Wednesday, January 23, 2013

Bead Art at the Joint Mathematics Meetings 2013

Last year, I wrote about the history of bead art at the Joint Mathematics Meetings through 2012.  So, I thought I'd write an update about the bead art the 2013 Joint Mathematics Meetings, including the 2013 JMM exhibit of mathematical art.  The meetings and exhibit were held in January 2013 in San Diego, California.  I'm going to show you pieces from the exhibit, but I also include other related works of mathematical bead work from 2012 so you can see them in context. 

Since this is my blog, I'll start with my own pieces.  I had two works accepted into the exhibit, each a set of three beaded beads.  The next two photos show two views of each beaded bead.

Beaded Borromean Links by Gwen Fisher

The Borromean link is a set of three rings, which are woven together into a single symmetric piece of art. The three rings are linked collectively, despite the fact that no two of them are linked to each other.

Each component is woven using cubic right-angle weave (CRAW), which is the three-dimensional version of right angle weave (RAW). RAW corresponds to arranging beads on the edges of the regular tiling by squares, and CRAW corresponds to arranging beads on the edges of a rectangular array of cubes. I stitched rows of cubes, and I used larger beads where I turned corners. A second layer of seed beads embellishes and buttresses the CRAW to make the pieces stiffer.  Read more on Intersecting Links in Bead Weaving.  If you would like to learn how to make one for yourself, check out the pattern and kits for the beaded Borromean Links

Hyperbolic Beaded Angle Weaves by Gwen Fisher

These three pieces combine Vi Hart's idea of beading a hyperbolic tiling with the idea of strategic tacking, as in Daina Taimina’s crocheted hyperbolic planes. Instead of tacking the edges together, however, I used larger beads within the folds of the hyperbolic surfaces to help them hold their forms. Also, instead of using edge-only angle weaves as Hart has done, I used across-edge angle weaves because they are tighter weaves that make the bead work more rigid.

This set includes two different patches of the uniform hyperbolic tiling that goes by many names, including (, which describes its arrangement of squares and pentagons. These two different patches of this hyperbolic tiling identify different subgroups of symmetries of the tiling.

The order (4.3.3) dual tiling is composed entirely of hexagons. While the center has four hexagons meeting at a point, and there are also places where just three hexagons meet at a point. Learn more about these hyperbolic beaded angle weaves here: ( small, ( large, and order (4.3.3) dual.

Tessellation Evolution by Susan Goldstine 

Susan showed a bead crocheted necklace in which the pattern progresses over the surface of the design.  It even won an award, the very first prize in the history of this exhibit to be won by a piece of beaded art!  Go Susan!
Susan says, "The necklace contains 4517 beads; the entire work (including the woven embellishments) contains 5669 beads.  (Not like I counted, but the info can be recovered from the pattern and the size of the woven blocks.)  In case you're wondering, that necklace is 28 inches long, not counting the clasp."  Previous to the piece in the exhibit, Susan made the next piece in 2012.
Notice how the design morphs across the length of the necklace.  Mathematicians call that a Parquet deformation, which is a tiling that slowly deforms across space.
Susan was kind enough to share this "photo of the bead strands, complete with the paper tags I used to keep track of where I was in the pattern." 
It contains 11 three-color tessellations with a common repeat length, and the total length of the rope is 29.5 inches.  The necklace has 3148 beads.

I wrote about Susan's bead crochet in past math art exhibits, often created with Ellie Baker.  Together, Susan and Ellie have a forthcoming book, "Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist," to be published by CRC Press/AK Peters.  It's a how-to-design book on bead crochet written by a mathematician and computer programmer.  You've never seen a bead crochet book like this one.  They expect both necklace patterns to be in the book and lots of other cool stuff about bead crocheted bracelets you've never seen before.  What's not to love?

Susan got her inspiration for these tessellation evolutions because I suggested she apply Parquet deformations to the ideas in her forthcoming book.  I got that idea while beading Chaos and Order.
Someone, I believe it was Robert Bosch, told me that Chaos & Order reminded him of a Parquet deformation.  Through the power of Facebook and the internet, that reference inspired Florence Turnour's beaded Parquet deformation below.  Florence's piece was not part of the exhibit, but it also influenced Susan's piece. 
Susan also cited work by Craig Kaplan in the creation of her necklace, and Craig's work also inspired Florence.  Of course, Craig's work was inspired by M C Escher, who also inspired Florence, Susan, and me!  I digress.  Back to the exhibit... 


