Doodle No. 33 Tribert Makes Thymine

The design of this is based upon the Poincaré model of hyperbolic space. Ask a mathematician about it. It’s probably their favorite model of hyperbolic space. 

5.75” circle

A lot of people photograph their oakk in brings of flowers with real flowers. I don’t paint flowers. So I thought these Polyhedra seemed appropriate. 

NFS 

Thanks for looking. 

Doodle No. 32 Adam Explains How Holes Work

These weren’t the colors I thought I would get, but in the end I’m happy with how they turned out. It took a lot of layers of colored pencil to get there

. 

This piece is for someone who likes old wooden furniture and modern art.  It would look nice in a fancy guilded frame.

6.5” by 9.5” oval 

Not for sale

Doodle No. 31 Adam Makes an Atom

This is what happens when I let Melanie Schrader pick the colors.

Mike Ryan pointed out that the atom has three electrons, so it must be lithium. He cares for bipolar. Here is a detail of Adam and his lithium atom and the flash of the mica paint. 

4” square

Prismacolor black ink, Faber-Castell Polychromos pencils, Finetec mica watercolor paint on Stonehenge 250 GSM 100% cotton paper

 NFS

Doodle No. 30 Too Many Holes

In Doodle No. 24 Six Holes, I asked, “How many holes are too many?” The answer I was given was infinitely many. So then I spent the next several days trying to figure out how to draw infinitely many holes. The answer lied in the Poincare model of hyperbolic space. Here, I present “Too Many Holes.”

5.75” diameter circle.

Drawn with archival Prismacolor black ink, Faber-Castell Polychromos pencils, Finetec mica watercolor paint on Stonehenge 250 GSM 100% cotton paper. 

Sold

thanks for looking. 

Doodle No. 29 Green Hole with Shrimp

If you ever visit a green hole, be sure to get a souvenir while you’re there. 

Someone challenged me to draw a black hole using a tiling other than the regular square one. That’s why there are triangles. Plankton and 2D RNA structures decorate the surface. The gold planet and most of the little dots shimmers in the light with mica paint. 

4” square

Prismacolor black ink, Faber-Castell Polychromos pencils, Finetec mica watercolor paint on Stonehenge 250 GSM 100% cotton paper. 

Original available: https://www.etsy.com/listing/704139530/

Hyperbolic Surface Tilings Woven with Beads and Thread

I've been beading hyperbolic tilings all week, and I can't stop!

I've seen lots of people crochet hyperbolic surfaces, most notably at the Institute for Figuring.  The typically technique is to crochet around and around the edge adding lots of extra increases in every round to make the edges ruffle.  Beaders sometimes do the analogous thing, making ruffled bracelets and necklaces that incorporate increases on each round.  But for these beaded pieces, I'm doing something a bit different.  I use hyperbolic tilings, also called tessellations.

Flat Bead Weaving

But before I go on, I want you to understand what I'm doing, so I'm going to digress a bit.  Consider flat bead weaving, like you might use to make a bracelet.  For example, you might bead a flat bracelet by using a tiling of squares.  With bead weaving, you can place one bead on each edge of a square tiling, and you get right angle weave (RAW).  This picture shows a few different flat bead weaves and the tilings used to generate them.

The bottom illustration in the picture above suggests that you could use the square tiling to make a different weave from RAW.  In particular, you could weave four beads in a loop for each square, and then add one extra bead on the edges to connect the loops.  That describes super right angle weave, or SRAW.  (I call that an across-edge angle weave.)  If you've ever done RAW or SRAW, you know that four loops at each corner make the beadwork lie flat.

Round Bead Weaving

If you use loops of four beads with three loops around each corner you end up with a beaded cube (generally called cubic right angle weave or CRAW).  If you start with SRAW and weave three loops around each corner you get a the photo below (which I named cubic super right angle weave or CSRAW).  You can think of this as the across-edge weave of a cube.  (It's also an edge-only beaded truncated octahedron, but that's not important right now.)

