Octahedral Space Grid Sructure Beaded with Bugle Beads

This large beaded bead is composted of 48 tetrahedron and 36 octahedrons, all arranged into a large octahedral form. It has 8 large triangular holes and is very hollow. The design is based upon the work of J. Francois Gabriel, who wrote about the use of polyhedra in the architecture of high rise buildings.

This beaded object is composed of somewhere between 1300 and 1400 beads.

The beads include sparkling dark bronze bugle beads, and a variety of smaller beads. This beaded art object is very light and a bit fragile, but not so fragile that you can't hold it and play with it. If you drop it on carpet, it should survive the fall unscathed. All of the beads are made from glass. So don't step on it.

It measures about 3 inches across, making it too large for jewelry, but it would make a nice hanging ornament or a piece of jewelry for your computer for you to gaze at while you consider the state of the universe and everything in it.

I started weaving this piece a long while ago, longer than I'd like to admit. It sat half done in a box, for years. Recently, I picked it up, washed it off, and finally finished it. As much as I'm pleased it done, I doubt I will ever make another one without a good reason.

If you would like to have it, you can find it in my Etsy shop. Thanks for looking.

Coxeter Bead in Pink and Silver

This Coxeter Bead is a beaded bead, woven from pink and silver glass seed beads. This ornate cluster is composed of over 400 beads, each one precisely woven into place. This beaded bead is very round and hollow and has a bit of a satisfying squish to it without being droopy. The shape is like a Buckyball virus.

Coxeter Beads are named after the great mathematician Harold Scott MacDonald Coxeter in honor of his extensive work on symmetry, especially four dimensional polytopes, on which this piece is based. It is woven like cubic right angle weave but with tetrahedrons and prisms instead of cubes. It has a fascinating internal structure that you can see when you look at it closely.

Beaded bead is 26 mm (1 inch) in diameter, suitable for a focal bead on a necklace. The largest hole is 2.5 mm wide, wide enough to accommodate a thin cord or chain.

If you would like to learn how to make your own Coxeter Beads, I have a tutorial for a couple variations. Thanks for looking.

Bridges Paper - Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving

I have been going a little crazy saving a surprise for you all, and today is finally the day to share it!  Meet the Highly Unlikely Tetrahedron.

These are photos that I will be presenting with my paper at the Bridges Conference in Baltimore this month.

My paper is called, "Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving." It is now available from the Bridges website. You can download the free PDF file here.  I hope you enjoy it!

Be sure to browse the entire collection of papers in the 2015 Proceedings of the Bridges Conference.  There are so many great contributors this year; it will be impossible to pick a favorite. Seriously, go brew yourself a pot of coffee, start at the beginning and click on any title that looks interesting. Candy for your brain. If you can make it to University of Baltimore on July 29 - August 1, 2015 (Wednesday - Saturday), you can even go to the Bridges Conference and listen to the authors talk about their papers, which is way more fun than reading them because you can meet the people who wrote the papers and ask them questions.

If you don't know about the Bridges Conference, neither did I until about 2003, when I heard a talk at the Joint Mathematics Meetings by Reza Sarhangi, a Bridges founder and generally fascinating fellow. Reza talked about this meeting where people discuss connections between mathematics, science and the arts, including visual art, architecture, music, poetry and theater.  As I listened, I thought, "I think I have found my people." It was a big moment for me, listening to Reza speak.  So, I went home and wrote a couple short papers on quilting and math, and submitted them for review.  Next thing I know, I was presenting my work to a group of like-minded people, other mathematical artists and mathematicians who loved art. They had many fascinating ideas to share, and they educated each other, plus they had interest in my work and opinions about it. I was in math-art-nerd heaven. I went to four Bridges conferences in so many years, and then I left academia to be an artist, and stopped going, and started going to Burning Man instead and doing art there with that community. Then, after last year with the Genie Bottle, I decided to take a year off of Burning Man, and go back to Bridges this year instead. So I wrote a paper on beading impossible figures, they accepted it, and I booked my tickets.

Then, Kelly Delp and the other organizers sent me an email. They thought my paper was so swell that they asked me to give a keynote address to the whole conference.  That means that I get more time to talk and show slides, and there will be no other concurrent sessions while I will be speaking.  I also get a little spot on their website here among the other keynote speakers, including John H. Conway, Ingrid Daubechies, and Alan C. Kay, who all have their own Wikipedia pages, by the way. So, you get that I'm excited to go and see and meet all the people. They asked me to make a mosaic for their website, which you can see here. 

