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?

Engineering a piece 2.0 (5.0?)

 One of the reasons I write a blog:
When I wrote the last entry, I thought I was through designing this piece and now I would just finish making it.  But writing the post got me thinking again, even as I was making it.  Unfortunately I had made more than half the necklace before things crystallized, and I started over.
The top picture is the same one that is in the last post.  I had achieved firmness, but at the  cost of  closing in the "donut".  Then because the top of the donut was small I had a fairly long extra oct as a link between donuts.  I had said in the post that I didn't want to make a unit have 4 octs ( one attached to each end of the stone structure and a chain of 2 joining them) because that would be lots of short tubes, and would be kind of busy.  But the structure ends up with 4 octs in each unit anyway, 3 in the donut and 1for the link in between.  I realized that if I made the 2 octs that attach to the stones angle out instead of in, I could have a more open structure without the extra link in between.  I think the version on the bottom (sorry about the blurriness; I actually did use a tripod, but I must have had the camera too close) is cleaner, simpler, and reads bigger, although it's the same number of beads.  In general, I like it much better.

Engineering a piece

I thought I'd post something about the way I design a piece, which is a very slow, trial-and error process.  I've been doing more with my idea of a structure that goes back and forth between the hex-based structures for round beads and the triangle-based tube structures.  I've done several along the lines of the one in my last post, where the necklace is a chain of polyhedrons, going back and forth from stone shapes to tube ones. Here I wanted to vary that a bit, and create individual "links" that combined tubes and stone, and then join them together.  I started by creating the stone shape (what you see is actually the 3rd version).  At each end it has a triangle of 3 10mm tubes and that becomes the base for a tube octagon.  The problem, then, was to join the 2 octs together to form a round shape and then link those shapes together.   You can't join them with an octahedron, because in those the triangles on opposite sides point in opposite directions, whereas these both have their point on the bottom.  You could use a chain of 2 octs, but that would entail lots of short tubes, and I didn't think that was what I wanted.  So I simply joined them with 3 parallel tubes at the top ( that's the one on the left).  But all those parallels, as I should have known, allow lots of wobble--rectangles becoming parallelograms, etc.  In the one on the right I added 2 more parallels down at the bottom of the tube part, but still had the same problem.

    After some screwing around I realized that there was a way to make the structure out of 3 tube octahedrons.  If I changed the shape of the ones coming off of the stone form so that they touched each other, then you could have an oct in the middle because you wouldn't be attaching to triangles on opposite ends of it, but to triangles that meet at a vertex.  It makes the "donut hole" smaller than I wanted, but that's what I ended up with.  I think you can see that I've taken advantage of the flexibility in the stone part of the link.  The stone structures in the top and bottom pictures are the same, but in the bottom one they look longer and shallower than in the top one.
    Another issue is color.  I think it dates back to my rugweaving days, but I always tend to think that if 1 version of a color is good than 5 are that much better.  That is, If I'm using green stones, I want each iteration to use a different green.  In designing these stone-and-tube pieces, I've so far fought that urge.  For one thing, I think a piece looks less formal when it uses several colors than when all the stones "match."     Also I'm beginning to think that just because a piece of jewelry is necessarily small, at least compared to a rug, more cohesiveness might be good.  However, in this piece, just because the color brown is itself so neutral, I went back to my old ways and used several browns and brown-reds.