building with tets

This is  a continuation of an idea I talked about over a year ago ( Jan, 2015 to be exact).  I've made several pieces that are chains of tetrahedrons.  What I've learned is that instead of making 1 tet, adding the next and then the next, I can plan it better if I first make a chain of all the bases of the tets (far left in the first picture), then turn each base triangle into a tet (middle) then join the tets at the tops.  The length of the tube that joins the tops will determine the curve of the structure.  Here I thought of not using a tube to join the tops, but rather a string of beads.  The length of the bead string determines the maximum curve, but within that limit the necklace can take the curve it wants.  And the curving bead string gives the piece a ruffly look that I sort of like.  The tighter the curve the piece takes, the less ruffly it will look.  I assumed it would want to take a tighter curve at the bottom of the necklace than along the sides, so I added 1 bead to each chain for the links at the bottom to preserve the ruffliness.

As a side note, I have to say that this blog has turned out to be a help to me in keeping track of what I do.  I've now had several occasions where I've had to reproduce something I've done before, or a variaton on it.  I've never been too good at keeping records of what I do, but looking back at what I've blogger about has been a help.  When I come up with a new idea, it seems interesting enough that I won't forget it, but 6 months down the road,  I'm trying to remember what the actual lengths and arrangements of things were, and it's gone.  Ah...age.

Size matters

 Here are 2 variations on a theme.  The one with the green stone beads (African jade) was made a couple of months ago, and I was pretty happy with it.  I like that it is regular, and yet the sawtooth edge breaks up the symmetry a bit.  But recently I started wondering how I could take the idea and make it a bit more dramatic.  The answer turned out to be size.  By increasing the size of each unit from around 1.5 inches to over 2.5 inches,  I made a necklace that, to me at least, has more impact.

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.

pinterest

I've just started playing around with pinterest, as a way to get my work "out in the world" a bit more.  I put up a board with some pieces that have been on the blog before, and some that haven't. I haven't spent much time on pinterest, so I don't know the system too well yet.  Seems like there ought to be labels, like a blog has and like Etsy has so that if someone is interested in geometric jewelry or design, my work might come up, but I don't see anything like that.  Any suggestions?

Also, I should mention that one of my necklaces was chosen for the CraftForms exhibit at the Wayne Art Center, in Wayne, PA.  Also, I have work in the Artistry exhibit at the Guilford Art Center in Guilford, CT.  Both of these last into next year.

new tet structure

I've been working on a few pieces based on a new structure built of tetrahedra (tets).  A lot of my necklaces are more or less chains of either octahedra or tetrahedra.  A chain of octs is relatively straightforward, because octs have opposite sides that are parallel to one another, so you can line them up end to end.  The trick there is to get the chain to curve, so as to make a necklace.  But with tets it's more difficult,  Because no side is parallel to another.  Still, I've managed to come up with 2 different chains of tets that create curves that can become necklaces.  But, I wanted something that was more complex than a simple chain.  
The question is whether you can tile tets in 3 dimensions.  The simplest is a group of 5 tets forming a sort of pentagonal circle (how's that for inexact terminology?), i.e. the shape pictured here.  It seems to work, and I'm sure I read that Archimedes, or one of the other big-time Greek mathematicians,

thought that it worked.  Actually, it doesn't quite work; if you make 5 perfect tets sharing a single central edge, there will still be just a bit of space left over.  But as we know, in beadwork, close is often good enough, hence the structure here.  The trouble is that that little bit that's off tends to build as you extend a structure.  If the structure on the left actually worked, you could also make an icosahedron out of 20 tets.  I've done it, but you can tell that all the internal spokes are just a hair too long and tend to bunch up at the center.  Still, I thought I could make the idea work for just a couple of layers before the slight errors built up too far.  So I built this necklace. I like the way it goes from being a chain of single tets on the sides to a more complex structure at the bottom. I broke up the open structure of tets

with cubes made of freshwater pearls, and I was pleased with it.  As I looked at it, though, I noticed that there was a line of tubes going more or less straight from the point at the center bottom of the piece all the way up to the back on each side, and that might be another way to accent the piece. 
 I thought about buying some gold tubes, but decided to do the first experiment with colored aluminum ones.  The aluminum tubes are much bigger in diameter than the silver ones, so I always use seed beads to close off the ends.  I decided to use them at the ends of the silver tubes too.  That made the 25mm tubes effectively 28mm, and so made the necklace a bit bigger, whch I liked.  
The one pictured here is actually my 2nd try.  

The first time I tried making the line of colored tubes go around the color wheel from red at the bottom thru rose, purple, blue, green etc.  But I found that the blues and greens didn't stand out enough from the dark silver, and so you lost the continuity.  So I stuck with just red.  I also added a second row of red tubes, going crossways at the bottom and then up to meet the long line.  This is actually a pretty lousy picture.  You can't see, for one thing, that the  crossways line is actually a different red from the long vee line.  Also, although I really like the way that black form shows off the shape of the piece, it makes the silver tubes look much lighter than they are.  However that's a problem for another day.

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?

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.

Combined structures

I've been thinking about the fact that I've been working with 2 different structures lately.  With the round gemstone beads I work with a system based on hexagons that is the outgrowth of what I learned from the Beaded Molecules blog.  Then I have another set of shapes based on triangles that I use with the silver tube beads.  I really like the openness of the tube structures, but it occurred to me that I could accentuate the not-there-ness of the tubes by having areas of more there-ness using gemstone beads.  I've done that in several pieces by having structures of gemstones and having them interlock with other structures, made of tubes.  I've liked that, but I wanted to find a way to integrate them into a single structure that goes from tubes in triangles to stone beads in 6 bead circles (functionally hexagons) and back again.  This does that, and I think I'll do more along this line.
  I've also been playing around with photography again.  This picture was done by hanging the necklace (with loops of monofilament line that I can then photoshop out) from a hoop of  steel cable with the background behind it but not touching it.  Here it's close enough that you still see the shadow of the piece on the background.  Can't decide if I like that.  I did it again with the background farther away, and that may be better.  I hadn't actually planned on putting that picture in the blog, but now that I've mentioned it I guess I will.  It's a more dramatic
picture, but for some reason I couldn't get it so that you could see the light aqua of the beads very well.  I think I need more light.

stars

On my recent boat trip, I was playing around with  some new ideas.  One of them was stellations.  I made an ornament by making an icosahedron and stellating it by turning each triangle into a tetrahedron.  I liked it, but as it was almost 3" in diameter it wasn't very practical as jewelry.  Then something occurred to me. The icosahedron at the center of it was made up of groups of 5 triangles that formed pentagons coming to a point at the center.  If I took one of those groups of triangles, and made it point inward instead of outward, I'd have a flat side that could lay against the body.  Then I could stellate all the rest of the triangles.  Voila!  Of course, these star forms still extend out quite a ways from the body, but I kind of like that.  I also made a few where the 5 tetrahedrons at the center of the star are made from shorter tubes, and so don't stick out so far, but for the big necklace I decided to stick with the long tubes.