Beadwork, Polyhedra, Tutorials

New Tutorial: Rhombic Mosaic

A new tutorial is available in my Etsy shop for the Rhombic Mosaic icosahedron! This icosahedron is Not Made From Triangles! Instead it uses peyote diamonds for a new take on this basic geometric shape!

BeadMechanics_RhombicMosaic2

This method of making an icosahedron means than you get distinct triangular faces rather than the diamond shaped faces you get if you use triangles. Here’s a comparison of two – Rhombic Mosiac is on the left and an icosahedron made from peyote triangles on the right:

BeadMechanics_RhombicMosaic3

I really like the effect this construction method gives! I started working on this idea last year with my initial Not Made From Triangles tetrahedron:

BeadMechanics_NotMadeWithTriangles2  BeadMechanics_NotMadeWithTriangles1

Since then I’ve tried a few other shapes as well – here is a Not Made From Triangles octahedron along with the triangle version:

BeadMechanics_RhombicMosaic4

I really enjoy making polyhedra using this method and have a number of other shapes already planned!

The pattern in the tutorial uses five different colours for the faces of the icosahedron and has every possible combination of each five at each vertex exactly once. Both colourways are in the tutorial too!

Happy Beading!

BeadMechanics_RhombicMosaic1

Beadwork, Polyhedra

Augmented Truncated Dodecahedron J68

A little while ago I wrote about the Beaded Johnson Solids Project set up by Diane Fitzgerald, a project to make all 92 Johnson solids out of beads. I volunteered to make number 68, the Augmented Truncated Dodecahedron. After a lot of time spent making decagons here it is!

BeadMechanics_J68_1

I’ve named the beadwork version Reflecting Pool. In total it’s made from 11 decagons, 1 pentagon, 5 squares and 25 triangles. To give a better idea of the shape here’s an animation of the polyhedron made using Stella4D Pro:

J68

Here’s the net of the beadwork shape before it the final assembly. I think it looks like a series of connected pools, which is where the name Reflecting Pool came from.

BeadMechanics_J68_Net

Before I started joining the beadwork net together I did a trial run with a paper model – fortunately my beadwork skills are better than my papercraft skills!

BeadMechanics_J68_PaperModel

I really like how the shape turned out. The decagons seem quite sensitive to even the small size variations in the beads and so ended up slightly concave rather than as flat as the ones I made initially. However, I really like how they end up looking when joined together.

BeadMechanics_J68_2

I’m tempted to make a plain truncated dodecahedron, with just decagons and triangles, however it might have to wait a while until I manage to make 12 more decagons!

BeadMechanics_J68_3

 

Beadwork, Polyhedra

Augmented Dodecahedron

Making polyhedra using round beads and polyhedral angle weave is my current favourite bead technique! Here’s an augmented dodecahedron made using 4mm beads:

BeadMechanics_Dodecahedron1

This is a dodecahedron with extra dodecahedra added to each face (augmentation). In theory there should be a slight gap between each neighbouring dodecahedron, but with the beadwork you can merge them together to end up with this shape.

It did require quite a lot of concentration to weave but it was still an enjoyable experiment. I’m definitely going to be trying more shapes like this!

BeadMechanics_Dodecahedron2

 

Beadwork, Polyhedra

Near-Miss Johnson Solids

The Johnson solids are strictly convex polyhedra with regular polyhedra as faces – that is polygons with sides and angles that are all the same. Near-miss Johnson solids however are strictly convex polyhedra that almost have regular polyhedra as faces, but not quite. There are actually a lot of interesting polyhedra that meet this definition. And since they are almost regular you can try making them using same sized beads and let the beadwork distort slightly to make up for the slight differences needed.

Here are a few of them made with illusion cord and 4 mm beads using “polyhedral angle weave” (which is just regular angle weave used to make the various polygons that make up a polyhedron).

BeadMechanics_JSNearMiss11

The first one is a truncated triakis tetrahedron, which has 12 pentagon and 4 hexagon faces:

BeadMechanics_JSNearMiss2

This was easy to make and only needs 42 beads. It’s fairly small and makes a nice little beaded bead!

The next is a chamfered dodecahedron. This is similar to a truncated icosahedron but with ten more hexagons:

BeadMechanics_JSNearMiss4

This one has 120 beads and works really well. It’s a bit bigger than a truncated icosahedron and looks very round, definitely one of my favourites!

The third is a rectified truncated icosahedron. This is basically a truncated icosahedron with triangles added between all the faces:

BeadMechanics_JSNearMiss6

This one has 180 beads and is less round but is still an interesting shape!

The next is an expanded truncated icosahedron, which is sort of like a truncated icosahedron version of a rhombicosidodecahedron. It has triangle, square, pentagon and hexagon faces:

BeadMechanics_JSNearMiss7

This has a lot more beads – 360 in total – and is much bigger than the others. It was a struggle to keep it looking reasonably symmetric, but the patterns made up by the combination of pentagons or hexagons surrounded by triangles and squares are really quite pretty.

The last one is a snub rectified truncated icosahedron and is like a truncated icosahedron version of a snub dodecahedron. It’s made up from triangles, pentagons and hexagons:

BeadMechanics_JSNearMiss10

This is larger still at 450 beads and does not work well at all! The faces are just too far away from regular to work with identical beads and it just wasn’t possible to get it to be symmetric. Well, not all experiments work! I’ll definitely be making some of the smaller ones again though!

