necklace part 2

    I finished the elements I wanted to make for my necklace--1 big and 6 small.  Now how to join them.  My plan was to just make ladder-like circles of beads to join the elements.  I made one and joined the big structure to the bottom left one that way. But I didn't like it so much.  After some playing around I realized that the corners of the big piece had an angled tube that was at right angles to  the tube at the corner of a small link.  So they could be opposite edges of a tetrahedron.  I did that at bottom right, and it makes a much cleaner connection.  It will also work between small elements, so I think that's what I'm going with.
   Actually something just occurred to me as I write this.  The loops I originally planned, because they're less structured, would allow the links at the back to lie next to your neck.  I think the tets will force the whole necklace more or less into a plane, which means the back links will tend to stand up.  Double tets at each join would give more flexibility, but would (I think) only work if the tubes on the adjoining links were parallel, not at right angles.  Hmmm.

More right angle tetrahedra

    I'm still learning a lot playing with tets (tetrahedra) that include right angles.  The first picture here, which I used in an earlier post, shows the kind of ring structures I've been building for a while with equilateral triangle tets.  The inner ring is a row of triangles alternating point-up and point-down.  Then each of these triangles becomes the base of a tet.  Then, for the outer edge I join all the top points of the tets. 
  In the last couple of posts I switched from equilateral triangles to right angle ones.  The inner ring is a row of right angle isoceles triangles that alternate hypotenuse-up and hypotenuse-down.  Then I went on to see what structures I could generate.
  But the problem with both of those structures is that all the tubes on that inner rim are at an angle

to the overall structure, so you can't easily break up the ring and put a clasp in, which you sometimes want to do.  So I started thinking about making the inner ring out of right triangles, but using the right angle so that you have tubes that are perpendicular to the overall direction of the ring.  I don't know if that makes sense, and I was in a hurry to write this down, so my picture may not show things well enough.  But I think this is going to be useful.  Here I won't be using a clasp; the plan is to make 6 (possibly 8) of the small shapes, with the big one at the bottom.  It'll be a  pretty large necklace--the big shape is over 4" across and the small ones are about 2 1/4" across.  So it should have a lot of impact.  You can't tell too well from my bad picture but there's a sort of pinwheel of orange tubes on one side and yellow tubes on the other.  I'll probably put 1 color on top for all the small shapes and the other color on top on the big one.  Stay tuned.

Math is SO COOL

I had such fun playing with beads/math today.  The other day I was looking at something that was showcasing geometric jewelry.  I'd tell you the source, except that I totally can't find it any more, and I don't know whose work it was that got me started on this.  But anyway, I saw some jewelry that had painted cubes in it.  The cubes faces were 2 colors, with the color change happening on the diagonal of the square.  The way the colors were arranged, you could see that cubes have tetrahedrons embedded in them, but tets that use right triangles instead of equilateral ones.  So I started playing around.  I was using 20mm tubes for the sides of the cubes, so for a hypotenuse I needed around 28 mm.  I used 25mm tubes with a #11 seed bead on either end, and that worked pretty well.  Turns out that you do that and make an octahedron with a tet at either  end.  That's a structure that I've used over and over, but not with right triangles.  When you make either equilateral triangles, of just irregular triangles, you get things like this:

But if you do it with right triangles, you get this:  a cube.  How cool is that? 

I hope you can see it; it's hard to photograph 3D stuff like this.  There's a tet at the top and another at the bottom of the picture.  In between is an oct with hypotenuse edges at top and bottom and side length edges zigzagging around the middle.

  In that cube some of the diagonals line up end to end and some don't.  I realized that I could make them all go line up end to end, and I wondered what the internal structure of a cube like that would be.  I guessed that it might be 5 tets, but I couldn't envision it.  The reason I guessed that it might be 5 tets if that I've learned that when you make an oct, at the time when you've put in 11 of the 12 tubes, and you're getting ready to add the 12th tube, there are 2 possible ways you can orient the 12th tube.  One produces an octahedron and the other produces a cluster of 3 tets.  So I guessed that it would be possible to make a cube out of 5 tets.

Voila!  It turns out that there's an internal equilateral tet made of all hypotenuses.  Then each of the 4 faces of that tet is the base for a tet made of sides of the cube.

You might say this is interesting, but so what.  But for me it could be big.  I really like building things using RAW, but I couldn't use it with the tubes, because the square sides weren't rigid.  Everything could collapse.  So I had to work with tets and octs.  But right angles are so much more intuitive.  We've been stacking blocks since we were kids, and we know how to assemble things using right angles.  60 degree angles and equilateral triangles are much wierder.  So now I can use structures that work with tubes, but still build cubes.  At the moment the cubes I've made are pretty big, but who knows where this will lead?

New octahedron piece

This is sort of a follow up on a post I did in January.  In that post I showed and talked about a necklace I had made in which I built octahedrons sized so that adjoining faces were almost at right angles to one another.  Each oct had a "waist" made of 2 short ( 10mm) tubes and 2 long gold filled tubes.  The angles weren't quite right angles because the 25mm gold filled tubes weren't quite long enough to be hypotenuses.  That would have required tubes of about 27mm, and I like to use stock sizes where I can.  So  on the sides of the piece the octs zigzagged a bit.
  In that post I mentioned wanting to use this idea to make a piece that was something on the order of a tic-tac-toe grid.  When I started to do it with the zigzagging rows of octs, I found that the intersecting rows weren't quite at right angles to one another.  To get that, I really did need octs that had right angles between their faces, so using the same 20 mm beads on the non-waist edges, I really did need 27mm waist tubes.
  It turns out the perfect solution was to use a 25mm tube with  a #11 seed bead at each end.  I wanted to use my anodised aluminum tubes instead of the gold filled ones, so that each chain of octs would have a different color, and I've found that with the aluminum tubes I'm pretty much always happier if I put a seed bead at each end of the tube.  That's because the aluminum tubes are quite a bit fatter than the silver ones, and so the open hole at each end of the tube is unattractive.  The addition of the seed beads solved both problems.

New Shape

I've been working again on creating different torus shapes based on the tube structures I learned from the Beaded Molecules people.  Since they're based on 6-bead circles, which serve as hexagons, they lend themselves to circles, triangles, diamonds.  shapes with right angles come harder.  I did a post last September where I came up with several shapes.  One of them, which I did in dark brownish red, was vaguely right angled, because I used 4 9-bead circles in the middle to get the curve.  It didn't sit flat, though, and just generally wasn't satisfactory.  I was playing around the other day and came up with a much better structure.  It still uses 4 9-bead circles in the middle (they're the darker green ones) but they're arranged differently, and I like the shape much better.  As a more general thing, it's a way to make a clean right angle turn in a tube structure.