two distinct sides of the disk, the inside and outside.  There are three ways to glue these disks together:
(i) inside of A to inside of B
(ii) outside of A to outside of B
(iii) outside of A to inside of B or inside of A to outside of B
The top example is (i) and the bottom is (ii).  (iii) results in a non-orientable surface.  Even though (i) and (ii) result in the same topological surface, the cycles used are fundamentally different because we have used a geometric idea of inside and outside.  If I were living in the interior of this surface, the holes would look like handles and vice versa.  [mind blowing sounds]  In other words, given an orientable surface and a defined inside and outside a handle is a cycle that you can contract in the inside of the surface, and a hole is a cycle that you can contract in the outside of the surface.

Topology through crochet

crochet mobius stripA good friend taught me how to crochet on Sunday night.  I started with the classic square-to-be-used-as-a-dish-rag project and moved onto the spiral-to-be-used-as-a-pot-stand project. Project number three? The Möbius Strip. Now, I understand the Möbius strip well. What kid has not taken a strip of paper, twisted it, taped the ends together and then drawn a line starting on one side of the paper only to seamlessly (or edge-lessly) reach the other side? I started crocheting the Möbius strip by creating a line and then joining the ends of the line, adding a twist to the line (a 1D Möbius strip?). I then continued crocheting along one edge of the cycle. But of course, since a Möbius strip has only one boundary, I could continue extending the thickness of the strip by spiralling outwards.

I thought: what would happen if I started dropping stitches, making this boundary shorter and shorter? Well, the Möbius strip is a non-orientable surface with one hole – that forms this boundary. I reminded myself that the Möbius strip is surface obtained by puncturing a projective plane – the non-orientable surface of minimum genus – however, I never had a good intuition of what the projective plane is. Let me tell you though, after dropping every fourth stitch on the boundary of my Möbius strip, I started having a pretty good idea of what a projective plane does.

crochet punctured projective plane 2crochet punctured projective plane 3crochet punctured projective plane 1

I couldn’t continue until the boundary closed up – for obvious reasons – so I still have a Möbius strip, but the physical surface is close enough to the projective plane that I get a much better feel for what that means.

I’m not the first person to crochet a Möbius strip – apparently it is a popular scarf design – nor am I the first to explore geometry with crochet. But I have to say that actually creating this surface adds an intuition I’m not sure you could get elsewhere. Finally, if you haven’t seen Margaret Wertheim’s TED talk on math, crocheting and coral including how to crochet a hyperbolic geometry and in which she says

So here, in wool, through domestic feminine art, is the proof that the most famous postulate in mathematics is wrong.

I highly recommend it.