Sorry, that was a lot of jargon I just threw at you.  Forgive me.  What's important here is that you have flat weaves that can go on forever like a plane (e.g., RAW and SRAW), and you have round beaded beads that close up on themselves (like a single unit of CRAW and CSRAW).  Mathematically, if flat curvature is zero, and beaded beads like round spheres have positive curvature, then it reasons to question: What beadwork has negative curvature?  Hyperbolic surfaces have negative curvature.  Intuitively, you can think of negative curvature as ruffles.  Mathematically speaking, ruffles are the opposite of spheres.  And flat sheets are in the middle.

Hyperbolic Surfaces

Hyperbolic surfaces are really interesting.  In fact, they have their very own hyperbolic geometry, quite different from the Euclidean geometry you probably learned in high school.  For one thing, in hyperbolic geometry, the parallel postulate is false. But what's most interesting to me, as an artist, is that there are lots of different ways to represent hyperbolic surfaces.  For example, this circle uses the Poincare disc model of hyperbolic space.

The square tiles are colored in pink, purple, blue, green and yellow.  That's right; those are squares (or maybe they're rhombuses).  I know they don't look like the regular squares you're used to, but that's just the Poincare model doing its thing.  Imagine that those four sided things are squares, and every black side is straight and the same length.  If you make this with bead weaving, you can make all the edges the same length.  For example, you could put one bead on every edge and weave a loop of 4 beads for each tile (an edge-only weave of the drawing).   I didn't do that.  Instead, I used an across-edge weave, something akin to SRAW.

In particular,  I weaved loops of four beads of the same color for each square (rhombus), and then attached the loops by one bronze bead on the edges.  Notice I used five colors just like the illustration above. The bronze beads are on the edges with the holes are perpendicular to the edges.  Here's another view of the same piece. 

And here you can see how big it is.  This little guy is looking for a new home if you'd like to adopt him: https://www.etsy.com/listing/188233621/

I used to think that a beaded hyperbolic surface looks like just a ruffled mess of beads.  I beaded a few in 2012, and I went to great length to try to bound them into symmetric submission by adding bigger beads into the folds.  Like this:

I showed this piece to Vi Hart, and she encouraged me to bead a different tiling without the extra big beads holding them in place.  That's why I beaded the...

Snub Tetrapentagonal Tiling

Ah, the beautiful snub tetrapentagonal tiling.  No, I didn't name it.  That's what everybody calls it.

Here is my beaded version.  I used pink beads for the pentagons, green beads for the triangles and yellow beads for the squares. Relatively speaking, this piece is flat-ish.  What I mean is it has less negative curvature than tiling one above.  I had to add a lot more beads before it started to ruffle.

Vi likes this tiling because it's chiral, which makes it unusual.  See the little pinwheels in the holes below?  If you look at the other side, you'll see the mirror reflection with the pinwheels spiraling in the opposite direction.

Then, I beaded the...

Rhombitetrahexagonal Tiling 

which I first noticed in John Conway's book, "The Symmetry of Things."  But I got this drawing from Wikipedea because it's in the public domain, and Conway's book isn't.

This is called the rhombitetrahexagonal tiling.  I didn't name this one either.  Notice the blue and green checkered stripes.  I like those stripes.  I wanted to emphasize those stripes in my piece, so I made the blue and green squares the same color.  They're all green in my beaded version below.  Maybe it's just me, but it seems a little peculiar to have a ruffled thing with stripes.  I guess you could make a ruffled skirt out of striped fabric, and then have striped ruffles.  Anyway, here it is. 

In my beaded version, I made the hexagons pink, and the squares green and purple.  The edges are a few different colors depending on which tiles they touch. Here you can see how big it is.  It's for sale so you can enjoy it in the comfort of your own home.

It's got a lot of personality, this little fellow. Now notice that this tiling has three squares and one hexagon around every vertex.  It's probably easiest to see that in the red, blue, yellow drawing above.

