Some math, some design

 In some earlier posts, I've talked about some necklaces I've made out of octahedrons, where a long tube, often a colored aluminum one, acts as a hypotenuse to create octs that can stack in a way similar to the cubes in RAW.  This works well for necklaces, where it's nice to have the structures lying somewhat flat, but it doesn't work so well in a bracelet, where you would generally want them to stand up.  I needed a way to join 2 standing up octs so they'd be at an angle to each other.  After some trial and error, I  came up with a solution.  If I made an oct so that the triangle on the top used tubes that were half as long as the tubes on the bottom, I'd get an oct that if you looked at it from the top would look like one of the pictures
below, with the little triangle in the middle being the top of the oct and the bigger outside one being the bottom.  And the triangle "walls" that connected them would be perpendicular to the top and bottom ones ( actually there are  are 6 triangles connecting them and every other one is perpendicular.  If I used the pattern on the left to join 2 of my squarish octs, they'd be at right angles to one another; if I used the other, equilateral one they'd have a 60 degree angle.  That's what I used, and started to make the bracelet  shown below.  I was pleased

 with it from a structural standpoint, but it didn't seem to have enough color, and the aluminum tubes seemed to sort of get lost in all the dark sterling.  I decided that the problem was that the joining octs didn't have any colored tubes, and also that the colored tubes on the inside of the bracelet were kind of lost.  So I took it apart  and moved all the tubes to the outer wall of the bracelet, using them in both the squarish octs and the joining octs.  Both the bracekets are incomplete because I was gradually taking apart the first one to build the second one.  But I hope  they're complete enough that you can get the idea

of what they look like.

more geometry games

     Back in September, I wrote about using right triangles in building tetrahedrons and octahedrons, and how really interesting things happened.  Mostly I was using 20mm tubes as the sides of my right triangles and 25mm tubes with #11 seed beads at each end as the hypotenuses.  Effectively, that made the hypotenuses 28mm.  That worked because to have a right triangle where the 2 sides are equal, the length of the hypotenuse has to equal the side times the square root of 2 (I wish I had a font with a square root sign in it).  So for a 20 mm side you need a roughly 28mm hypotenuse.  I had been cutting my tubes in lengths of 10, 15, 20, and 25mm lengths.  But it occurred to me that if I changed from cutting 15mm tubes to cutting 14mm instead, and added a 28mm length, then each length (except the 25mm) would be a hypotenuse length for the size below it, i.e. I could make right triangles with sides of 10 mm and a 14mm hypotenuse, or 14mm sides with a 20mm hypotenuse, or 20mm with a 28mm one.  I kept the 25mm size too because it's roughly an inch and sometimes my non-scientifically minded brain just reverts to inches.
    Anyway that led to lots of playing around with  right triangles.  And here's another place where cool things happened.   I've made several bracelets out of chains of tetrahedrons.   As you can see

from the first picture it's not a perfect fit.  There's a repeating 4-tet unit, and six of them, pictured here, don't quite come together.  You can pull them together as  in picture 2, where there's a hexagon at the center of the circle.  Or, depending on the length you want, you can force them apart a bit and make the circle out of 7 units with a heptagon in the middle.  I think in my bracelets I've actually pushed them apart enough to use 8 units to make the circle big enough to fit over the wrist.
    But anyway, for purposes of this exercise I was acting as if 6 units made a circle.  But then I tried substituting one size longer tubes on some of the outside edges to make the structure curve faster.  In picture 3, I alternated 2 units made all of 14mm tubes with 2 units made of 14mm tubes but with 20 mm tubes on the outside.  I had no idea I'd get anything regular, but it turned out that I got a circle made of 4 units (16 tets).  The figure at the center is a square on one side, the bottom in picture 3, and a diamond on the other, top, side.  And if I made all the outside edges 20mm and all the others 14mm ( last picture) I got a circle with 3 units.
    My favorite of these shapes is the 4-unte one, especially because the front and back are different.  I'd like to make a piece using these and using some colored tubes to highlight the patterns on front and back.  Stay tuned.

Necklace "upgrade"

Back in April I did a post about playing with octahedra that had a square shape, i.e. that had 2 sides, in front and back that were basically squares with a long edge forming a hypotenuse.  I showed this piece, with the colored tubes acting as the hypotenuses.  I liked the piece because of the color, and because I liked the squar-ish shapes, but over time I came to think it was just a bit boring.  My work tends to be pretty minimalist, and I like that clean, simple look, but sometimes it's hard to find the line between simple and boring.
  I kept looking at the piece, and finally decided it needed  a change.  I wanted to break up the symmetry just a bit.  Actually all I did was add 5 more octs to the red section in the middle, but I thought it made the necklace a lot more

interesting.  And I guess I was proved right because I sent it to a sort of trunk show that I participate in, and it sold the first day.
  With that piece gone, I'm working on another one using the square octs with colored aluminum hypotenuses.  If it works out I'll post it when it's finished.

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.