# Gyroelongated Square Bipyramid Kaleidocycle

Here’s my latest kaleidocycle!

The book “A Mathematical Tapestry” by Peter Hilton and Jean Pedersen has a discussion of the various rotating rings (kaleidocycles) of polyhedra that are possible, including a diagram of one made of 14 hexacaidecadeltahedra – better known as gyroelongated square bipyramids. It was such an intriguing shape I decided to try and construct one from bugle beads. The finished ring is fascinating – in one configuration it’s rigid but in others it’s completely flexible with many degrees of freedom.

It also makes a great bracelet as it will flex enough to fit over your hand but can then be rotated into the rigid configuration to stay on your wrist!

I made the original version with 12mm beads (Matsuno size 5 twisted bugle in Silver-Lined Bronze), but it works with other sizes. The one above is made with 9mm beads (Toho size 3 bugle in Silver-Lined Teal, Opaque Turquoise and Opaque Jet). The bugles just need to be large enough for several thread passes! I use 0.25mm monofilament nylon illusion cord as the thread, which is strong enough not to be damaged by the bugles but thin enough to allow enough passes through each bead.

I did briefly try making a peyote version using triangles (in this case the units are Eva Mari Keiser’s “gyro-eggs”), but unfortunatly it didn’t work very well as the shapes lose their defining sharp geometric shape.

So what is a Gyroelongated Square Bipyramid? It’s two square pyramids (the square bipyramid part) connected with a strip of 8 triangles formed into a ring (the gyroelongated part). Here’s a square pyramid and a square bipyramid (aka an octahedron):

Here’s a strip of 8 triangles which can be made into a ring to make a square antiprism:

Put this in the middle of the square bipyramid (octahedron) and you get a gyroelongated square bipyramid:

It’s an interesting shape! The two pyramids are at angles to each other and you can find pentagons made from 5 triangles at almost every corner.

They can put together into a kaleidocycle by using evenly spaced bugle beads from the middle (the gold ones in the photo above) as shared hinges. These hinges will be at an angle to each other if you look at the shape from the side, rather than parallel. Turns out that this is the critical feature for getting a kaleidocycle to work, and it’s why you end up with sets of mirror polyhedra in a complete cycle.

Below is a tutorial on how to make a four-colour version of this kaleidocycle! Please be careful with it though – remember that it’s made from fragile glass beads which may have sharp edges, so should be treated with care!

# Mookite Rick-Rack

Work has been slow on my next geometric shape, so I thought I’d revisit a piece I made about this time last year: a double rick-rack with a Mookite cabochon. There are some great bangles with tear-drop stones in the Contemporary Geometric Beadwork project — see the CGB facebook page for a photo of an amazing one by Cate Jones! When I saw this cabochon I knew what I had to make with it!

It’s a bit of a departure from my normal work as it’s the only bangle I’ve made in the flat and the only one I’ve made with size 11 seed beads instead of delicas. I really like the effect of the seed beads though, and their larger size made it possible to mix in a row of 2mm stones as well.

It’s a standard rick-rack built off an MRAW base, with a slightly larger peak in the centre than the sides. The bezel for the Mookite cabochon is also a simple MRAW band with a few rows of delicas and size 15 seed beads on the back and front. It’s joined to the band using a few extra stiches and beads at the sides of bezel.

When working in the flat there’s always the question of what to do at the ends. I decided to finish them straight and use that edge as a base for a few rows of herringbone stitch on both the front and back. I then joined the band to a simple toggle clasp with a couple of jump rings.

I was worried that it might not sit well when worn, but the cabochon seems to balance quite nicely and stops it twisting round. I’ll definitely be making more bangles with seed beads in the future!