Polyhedra, Tutorials

Truncated Tetrahedron

When I made the Sunburst dodecahedron I thought that the technique could be easily adapted to make other polyhedra. The flexible nature of the edges make it easy to adapt to shapes with different angles between the faces. I recently tested this idea by making a truncated tetrahedron, the piece below is the result:

tetra2_beadmechanics

A truncated tetrahedron has four hexagonal faces and four triangular faces, so the resulting shape looks quite complicated, and looks very different from different angles!

tetra1_beadmechanics

It’s a bit smaller than the original dodecahedron, but not by much. Here they are side-by-side for comparison:

tetra_sunburst_beadmechanics

I really enjoyed making this piece and I was pleased by how easily the components could be used to make both triangular and hexagonal faces. I have a lot more ideas for other shapes now too!

Instructions for both these pieces are in the Sunburst tutorial in my Etsy shop! (And a huge thank you to Sue Harle for permission to use her original diagonal tubular peyote technique in the tutorial!)

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

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!

Beadwork, Polyhedra, Tutorials

New tutorial!

I’ve finished the tutorial for my first beaded icosahedron – now named Whirlwind! You can find the tutorial in my brand new etsy shop: www.etsy.com/shop/beadmechanics.

whirlwind_beadmechanics_1

I’ve been working on this for a while – it’s been quite a learning experience! The tutorial is 21 pages with more than 60 photos and diagrams – there’s also a net for a paper version of the model you can cut out and make to help with putting the beadwork together!

I’d always intended to make this icosahedron again so I took the opportunity to take photos as I went along so I could write a tutorial. The new version is actually the mirror image of the original – so now I have a matching pair! (Some brief instructions on how to make a second one so you have a matching pair are also included in the tutorial!)

whirwind_beadmechanics_2

Happy beading!

Beadwork, Polyhedra

Rhombicosidodecahedron

Rhombi-what? Like a dodecahedron, but with some extra squares and triangles between the pentagons! My beaded version looks like this:

beadmechanics_diamond7

Technically it’s a small rhombicosidodecahedron, since there’s also a great rhombicosidodecahedron, which has hexgaons instead of triangles and decagons (I think that’s the right word for a ten-sided polygon) instead of pentagons.

It’s actually based on my previous icosahedron model, although it ended up being a slightly different shape in the end. It took me a while to work out which polyhedra it corresponded to, but a rhombicosidodecahedron is an expanded icosahedron so that makes sense! Here’s photo of the two together:

beadmechanics_diamond2

Another thing it turned out to be is really difficult to photograph! Not helped either by the lack of sunshine today (why is it always cloudy every time I finish a piece?).

beadmechanics_diamond3

It’s made using 30 individual diamond-shaped pieces. These are made using some CGB techniques – each one is made up of two layers built from an MRAW band, with two side increases on the bottom side and four on the top. It was definitely a bit of a marathon making 30 MRAW bands though!

beadmechanics_diamond4

Hopefully the weather will improve and I’ll be able to get some better photos soon!

Beadwork, Polyhedra

Fractal Tetrahedra

So I was playing around with beaded triangles thinking about making some Sierpiński triangles. These fractals are simple to make – you start with a triangle (the first iteration) and remove an inverted half-size triangle from the centre, leaving three smaller triangles joined together to form the larger one (the second iteration). Then you do the same with each of these three triangles to make the third iteration. Keep doing this and you end up with a series of fractals like this:

beadmechanics_triangle

I was looking at these and thought – can you do something similar, but with tetrahedra? A quick search told me that yes, you can! It’s called a Sierpiński tetrahedron, or a tetrix. I went and found some beads and started beading straight away!

The first iteration of a tetrix is just a plain tetrahedron:

beadmechanics_tetrix_1

The matte black beads I used here are some of the first delicas I ever brought, over a decade ago!