The mosaic includes photos of our jungle gym Bat Country and the Genie Bottle, two Burning Man art projects that I created with the help of my friends in Struggletent.  The beadwork includes a circular Celtic knot, my cover of the Journal of Mathematics and the Arts with an Octahedral Cluster and a Seirpinski tetrahedron, bacteriophages, DNA, a Highly Unlikely tetrahedron from my Bridges paper, and a few photos of cellular automata, which is the project I'm currently working on.  So stay tuned for that. It's going to be cool. You'll like it.  I promise.

I hope to see you in Baltimore!  As always, thanks for looking.

Coxeter Beaded Bead in Aqua

Here's the latest piece of my bead mat, a Coxeter Bead. It's a little over an inch wide, quite hollow, and it has big holes for stringing it on cord. The symmetry of this piece makes me think of a virus.

The dominant color is the aqua bugles, and then I added matte blue half tila beads to soften the bright aqua.  The tiny drop beads inside the circles came from a mixed box of beads, and I separated the colors when I wove them into the beaded bead. 

If you'd like to learn to make your own, you're in luck because I wrote a tutorial that explains how to weave a Coxeter Bead.  Thanks for looking!

New Tutorial -- Coxeter Bead

 This is my newest beaded bead tutorial, the Coxeter Bead. 

Coxeter Beads are named after the great mathematician Harold Scott MacDonald Coxeter in honor of his extensive work on symmetry, especially four dimensional polytopes, on which this piece is based.

You weave it like cubic right angle weave, but with tetrahedrons and prisms instead of cubes. This tutorial is designed for experienced beaders, and it includes charts like those found on my blog here. This tutorial assumes you already how to do cubic right angle weave and know how to connect two ends to make a continuous strip. If you don’t, check out this link at my blog to learn how. You should also probably already know how to bead a dodecahedron or at least know what a dodecahedron is before trying this design. This is a dodecahedron.

This is a spinning dodecahedron.

If you want to learn how to bead a dodecahedron, Cindy Holsclaw wrote a free tutorial.  

With most of the same materials, you can make Coxeter Beads in two sizes (26 mm and 20 mm).

This is the main design, the larger version that I used in the step photos.  It uses 3 mm Toho beads and half Tila beads, tiny drop seed beads and some size 15° seed beads.

And this is the smaller version that I describe at the end of the pattern with some extra drawings and photos.

As a beaded bead, six large holes run through the center of a Coxeter Bead.  So you can easily string it on chain or cord.

Although it might sound complicated from that introduction, the structure of this thing is actually quite elegant. Once you get the hang of it, it's quite intuitive, and my tutorial is designed to give you that intuition. Click on the photo below to see the materials list. 

The tutorial is 14 pages, including over 100 illustrations and photographs. The tutorial is a PDF file that gives charts and explanations for reading the charts to make Coxeter Beads in two sizes.

Thanks for looking!

Conway Bead with Dodecahedron 5-Coloring of Vertices

This Conway Bead is a beaded bead, woven from blue, green, purple, and silver seed beads in different shapes including drops and O-beads. This ornate cluster is composed of nearly 400 beads, each one precisely woven into place. I used five different colors for the background, giving the piece an interesting symmetry where every hole touches each color exactly once. 

If you look at just one of the colors, say lime green, for example, they form the vertices of a regular tetrahedron.  Since there are five colors, there are five tetrahedrons.  Thomas Hull has made an origami model of five intersecting tetrahedrons that exhibits the same coloring as my beaded bead.

You can see Thomas Hull's folding instructions here: http://mars.wne.edu/~thull/fit.html.
Here is a 3D printed version of the same model by George Hart. 

I tried beading one of these orderly tangles a long time ago, and failed after several attempts, but that shouldn't stop you from trying!  It's a real puzzle.

Anyway, my little Conway Bead is very round and hollow. The shape reminds me of a virus. 

It measures just 21 mm (> 3/4 inch) in diameter, suitable for a focal bead on a necklace.

There are a couple of different ways to string this, and the largest hole is 2 mm wide, wide enough to accommodate a thick cord or chain.  I also made a pair of matching earrings using my Iris Drop Earrings Tutorial.

