# The Beaded Johnson Solids Project

The peyote octagon and decagon make it possible to bead a lot of polyhedra. For example here’s a truncated cube – one of the Archimedean solids – made using triangles and octagons:

It’s a fun shape – I think it looks like it’s made of flowers!

As well as the Archimedean solids it’s also now possible to make all the Johnson solids, and Diane Fitzgerald has set up a project to do just that!

The Johnson solids are all the strictly convex, regular polyhedra that aren’t uniform. A convex polyhedron is one that has no “valleys” on it’s surface, like the truncated cube above. Strictly convex means that flat surfaces formed by polyhedrons don’t count as convex either – so a polyhedron that is essentially a cube with each square face split up into four smaller squares would not be strictly convex, since the larger square made from the four smaller ones would be flat. Regular just means that the polyhedra are made from regular polygons, which have equal angles and sides. A uniform polyhedron is a regular polyhedron that has identical vertices – that is, each vertex is made of the same combination of faces meeting in the same order and at the same angles. The Platonic solids, Archimedean solids, prisms and antiprisms are all uniform convex polyhedra. All the other non-uniform regular convex polyhedra make up the Johnson solids.

There are exactly 92 of these shapes, and they were first listed by Norman Johnson in 1966 in the paper Convex polyhedra with regular faces (Canadian Journal of Mathematics, 18, 169). This list was then proved to be complete shortly afterwards by Vicktor Zalgaller (Convex polyhedra with regular faces, Seminars in Mathematics Volumne 2, V. A. Steklov Mathematical Institute 1966, English translation: Consultants Bureau, 1969). They’re numbered as J1 through to J92, and each has it’s own (often very long!) name. Although there are 92 different shapes they’re all combinations of just triangles, squares, pentagons, hexagons, octagons or decagons!

Diane’s project is a call to beaders internationally to join in making the 92 Johnson solids out of flat peyote shapes, just for the fun of it! Once complete they will be strung in order and be available for display.

If you volunteer for the project you get to pick the shape you want to make (and then give a beadwork name to!) and you’ll get a (free!) copy of the instructions for the basic shapes and a guide on how to make the Johnson solids. It’s a great opportunity to learn some new beading skills! There’s a facebook group for the project here, or you can contact Diane directly for more information.

At the moment more than half the shapes are in progress or complete. Here are some photos of a few of the finished polyhedra!

J5 and J16 by Ina Hascher

J13 and J59 by Vee Pretorius

J46 by Maria Cristina Grifone

J57 by Diane Fitzgerald

J58 by Nancy Kooyers Jenner

J63 by Carol Romano Geraghty

J91 by Sylvia Lambourg

J92 by Gerlinde Lenz

They’re all fascinating and beautiful! Here’s the complete set of Johnson Solids, J1 through to J92 in order left to right, then top to bottom. Please join in and bead one!

# Flat Peyote Decagon

Following on from the peyote octagon, here’s a flat peyote decagon!

A decagon is a 10-sided polygon. The formula derived to calculate the increases for an octagon can also be applied to a decagon by setting n = 10, and it says 0.4 extra beads are needed per row. That’s 2 beads every 5 rows, which is actually easy to do in peyote with the increase pattern 2-1-0-1-0.

However, I decided to stagger this pattern slightly in order to make the increase in row length a bit more gradual, which also allowed me to start with a ring of 5 beads instead of 10. I slightly changed the pattern in the last two rows to be 2-1 all around (rather than alternating different increases) so that the last row is the same on all edges, which makes it easier to join to other shapes.

The result is as you see above – a flat peyote decagon! I’m very pleased with how it turned out – and that the formula worked for this shape as well!

Diane Fitzgerald has written some great detailed instructions for all the flat peyote polygons – including the octagon and this decagon! They’re available here in her Etsy shop DianeFitzgeraldBeads.