Hyperbolic Beaded Angle Weave (4.5.4.5)

Last week, I attended the Gathering for Gardner an event held in honor of the late, great Martin Gardner.  In case you have never heard of him, Gardner is generally considered to be the most famous recreational mathematics writer of all time.  He wrote about puzzles and games, optical illusions and magic, mathematical art, poetry, and juggling; he also wrote the definitive annotated Alice in Wonderland, and the list goes on and on.  I grew up reading many of his books. My very first quilt (using a Penrose tiling) was inspired by one of his essays.

This year I got an invitation to the Gathering.  So I eagerly traveled to Atlanta, GA to meet this wonderful community of puzzlers, mathematicians, artists, magicians and so forth, all of whom love and have been inspired by Martin Gardner's writings, just like I have.  It was my first time attending, and I gave a short talk on mathematical bead weaving.  Although I only had five minutes to speak, I presented twenty something slides across a wide gamut of mathematical concepts that I have represented with bead weaving over the years.  (Now, I'm trying to turn my five-minute talk into a four-page paper. Wish me luck.)  I didn't realize until the day after my talk that it was the largest group I've ever addressed, maybe 300 people.  Fortunately, the talk was over so quickly that I didn't have time to get nervous.

At the Gathering, a few women showed their versions of hyperbolic planes.  These included the crocheted coral reef by Margaret Wertheim, the director of the Institute for Figuring; Daina Taimina's crocheted Geometric Manifolds; and the hyperbolic bead weaving of Vi Hart, who you might know from her videos about doodling in math class.  Inspired by their work, I thought I'd take a new try at bead weaving a hyperbolic surface of my own.  To do this, I first noticed that Vi Hart's version shows an edge-only angle weave of (7^3), that is, she uses one bead on every edge of a tiling with three 7-gons around every vertex.  Her version is sparkly, and fun to fiddle with, but it's very squishy and something of a ruffled mess.  It's nice to hold, but difficult to photograph as it doesn't hold its shape. It was exactly this kind of uncontrolled ruffling that had prevented me from trying to bead hyperbolic tilings in the past.  I had seen this kind of ruffled confusion before, in such works as Helaman Ferguson's hyperbolic quilt, and I didn't give it much thought because I like my beading (and quilts) to look more organized than that.

But then I had an epiphany. You see, at the Gathering, Daina Taimina exhibited crocheted hyperbolic planes in a way I'd never seed before.  She used strategic tacking to turn a ruffled mess into an organized structure like I had done in my Dancing Fan beaded bead.  Her crochet was stiff enough to keep the whole piece from collapsing, and the tacking kept the ruffles in place.  It was easy to see the symmetry in Taimina's crochet.  I noticed this tacking immediately as I had never seen someone do that before on a crocheted hyperbolic surface.   I decided to combine Hart's idea of beading a hyperbolic tilings with strategic tacking.  Instead of tacking the edges together, however, I would use larger beads within the folds.  Also, instead of using an edge-only angle weave as Hart had done, I tried an across-edge angle weave because it would give a tighter fit and thus make stiffer beadwork.  I made a patch of the tiling below, namely the uniform hyperbolic tiling that goes by many names, including (4.5.4.5).  It has squares in yellow and pentagons in red.  I chose this one because 4 and 5 are small numbers, so the beads would fit tightly.

What you see in the first photo in this post is three views of the finished beaded bead.  Below you can see what it looked like in progress before I added the largest beads.  I show five different ways to orient my little patch of this hyperbolic tiling, but these are not all of them.  Each illustrates a different subgroups of symmetries of this patch of the tiling.   I could have used any of these as the symmetry of my beaded bead above, and I ultimately chose the one on the bottom right.  A different patch of this (4.5.4.5) tiling could also be used to show five-fold symmetries.

This little experiment made me realize that there are a lot of interesting possibilities for hyperbolic beading that are yet to be explored, an infinite number in fact.  Many infinities.  If you thought there were a lot of different polyhedra to bead, that's only because you haven't tried beading hyperbolic tilings yet.  Try it, because with infinitely many tilings to go, I know for sure that I won't have enough time to bead them all myself.

I beaded another patch of (4.5.4.5) here.