Let me say that again: three squares and one hexagon around every vertex.  So does this piece of beaded Faujasite have three squares and one hexagon around every vertex.  You have to be careful where you look to see that because some places appear to have two hexagons and a square.  Those are places where I stopped adding beads.  If I kept going and made this piece infinitely large in every direction, they'd finish with three squares and a hexagon just like the rhombitetrahexagonal tiling above.  (I'm going to need more beads for that.)  So, this piece below is a different representation of the same hyperbolic tiling right above.  Wacky. 

There are some fascinating artistic implicatons to that last thing I said.   So stay tuned, 'cause I'm playing around with that idea.  And if you actually made it this far, thanks.  You're awesome.

Infinite Skew Polyhedron Faujasite (4.4.4.6)

I beaded another infinite tiling. This one represents the crystalline structure of faujasite. 

This piece of beadwork has nearly perfect tetrahedral symmetry, but I left out a few of the beads for aesthetic reasons.  If you look closely at the photo below, you will see the bottom edge is different from the other five.  For one thing, leaving out the extra beads makes it much easier to balance the piece on edge.

This piece contains over 19 grams of size 11° seed beads.  That's almost a whole box of beads, and as far as I know, it's flawless.  No mistakes.

You can think of this piece as a tiling of squares and hexagons in 3D.  Thought of as a tiling, every vertex is the same type, 4.4.4.6. That means there three squares and a hexagon around ever corner.   Consequently, this piece contains loops of 4 beads and loops of 6 beads. 

When I made it, I thought of it as a bunch of polygons glued together.  We have truncated octahedra and hexagonal prisms glued together on the hexagonal faces.  All of the hexagons on the prisms are glued, but only half of the hexagons on the truncated octahedra are glued.

My inspiration for this piece came from Figure 7.41 in the book, Crystal Structures I: Patterns and Symmetry by M. O'Keeffe and B. G. Hyde.  The illustration above (Figure 7.41) is what I used from that book.

Here you can me holding it showing off a triangular face of the tetrahedron.  This qualifies as one of my larger non-wearable pieces of beadwork.

A tetrahedron has six edges.  On this tetrahedron, I made one of the edges is different from the other four. It's the bottom edge in this photo.

And it's the front edge in this photo.  I like the way it looked without the extra beads.  It's adds variety, and the piece doesn't need them to hold itself in position. 

 

For comparison, the other five edges look like the front of this.

 

Next I show you a few process shots so you can see how the piece started.  First, I made a ring of six truncated octahedra and six hexagonal prisms.

After adding more beads, I had two of these rings joined together.

With more beads came three joined rings. If this were actual faujasite, the inner cavity would have a diameter of 12 Å.   It has tetrahedral symmetry at this point.  This would be a nice place to stop if you wanted a little beaded bead to wear as a pendant.  

But since I knew I was beading a repeating pattern, I couldn't help but make more repeats. This is like two tetrahedrons glued face to face, but with a half turn rotation first.  The symmetry of this is an antiprism with a 3-fold rotation.  It's a very weird symmetry. 

 And then this...

And one last photo of the finished piece.  This piece is SOLD!

If you liked this post, you might enjoy these posts on beaded infinite polyhedra:
http://gwenbeads.blogspot.com/2014/03/infinite-polyhedra-and-cubic-honeycombs.html
http://gwenbeads.blogspot.com/2014/01/infinite-skew-polyedra-pendant-with-craw.html
http://gwenbeads.blogspot.com/2014/01/infinite-skew-polyhedron-4448-w8.html
Thanks for looking!

Ionic Octahedron, Star Cluster and Hyperbolic Surface

Beaded beads that sparkle...

Ionic Octahedron Beaded Bead in Blue and Silver

 

Star Cluster Beaded Bead with Redside Designs Borosilicate Glass

Ionic Octahedron Beaded Bead in Mixed Metals

 Hyperbolic Beaded Angle Weave in Green Gold

Thanks for looking!

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 (4.5.4.5), 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: (4.5.4.5) small, (4.5.4.5) 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.