While at the 2011 Bridges Conference, I learned a bit about the mathematics of weaving. One of our projects was to plan and weave a tiling of the plane. For some unknown reason, I chose a Hallowe…

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.

Familiarize yourself with some heady math concepts by way of soothing knitting and crocheting projects.

Familiarize yourself with some heady math concepts by way of soothing knitting and crocheting projects.

Math art beads tutorial pattern sewing weaving hat gwen fisher

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.

Familiarize yourself with some heady math concepts by way of soothing knitting and crocheting projects.

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.

A few posts ago, I compared Pondo Stitch with Super Right Angle Weave (SRAW), and found that they create a similar flat structure of beads, albeit with very different thread paths. Soooo, in the comments, Emilie had the insight to ask, "Have you done a cubic version of the super RAW?" Indeed, I had not. And always enjoying a good beading challenge, I made this. It's a row of three cubes using cubic super right angle weave, or CSRAW for short. I think I can pronounce that out loud. Csraw. K-sssssssssssraw. Here is one cube. It has 4 x 6 = 24 black beads and 12 gilded red beads. (Edited to add: I eventually realized with loops of 4 and loops of 6, this is a beaded truncated octahedron. See the hexagons and squares in the illustration below.) When you add another cube, the new cube shares one face, or 4 black beads with the first cube. Each cube shares 4 black beads with the cube next to it. It's a little squishy, and it shows more thread than I like, but it has the nice property that even though it squishes, it likes to pop back into shape. It has a nice springiness about it. Noticing the 4 black beads on each face, I realized that you can add cubic right angle weave (CRAW), on any face of SCRAW. Yup, CRAW and SCRAW play very well together! Here is a square using SCRAW on the corners and CRAW on the edges, and embellished with silver seed beads to fill in the spaces. The embellishment stiffens the weave significantly. That embellishment is essentially what I show at time 2:50 in the video below on Cubic Right Angle Weave. Now, if that weren't enough, there's more. Because these cubes in CRSAW have extra beads, you can attach them by the corners, like this. Each pair of consecutive cubes shares 6 beads instead of 4. That's 3 purple beads and 3 golden beads shared by each pair of cubes. And there's even more, but I got to save something for later, right? Okay, I'll leave you with two last photos. Here's what I had for Thanksgiving dessert. And this. My sister made these delicious delights. The candied cranberries were my favorite... super tart and just a little sweet. Thanks for stopping by.

Fiona Curren's designs for the rug company, are an example of some geometric/tessellation inspired musings. If you scroll down you'll se...

I enjoy experimenting with freeform and hyperbolic crochet techniques. For more information on crocheting models of hyperbolic geometry check out this link: Math & Fiber. Here is an ex…

I enjoy experimenting with freeform and hyperbolic crochet techniques. For more information on crocheting models of hyperbolic geometry check out this link: Math & Fiber. Here is an ex…

I've made another hyperbolic form...I think I'm addicted! I bought a nice 4ply gold mercerised cotton, and using a 2mm needle, I increased 1 in every 2 stitches. Here it is in the making!!