Beadwork, Polyhedra

Rick-Rack Polyhedra Variations

While I was making the first rick-rack dodecahedron I had an idea for a slight variation made by joining the rick racks together point-to-point instead of edge to edge. Since this would require a join between three edges, I first thought that I could use a warped hexagon instead of a warped square. However, I was completely wrong about that! The angle of the warped hexagon was far too small. After a bit of trial and error I found that three warped squares joined into a pyramid on two of their sides resulted in a triangular shape with the right surface angle. I then spent the better part of a day trying out different colour combinations and patterns to end up at the conclusion that they looked best just all in the dark blue colour. It’s a bit of extra work making the three squares for each join but they look great – the small pyramids create this extra spikey structure around the rick racks that I really like. Here’s the finished piece:

BluebellTwo

If you compare it to the original you can see the differences – in the new shape two rick-rack points meet at each join where in the original three points meet at each join:

rickrack_dodecahedron_2_beadmechanics

You can see the difference in how the two are constructed in the diagram below – on the left is the variation and on the right is the original:

The two polyhedra – the original and the variation – correspond to a dodecadodecahedron (no that’s not a typo!) and a ditrigonal dodecadodecahedron. (Although the triangular faces on the new shape are convex not concave, but close enough!). The original shape is also similar to a hyperbolic dodecahedron (which I believe is a dodecahedron that tessellates in hyperbolic space, rather than is entirely hyperbolic itself).

I was also asked if it was possible to make smaller shapes such as tetrahedra or cubes out of rick-racks and warped squares. The crucial factor here is the angle of the warped squares – their maximum angle (the angle between the ‘V’ shape each one forms) is about the same as the angle needed to join the rick-racks together on a dodecahedron. They’ll squash down enough to join together rick-racks at a smaller angle, which should make shapes with higher dihedral angles (i.e. less pointy edges) possible, such as truncated icosahedra, but not those with small dihedral angles such as tetrahedra. Fortunately there’s an easy solution! You can make something something very similar to a warped square by joining two triangles together one on side, as shown on the below on the left, compared to the original warped square join on the right:

Unlike the square however, the triangles will bend to any angle along this join, which should make any regular or semi-regular polyhedra possible! To test the idea I made a tetrahedron using this method:

BluebellTetra

The rick-racks here have three peaks instead of five, but otherwise it’s made in exactly the same way as the original dodecahedron, just joining each one together with the pairs of triangles instead of with a warped square. The resultant shape is very pointy (and hard to photograph!) but much quicker to make!

I’ve now made a set of podcasts with various different numbers of sides so I can make a whole set of these polyhedra!

BluebellRickRacks2

Beaded machines, Beadwork

Mobius warped square kaleidocycle

A while ago I came across a youtube video with instructions on how to make a Mobius kaleidocycle out of folded paper hypers – a kaleidocycle with a twist! So of course I had to try making one out of warped squares!

It worked! And it’s great fun to play with. It’s just a strip of 7 warped squares made into a Mobius strip (i.e. twisted once before being joined into a ring). But it also cycles!

BeadMechanics_MobiusWarpedSqKaleidocycle_Join

If the colours look familiar that’s because they’re the CGB prism colourway!

I don’t know if it would work with regular tetrahedra, but I think the warped squares distort more than a beadwork tetrahedron would and allow it to turn.

BeadMechanics_MobiusWarpedSqKaleidocycle

This is actually the second Mobius kaleidocycle I’ve made – while I was planning out a peyote version of the trefoil knot kaleidocycle I discovered that it also has a twist (which is awesome but did cause a few issues with the pattern I had planned for it!).

It would be interesting to see if other Mobius cycles can work – maybe with a longer strip of squares more than one twist is possible!

Beadwork, Polyhedra, Tutorials

Rick Rack Dodecahedron (with Tutorial!)

I was challenged a while ago to see if I could make a dodecahedron out of rick racks. After a bit of experimenting I ended up with this, a Contemporary Geometric Beadwork rick rack dodecahedron!

rickrack_dodecahedron_2_beadmechanics

It’s made from small 5-sided rick racks joined together with warped squares. The rick racks are the light blue beads you can see, zipped together at the top with dark blue beads. The warped squares joining them together are the dark blue diamonds you can see in-between each rick rack.

rickrack_dodecahedron_3_beadmechanics

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Beadwork, Beadwork objects, Polyhedra

Sunburst

Update: a tutorial for this piece is now in my etsy shop! Thanks to Sue Harle for permission to use her original diagonal tubular peyote technique in the tutorial!

I don’t seem to have had much time for beadwork recently, but a few months ago I did manage to finish a new piece: a sunburst dodecahedron!

photo1

It uses Sue Harle’s tubular diagonal peyote technique, which is just fantastic for geometric work like this as it’s beautifully flexible, but still strong enough to hold the shape together.

I’ve been playing with this idea for about a year but couldn’t really get it to click, I think mostly as I wasn’t sure that it would work (a feeling that stayed with me all the way up until it was finished!). There was also a bit of hasty re-engineering of my initial idea half way through (I might have forgotten how many edges a dodecahedron has…), but it did work in the end and I’m very happy with how it turned out. Its only downside is that it’s really hard to photograph!

photo2

I really want to try taking this idea further by trying a similar approach with other shapes – although I’m not sure yet if the angles will work out for other polyhedra. It’d also be fun to try two nested shapes, maybe a dodecahedron and an icosahedron, or two dodecahedrons, but I’m still contemplating how to join them together so they stay centered.

photo4

I also really like these metallic yellow delicas. I don’t often use yellow in my designs (you might have noticed that blue is my go-to colour!) but I’m glad I tried venturing out if my colour comfort zone to try this.

photo3

I have started writing a tutorial for this shape too! I’m still a bit apprehensive about drawing the diagrams for it at the moment, as it’s very 3 dimensional and hard to show on a flat page – need to spend some time thinking back to maths lessons about 3D projections!

Beaded machines

Trefoil Knot Kaleidocycle

A while ago I found an interesting paper about rotating rings of tetrahedra (aka kaleidocycles) by Jean Pedersen┬╣. Apart from some great instructions on how to make them by braiding two strips of paper together it also mentions that with enough tetrahedra, a kaleidocycle can be tied into a knot and still rotate.

So of course I had to try this! The paper says that the minimum number of tetrahedra required is 22, which is quite a lot. I decided to make them out of bugle beads to test the idea. I made a long strip of them using right angle weave (although in this case the angles aren’t right-angles) and illusion cord . When I had enough tetrahedra I tied the strip into a trefoil knot – this is just an overhand knot with the ends joined together. The completed kaleidocycle looks like a bit like 3 normal kaleidocycle merged together:

TrefoilKnotKaleidocycle_BeadMechanics_1

Now for the moment of truth – does it rotate properly?

The answer: yes! It took a few tries to work out how to get it to turn properly, but it’s great fun to play with. Here’s a video:

I think this is my favourite kaleidocycle so far! I want to make a peyote tetrahedra version, but the 88 triangles needed might be going to take me a while!

 

┬╣The paper is “Braided Rotating Rings”, Jean J. Pedersen (The Mathematical Gazette, 62, 1978).