If you would like to learn how to make your own Conway Beads, I have a tutorial here: https://www.etsy.com/listing/189075857/

This beaded bead is for sale.  Click on the photos to see the listing.  Thanks for looking!

TUTORIAL Conway Bead Beaded with Tetrahedrons and Prisms

I just posted my newest beaded bead pattern.   I'm calling it the Conway Bead.  I named after the great mathematician John Horton Conway in honor of his extensive work on symmetry, especially four dimensional polytopes, on which this piece is based. This particular design is taken from the 03-ambo polydodecahedron.  (Say that ten times fast!)  Alicia Boole Stott discovered this shape last century, along with a bunch of other 4D polytopes, built models of them in paper, and wrote about them.  I didn't bead the whole thing, just a small piece of it.

Although it might sound complicated from that introduction, the structure of this thing is actually quite elegant.  Once you get the hang of it, it's quite intuitive, and my tutorial is designed to give you that intuition.  It's beaded much like cubic right angle weave but with tetrahedrons and prisms instead of cubes.  This tutorial is designed for experienced beaders, and it includes charts like those found on my blog here. This tutorial assumes you already how to do cubic right angle weave and know how to connect two ends to make a continuous strip. If you don’t, check out the links here to learn how.

You should also probably already know how to bead a dodecahedron or at least know what a dodecahedron is before trying this design.   This is a dodecahedron.

One of the things I like about this structure is that it has large holes that run through its center so you can easily string it on chain or cord.

Another thing I like about this beaded bead is that the underlying structure comes from something that is four dimensional.  If you were to try to build the whole structure with bugle beads, it wouldn't work because the angles don't actually match up precisely.  Even the little piece I beaded probably wouldn't work.  It's close, but not exact.  But because seed beads are short and bead weaves are flexible, you just have to be close.  So bead weaving makes it possible to build a little chunk of this 4D thing in 3D, thereby making the impossible just unlikely.  Thanks for looking.

Math Cookies

We made math cookies!  More specifically, we formed mathematical objects out of shortbread and white chocolate. These photos show the combined confectionery efforts of Ruth Fisher, Andrea Hawksley, Vi Hart, and myself.  

Here you can see our set up.  We rolled out the colored shortbread dough using some plastic strips so that we would get a nice, evenly thick sheet of dough.  Here we are cutting (1 : root2) rhombuses with 60° rhombus holes.  We were able to cut the holes with that little metal cookie cutter, but we had to use a knife and straightedge to cut the larger rhombi.

With the little 60° rhombuses that were in the holes, we made a tiling.

Here's the shortbread tiling, before baking.

Shortbread tiling, after baking.

Can you guess what we made with the bigger rhombuses?

 Here you can see them being assembled on jig made from paper and tape.

We made a rhombic dodecahedron, called that because it has 12 faces, each of which is a rhombus.

Here's the complete rhombic dodecahedron assembled with white chocolate.

Then we started assembling our blue hexagons.  Here's the jig for a truncated octahedron.

Here's the jig for a truncated icosahedron.

Here we are assembling the truncated icosahedron with melted white chocolate.

Here you can see all three of the truncated regular polyehedra we made.  Notice we just used one cookie cutter for all of these cookies, a hexagon. In fact, it was BECAUSE we had a hexagonal cookie cutter that we decided to make these particular shapes.         

Quasiperiodic cookies taste better than nonperiodic ones.  We had to make our own cookie cutters for this piece.  We made them out of paper and tape.  This is called a Penrose tiling.  We wanted to decorate it with white chocolate to show an isomorphic tiling on top, but we found an error.  To anybody else who can find the error, you will receive two points and a gold star.

This is a short braid made out of shortbread.

With the last bits of dough, we made lots of little tetrahedral cookies.

We assembled these tetrahedrons into second generation Sierpinski tetrahedral cookies with white chocolate. This actually worked better than I thought it would. Unfortunately, we ran out of dough before we could make the third generation. 

They were delicious.  Thanks for looking. 

Updates: Vi Hart also created a video of our cookies making adventure.  She's a wonderful story teller, and among other things, she included my dog Walter's involvement in our day of making cookies.  Check it out. http://www.youtube.com/watch?v=_n1126GoxbU

We even made it on BoingBoing! http://boingboing.net/2014/03/04/cookie-geometry-with-vi-hart.html