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!

Infinite Polyhedra and Cubic Honeycombs in Beads

Here are a couple of beaded honeycombs.  In geometry, a honeycomb is a way to fill space with polyhedra, with no overlaps or gaps, like a tiling (tessellation), but in more than two dimensions.  A beaded honeycomb is a 3D weave of a honeycomb, where (a) beads are placed on every edge of the honeycomb and (b) two beads are connected if they are on adjacent edges of the same polygonal face in the honeycomb. First is this tetrahedral-octahedral honeycomb I beaded in 2006, and it's remained one of my favorite beaded beads since then.  Today, I finally got some better photos of it.

 At the time, I made this, I knew that it represented some sort of crystal structure, and I knew there were at least a couple hundred different molecular arrangements of crystals.  (In fact, there are 230 space groups.)  I was a little overwhelmed by the possibilities at the time. So I shelved the idea of looking at them until lately.   I recently beaded this bitruncated cubic honeycomb.

If you like this post, you'll certainly enjoy the beaded runcitruncated cubic honeycomb.  I like to imagine that I'll find time to bead more of these honeycombs in the future, now that we have Wikipedia as a resource.  If you have a favorite honeycomb that you want to see rendered in beads, let me know because I could really use some help picking.  There's just so, so many to choose from! Anyway, thanks for looking.

Infinite Skew Polyedra Pendant with CRAW

Last week I showed you a bunch of progress photos of this infinite polyhedron (4.4.4.8) and gave a little background on the mathematics I used to design this piece. I've since learned it's called the runcitruncated cubic honeycomb.

It's made with a variant of cubic right angle weave (CRAW) using cubes and octagonal prisms.  Since not all of the units are cubes and not all of the angles are right, I think it might better be described as prismatic weave or even better, a 3D angle weave. Just saying...

Anyway, this piece looks really different depending upon how you hold it.

Here you can see the back, with a tiny sterling silver tag that's stamped with my name nestled inside one of the squares.  You cannot see the tag from the front.

In case you're wondering what to do with such a thing, it's a pendant. Here you can see it hanging simply on a piece of cord.  You could also hang it from a nail in the wall and make it wall art.

It's for sale in my Etsy shop.  Thanks for looking.

Infinite Skew Polyhedron (4.4.4.8) W*8

I'm beading an infinite polyhedron, also called an infinite skew polyhedron.   This structure continues to be full of surprises. 

It is a patch of the infinite polyhedron (4.4.4.8), called that because every vertex is surrounded by 3 squares and an octagon.  I copied this picture below from Crystal Structures I: Patterns and Symmetry by M. O'Keeffe and B. G. Hyde, Figure 7.42 because it's precisely my inspiration for this piece of beadwork.  You can see it's made up of cubes and octahedral prisms.

I'm simply placing one seed bead on each edge of this structure.  Accordingly, I'm using cubic right angle weave to make this, with bits of prismatic weave.  Really, it's just a 3D edge-only angle weave where some of the loops have four beads and some have eight beads.  I also added little bicone crystals on both bases of each octahedral prism for sparkle and structure.

I think I first learned about infinite polyhedra (or at least the idea first sunk in), when I read this paper

Patterned Triply Periodic Polyhedra, by Douglas Dunham. 

Here are some action photos so you can see my process a bit.  Here's the first bit I beaded.

After I added a bit more, it looked like a little square crown.  I might modify this little guy to make a new pendant design.  I think it's got a lot of potential.  This is precisely what is shown in Figure 7.42 above.

And then it turned into this, but my thread ran out just before finishing the last crystal filled bit (i.e., octahedral prism) to make the whole thing look like a cube. It's the center front rightish that's missing, right where the big hole is.

And here is where it is right now (and the first photo).  I think I'm going to keep going because every time I add more, it surprises me and looks different.  Why stop now, right?

 

It's done.  You can see more here: http://gwenbeads.blogspot.com/2014/01/infinite-skew-polyedra-pendant-with-craw.html, and it's for sale here.  Thanks for looking.