Beaded Hilbert Curve by Chern Chuang, Bih-Yaw Jin and Chia-Chin Tsoo 

Chern Cuang and Bih-Yaw Jin have shown beadwork in past JMM exhibits.  You might know these men by their blog: The Beaded Molecules.  Here are four views of their beautiful Hilbert cube, woven with spherical beads and clear monofilament thread.
For obvious reasons, the piece by Chern and friends reminds me of the beaded Hilbert curve by Martina Nagele, who beaded the piece below in March 2010.  
I have had the unusual opportunity to see both of these beaded Hilbert curves in real life.  Martina's piece is quite tiny compared to that made by Chern, Jin, and Tsoo.  I don't remember precisely, but it is only an inch or two across. In contrast, Chern's piece is the size of a softball.
I lightly touched Chern's piece to see how stiff it is, and it's stiff.  Martina's piece is much more flexible than Chern's.  Martina's piece readily unfolds to form a closed ring that is about the size of a bracelet.  Here you can see it unfolded.  Once unfolded, it is really easy to fold it back into a cube.
Martina also sent me this step photo of her Hilbert's curve, woven with cubic right angle weave (CRAW).  You can see how flexible the bead work was before the black embellishment beads were added.  Chern's piece is really stiff, but contains no embellishment beads. 
Although Martina did not show her piece at JMM, I want to discuss it because it was the first Hilbert curve I had seen in beads, and it compares well with Chern's piece.  I asked Martina to tell me about it, and she cited this link of mathematical imagery as her inspiration.  She writes, "I was always intrigued by the possibility to build geometric objects with beads. On my inspirational hunts through the WWW I stumbled over this site by Torolf Sauermann in February 2010. The rendering of a Hilbert curve with a metallic look caught my eye. I had no clue what a Hilbert curve was or how it worked, but I knew I had to bead it! Some days later I had learned a bit more about it and started my piece, the round sterling beads were perfect for it. A little problem while beading is the silver 'worm' is very slinky. and you have to be very concentrated not to take a wrong turn. The most fun part was to embellish the edges with hundreds of tiny crystals and see the fold-up cube emerge from a seemingly chaotic loop."  Apparently, Chern and his friends missed the fun part.


Super Buckyball of Genus 31 by Bih-Yaw Jin

Bih-Yaw Jin was one of the fellows who made the Hilbert curve above, and he also contributed another beaded bead to the exhibit.  
Interestingly, the piece that Bih-Yaw Jin showed in the exhibit was not identical to the piece on the website.  I could tell because one is purple and the other is green.  They are identical beaded structures, but they are different sizes and colors.  The purple one I saw was huge and beautiful!  Apparently, the green one is bigger, and was too difficult to ship.  So I can only guess that it is huger and even more beautiful than the purple one.

Here you can see Florence and me standing in the middle of the exhibit.  We posed with Jin's purple beaded masterpiece.  You can also see my beaded beads in the two black frames on the bottom left.  Gosh, they seem really small in comparison.

Beaded Colored Platonic Solids by Ron Asherov

Mathematically, these beaded platonic solids are interesting.  Ron beaded the five regular solids so that every face shows bead of different colors on each edge.  Also, the path of the string connects every pair of adjacent edges exactly once.  If you look really close at the center of the photo below, you can see how the thread lies at the corners.


Limit Set I by Vladimir Bulatov

I'm not sure that Vladimir intended his sculptures to be beads, but with all of those holes, they certainly qualify.  Plus, I'm such a huge fan of his work, I had to include him.  Moreover, he gets bonus points for presenting a related talk called, Bending Circle Limits (PDF) in the MAA Session on Mathematics and the Arts: Practice, Pedagogy, and Discovery.  No other bead art was represented in the contributed paper sessions (that I know of).   This piece a 3D print, a sculpture made with a computer printer.  Although I'd love to wear it as a bracelet, it's too big for jewelry and the hole in the center is too small for my hand to fit through.  It's the size of a salad plate, and it's flipping gorgeous.  The holes are spheres. Imagine a wheel of mathematical Swiss cheese.

When I asked Vladimir if I could use his photo here, he sent me another.  This is a smaller piece he showed at his sales booth.   It's about 5 cm in diameter.  It is some sort of fractal hypercube.  Florence fell in love with it and took it home.  The funny part was that Vladimir didn't want to sell it to her because he had just printed it.  She told him to print himself another one, and that was enough to convince him to let her keep it.

JMM 2013 was a great conference with much to see and hear.   This post showed a mere sliver of all there was to see there.  If you really like mathematical artwork, and you don't want to wait a full year to see more at JMM 2014, you should consider going to the Bridges conference this summer.  You won't regret it.


  1. Excellent blog post - I just wish my mind and math got along better. I hope the bead crochet book happens. I've corresponded with Ellie Baker and both she and Susan do fantastic work.

  2. Terribly impressed with all the work here! Awesome! Sadly I am not a math brain, but I can appreciate the brain work involved. Just awesome!

  3. I'm sitting here drooling over all of these pieces. I would have loved to see Chern Cuang and Bih-Yaw Jin's work in person, since I've long admired their blog. I'll have to ask Martina to bring her Hilbert curve to Bead & Button this year too. Awesome post!

    1. I would have loved to have met them as well. At one moment, someone told me, "He was just here!" but alas, I missed them.


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