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.

more right angle explorations

I haven't blogged in a while; sometimes life gets in the way.  My husband and I have been in the sailboat business for 40 years, and we've just closed our store ( which was also my main studio) and are trying to figure out the future plan.  I've now sold all my looms, and the jewelry work is quite portable, but it still creates complications.  Right now I'm cutting tubes at a workbench in a storage locker we've rented, as we live in a townhouse, and the room I've set aside as my workplace is carpeted, and not conducive to sawing metal.  My plan is to cut tubes in larger batches so that I don't mind going to the storage place to do it, but I'm also looking at the possibility of an outside workbench here that won't look too industrial and irritate the neighbors.
   Meanwhile. though, I'm still playing with structures.  I find the blog is a great way to keep a record of what I'm doing.  I use my silver tubes to experiment, but it's too expensive to keep permanent structures made out of handcut and oxidized silver tubes just so that I can get ideas from them.  So if I blog about them, then I have pictures and notes, and I can reuse the tubes for actual jewelry.

    Ever since I made thebracelet above, I've been fascinated by how an alternating series of right triangle tetrahedra and equilateral ones makes these square structures.  I'm making a similar bracelet now, but I kept wondering if I couldn't make the series continue in a straight line instead of turning corners.  (This happened at about the time I realized that I didn't have enough 20 mm tubes to finish the bracelet anyway.)  So I tried it--and it works!

  Doing it just the way I had done the square structures, i.e. alternating 1 right angle tet with 1 equilateral one, I got the top  structure, which is flat on the bottom with a zigzag top.  But what was more interesting, I thought, was that since the zigzag tubeson top were at fight angles to each other I could make them into right angle tets too and then (bottom picture) I get a structure with a square profile that I can extend as long as I want.  Also if I end with an equilateral tet I have a tube at the end which

could serve as a hinge to join another structure. Actually, just as I write this and look at the pictures, I realize that this is really an octet truss, because if you left out the central tube that is the "hub" shared by 4 tets, you'd have an octahedron.  Stay tuned!
   I should mention that things don't always turn out that neatly.  When I made a chain using only right triangle tets instead of alternating, nothing interesting happened.  Also I suspect I could have found out the same thing just by creative use of the Pythagorean theorem, but then I wouldn't have pictures to remind me of what I'd learned.

Cubes

Ever since I figured out ( post in Sept., 2015) that I could make a rigid cube out of an octahedron and 2 tetrahedrons, by using right triangles instead of equilateral ones, I've been wanting to make a necklace like this.   Not much to say about it, except that I really like it, and I wore it for the first time this week and got several compliments. I think I'll list it on etsy. I'm working on a similar one with 1 cube made from 14 kt gold tubes.  I've been wanting to buy some gold tubing for a while now, and I've finally done it. It's not quite finished, but I love the contrast between the gold and the dark silver.

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.

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?

Pythagorus pendant

Great comment,Gwen, about adding the cube in the middle to indicate the right angle. The thought had crossed my mind, but I was afraid of detracting too much from the jewelry side in favor of the math side. But based on the comment I decided to try it, and I kind of like it. Thanks.