 

Beaded machines, Beadwork, Tutorials

Oval Kaleidocycle Tutorial

This video of a kaleidocycle made from peyote ovals was the first post on my blog almost four years ago.

The tape on my hands in the video is to cover up scrapes from rowing, not beading the kaleidocycle – and since I can’t go out to row at the moment I took the opportunity instead to finish the tutorial for it that I drafted several years ago to share with you all!

Tutorial

This tutorial is also available as a pdf!

This kaleidocycle is made from six tetrahedrons. Each tetrahedron is made from six peyote ovals. The ovals are all identical apart from the two accent colours used in the pattern. There are then two different combinations of the ovals to form the tetrahedra – pattern 1 and pattern 2, which is a mirror image of pattern 1.

Continue reading

Beadwork, Tutorials

PDF Tutorials

I’ve now finished creating pdf versions of all the tutorials here on the blog – added to the two from last week are the Folding Cube and the gyroelongated square bipyramid kaleidocycle, which I’ve renamed Solar Cycle since it’s easier to say and the shape makes me think of a simple drawing of the sun!

Here are all four tutorials – click on the name to download the pdf!

Folding Cube

BeadMechanics_FoldingCubePDF

Solar Cycle

BeadMechanics_SolarCycle

Spherical Harmonics

BeadMechanics_SphericalHarmonics

Trefoil Knot Kaleidocycle

BeadMechanics_TrefoilKnot

Happy Beading!

Beadwork, International Beading Week

International Beading Week

I’m really pleased to say that I’m going to be an Ambassador for this year’s International Beading Week! I’m already planning beading activities and putting together some patterns and tutorials to share with everyone to celebrate the week!

IBW Logo

This year International Beading Week (IBW) takes place from the 25th July to 2nd August. The aim is to bring beaders together and inspire people to take up the craft! More details of everything that will be happening can be found on the IBW website.

As a start I’ve been doing a bit of work creating pdf versions of the free tutorials here on my website! Hopefully this will make them easier to save and print. Here are the first two!

Spherical Harmonics

Spherical Harmonics is the Rick Rack Dodecahedron – which finally has a name! Spherical harmonics are a type of function defined on the surface of a sphere which have many uses in maths and physics. The visualisations of these functions remind me of the shape of the rick rack dodecahedron.

BeadMechanics_SphericalHarmonics

Trefoil Knot Kaleidocycle

Did you know that the trefoil knot kaleidocycle is also a Möbius kaleidocycle? If you use a different colour for one edge of the band of tetrahedra you’ll see that it has a twist when you join the ends together to make the trefoil knot!

BeadMechanics_TrefoilKnot

The pdf versions of the folding cube and gyroelongated square bipyramid kaleidocycle tutorials will also be available soon!

People Chain x 12

Beadwork, Polyhedra

Rhombic Dodecahedron

Here’s a fun shape – a rhombic dodecahedron made from warped hexagons and octagons. I wasn’t sure how well this was going to work, but I really like the way each face looks like a slightly folded square. A rhombic dodecahedron has twelve diamond-shaped faces, so I knew it wouldn’t look completely symmetric and it’s interesting to see how it’s turned out!

Rhombic_Dodecahedron_Vertex_Hyparhedron_Verrier_BeadMechanics

I also really like the colour of these green delicas – I don’t use green very often but glad I did this time! I was also very relieved that inclusion of a few matte delicas (the row of brown beads) did not result in disaster! I’m always a bit wary of matte beads since they seem to break very easily, but treated with a lot of care they worked out well.

I started this about 2 years ago just before a move but it got left forgotten afterwards for most of that time. Decided it was about time to finish it last summer! It’s quite similar to the Hypernova dodecahedron in some ways – both are what I’m going to call vertex-hyparhedra, because they are based on the idea of placing higher-order hypars over the vertices of a polyhedron. I think I’m going to start calling the other beadwork hyparhedra edge-hyparhedra (as they involves putting hypars over the edges), and Erik Demain’s original method face-hyparhedra (as it involves putting hypars over the faces of polyhedra).

Rhombic_Dodecahedron_Vertex_Hyparhedron2_Verrier_BeadMechanics

Whatever it’s called it’s definitely an odd little shape, but I like it!

Beadwork, Polyhedra

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:

TruncatedCube_BeadMechanics

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!

 

InaHascher_J5_J16

J5 and J16 by Ina Hascher

 

VeePretorius_J13_J59

J13 and J59 by Vee Pretorius

 

MariaCristinaGrifone_J46

J46 by Maria Cristina Grifone

 

DianeFitzgerald_J57

J57 by Diane Fitzgerald

 

NancyJenner_J58

J58 by Nancy Kooyers Jenner

 

CarolRomanoGeraghty_J63

J63 by Carol Romano Geraghty

 

SylviaLamborg_J91

J91 by Sylvia Lambourg

 

GerlindeLenz_J92

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!

JohnsonSolids2

 

Beadwork

Flat Peyote Decagon

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

Decagon_BeadMechanics

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.

Happy beading!