The second iteration is where it starts to get more complicated! This is four tetrahedra, half as large as before, assembled to make one larger tetrahedron like this:

beadmechanics_tetrix_2

Each outer face of the tetrahedron is a Sierpiński triangle!

I was worried that joining the pieces together would be difficult, but I just followed a threadpath as if completing the last row of each missing triangle on the outer faces. This seems to hold the pieces together well, and also means the top piece rests on top of the others at each corner, so it doesn’t collapse.

The third iteration proved to be more of a challenge – at this point my tetrahedra were made up of triangles with only three rows. I split it up into four separate groups of four tiny tetrahedra. Each group is made with one thread, and each face is added by working inwards from an outline connected to the rest, rather than by making each one individually. It was quite tricky to do, and there were a few broken beads – I regret picking a matte finish for the edge beads! – but I managed to stitch it all together in the end. Here’s the completed third iteration:

beadmechanics_tetrix_3

At this point I had to stop since I couldn’t make the tetrahedra any smaller. Should have started with a larger tetrahedron!

Here’s the completed sequence of beaded fractals all together:

beadmechanics_tetrix_5

Definitely going to try this again – what’s the largest tetrahedron I can start with??

Beadwork, Polyhedra

Warped polyhedra

So I’ve finally finished the pair of beaded shapes I was working on over the last few months! Here they are – a rhombic hexecontahedron and what is probably best described as a hyperbolic dodecahedron:

hexecontahedron_hyperbolic2_beadmechanics

So around the start of July I was reading about various polyhedra and I came across a rhombic hexecontahedron (the shape on the right) and realised that I could make one out of warped squares. I then realised that I could do a similar shape using warped hexagons and end up with the shape on the left. This isn’t really a polyhedron as the faces aren’t flat, but it’s similar to a hyperbolic dodecahedron shape, which is also known as spikey, the Mathematica logo (while a hexecontahedron is currently the Wolfram Alpha logo). I used Mathematica a lot when I worked in research, and spikey was one of the first ‘mathematical art’ polyhedra I encountered!

hexecontahedron1_beadmechanics

It seems that July was a month for making shapes out of warped squares though! While I was making this I saw Joy Davidson’s 3-star beaded box on facebook, and later saw Kat Oliva’s lovely patchwork rhombic hexecontahedron as well. I also ran across a photo of one on pinterest shortly after I finished it, which turned out to be a pattern by June Huber (Juniper Creek Designs). So it seems that I have just reinvented the wheel on this one!

hyperbolic2_beadmeachanics

I really like the hyperbolic dodecahedron, although it was at times challenging to make. I managed to make the tension a little too tight on some of the points and there were a couple of broken beads that had to be fixed by removing a section and repairing it, but I finially managed to finish it last week. I was also worried that it would be very difficult to stitch the last few pieces together, but it turned out to be much easier than I thought it would be (curved beading needles are an awesome invention!).

 

Beadwork, Polyhedra

Icosahedron

I’m busy making a new geometric sculpture, so I thought I’d take a break and take look back at an older piece. This is an icosahedron I made almost 2 years ago now.

icosahedron_verrier

I say icosahedron, but it’s really half way between an icosahedron and a dodecahedron. If you think of the beaded ovals as the edges, then you can find groups of three that make the triangular faces of an icosahedron, but you can also find groups of five that make pentagons and overall look like a dodecahedron. However, I based it on a icosahedron when I was making it, so that’s what I’ll call it!

Unlike my beaded dodecahedron, this piece has a chirality because of how the ovals are arranged to make each face. Chirality just means that a shape looks different to its mirror image. You can see that here – the ovals all point out clockwise around each triangle, but the mirror image would have them all pointing anticlockwise. One day I’ll make the mirror image version to match!

I’d also like to try this pattern as a truncated icosahedron (which is an icosahedron with the vertices cut off, so it’s made up out of pentagons and hexagons – like a football). I think it’d look good as a larger shape, but I’m slightly put off by the 90 edge pieces I’d have to make!