Topological Surfaces in Felt for the Math Nerd in You

You probably don't know this about me, but I love math jokes.  I collect them like I collect beads.  Unlike most people, I think they're funny, or at least, I think they're fun.  There's an old one that goes like this:  Why should you never have breakfast with people who study topology?  Because they can't tell the difference between a coffee mug and a doughnut.  While this joke probably is not very funny, it does illustrate a nice point.  Topology is the study of "properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing."  In other words, if they were made from soft clay, you could deform a coffee mug into a doughnut without any cutting.

I've seen a lot of people make topologically interesting objects from metal, wood, rigid clay or plastic, knitting and crochet.  But I think felt is a better medium to make topologically interesting objects because the felt is stiff, seamless and flexible; so they hold their shape, but you can still fold them and flip them inside out, like the one I'm holding below.

Since I was just making felted wool cuffs, I decided to make some topologically interesting objects, too.  The first set below are all topologically equivalent to a Mobius band with an extra hole.  Like a Mobius band, each piece has one face, but the extra hole gives them all two edges.  Each can be theoretically deformed into any of the other two.  In mathematical terms, each of these surfaces is homeomorphic to the other two.

Just like a Mobius band, each piece in the second set (below) also has one face and one edge.  However, these are not Mobius bands.  I don't know precisely what they are, but I know they're not that.  They're a little more complicated, like a Mobius band with a strap attached.

The brownish piece on the top right was just screaming out to be shrunk down and made into a finger ring. So I made a ring in pink and purple wool, and added some seed bead embroidery to make it look nice.

 Topology for your finger.

Then, I decided to do some homework to see what other surfaces I could make.   I made this pair of Seifert surfaces of a trefoil knot.  In other words, the edges (or holes) of these little guys form a knot and they each have two faces instead of one.  As you know, most holes in every day objects are (topologically equivalent to) circles.  So, a knotted hole is a very strange kind of hole, indeed!  The one on the left is a little bowl with a funny handle.

For this pair, I wanted to see what the edges would look like if I trimmed them with scissors.  I think it makes the edge a little more pronounced, but also slightly less durable.   It also let me make them a little more symmetric because I could trim off the wonky bits.   They're still very durable, but they might fuzz a little on the edges if you fiddle with them for a while, but I'm sure they won't rip with normal usage.  I'm not sure what "normal usage" is for such things.  I'll leave that up to you to decide.

They are art.  They are also mathematical models.  They are plushy mind games, cuddly toys for your brain.  These are all available in my Etsy shop.  Click on the photos to see the listings.