Hyperbolic Felt Lace Bracelets

My new friends and I met for our second "Make Stuff Sunday" yesterday, and we continued to play with felt.  As you can see from my last few posts, I devised a technique for making chunky felted lace which can be used to make surfaces of all types, and I've been playing around with it to see what is possible.

In this post, all of the (lone) pieces in the photos are examples of hyperbolic surfaces, meaning they have saddle points or negative curvature.  Informally, it's a saddle point if the concavity goes up in one direction and down in another direction.  One super cool fact about hyperbolic surfaces is that the geometry on a hyperbolic surface fails the parallel postulate, which is why I think everybody's been calling it a postulate for so long and not an axiom.... but I digress.  In terms of sewing, you might say that a piece of hyperbolic fabric has ruffles, or it flares like you get when you add gussets to a skirt.  This is in contrast to flat fabric (i.e., normal fabric) or something that curls into a ball, like the blue-purple bracelet here...

...or the green-purple bracelet on the bottom of this photo.  The top bracelet is hyperbolic, and the bottom bracelet is spherical with positive curvature (i.e., it's a ball).  I like the way you can stack them into a vase.

This purple and black piece resembles a hyperboloid, but I checked it for straight lines and it failed to have them.  So it's still hyperbolic, but a little too ruffled to be a true hyperboloid.

This is a little crown for a snow queen.  I stitched beads to the felt to look like snowy ice crystals. Like most crowns, this piece flairs out at the top, thereby making it hyperbolic.  It's more or less the same size as the others, and it still fits as a bracelet, but I intended it mostly to be worn on top of the head as a crown.   I think it would look nice with a wedding dress, something blue, and all.

This cute little ruffle isn't a bracelet, but it is an awesome little hyperbolic surface.  More precisely, it's a patch of the hyperbolic tiling by pentagons, where four pentagons meet at every vertex.  It's called the order-4 pentagonal tiling.  This piece was created by mathematical rock star, Vi Hart (she calls herself a mathemusician, but I kind of like mathematical rock star).  After our adventures last week making Seifert surfaces in felt, Vi wanted to felt a hyperbolic tiling.  I gave her some instructions to get her started, and she did everything else.  

In fact, it was her expressed desire to make this piece that led me to devise this technique for felted lace.  This is a nice example of how representing abstract mathematical objects as sculptures leads to more general techniques.  I really love it when that happens.  Math inspires art, which then gives me a reason to write about both.  

The yellow is wool, and the orange is recycled sari silk, which Vi used to emphasize the boundary.  After playing with hers, I kind of want one.   But I also want to make a felted hypercube.  Too many choices...