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.

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.

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.

dangles redone; learning about structures

   Well, I redid my dangles, as I mentioned in my last post, and they do indeed dangle more freely now.
    The thing I probably should have talked more about concerning this necklace is the tube itself.  It took several tries to get it right.  The nice thing about doing more random structures is that when you need it to curve, you can just start throwing in a few short tubes on one side and longer ones on the other side till you get the amount of curve you need,  But with a piece like this, where you keep repeating a structural shape, you have to get the shape right so that it will produce the curve you need.  My initial plan was that my tube would be octahedrons ( I call them octahedrons because they're made of 8 triangles, but, of course, they're not equilateral triangles as they would be in a true octahedron).  The cross-section triangle would be all short (15mm) tubes .  The tubes running more or less lengthwise would be 20 mm, except that to get the curve I wanted the triangle on the outside would have 25mm tubes.  I'm trying to use tubes of 5, 10, 15, 20 and 25 mm and make that work without cutting special lengths for a particular piece.  Anyway, it didn't work--the curve was too tight.  I tried 2 more variations before I came up with this one, where the cross-sectional triangle alternates between a 15mm equilateral one and an isosceles one.  That made a curve that was just a bit too shallow, but I had a few longer tubes (28-ish mm) so I could put them in the 2 outside triangles at the very center, i.e. bottom of the necklace and get a sharper curve just there.
  This brings up a whole new idea--that I have to move from thinking of a piece as a chain of units, and think more about the overall piece as a unit.  I came up against this same issue some years ago in a sphere I made and posted about at the very start of this blog.  My idea was to take the basic dodecahedron ( Plato bead in

some publications) and think of it as a bead to use in building a bigger sphere, a truncated icosahedron ( or Archimedes bead).  That's what I did here (sort of--Actually there are 60 small spheres, not 90, but anyway...).  I built it adding on 1 sphere after another, and when I got to the last 4 or 5 spheres it was really hard to get the needle in and out as the big sphere closed up.  After I was done I realized it would have gone much easier if after I had gotten, say, 2/3 done I had switched my viewpoint to the whole sphere.  Then I could have built the whole inside of the rest of the sphere, the very dark blue beads, then the whole middle layer, the clear beads, and finished with the outer cobalt blue layer.

    Now I'm have the same issue with my tube necklaces. Here's one in particular.  I've been using a chain of tetrahedrons to make either a bracelet or a necklace.  Mostly the only way I could make the structure go from making a bracelet to making a necklace was to make all the tetrahedrons bigger. 

  But then I rethought, and realized there are sort of 3 parts to the structure.  The last picture shows it (I hope) with bugle beads.  First there's a series of triangles at the top of the structure, which will be the inside of the curve.

In this example they're a sort of light iridescent gray, and you can see them by themselves on the left.  By playing with the size of these triangles you can vary the height of the piece.  I have tended so far to make it tall  and sort of dog collar-ish.  Then there are the tubes (here dark gray and spirally) that turn those original triangles into tetrahedrons, so that the dog collar becomes a spiked dog collar, as in the center.  Actually you could leave the piece like this, and one of these days I probably will. 
   Finally, you have the zigzagging tubes on the outside edge of the curve.  Here I have just 2 bright silver ones on the far right.  It turns out that the length of these tubes determines the shape of the curve.  Here I've used tubes that are shorter than the other ones.  By using shorter tubes (20mm instead of 30mm) you almost eliminate the curve in the structure.  The longer the tube the tighter the curve.  So now I understand the structure and will have much more control over what I build with it.

  This has been pretty windy, but spelling it out helps me get it into my head, so if you've gotten this far, my apologies for the length.  Also the paragraphing got real weird at the end and I don't have the energy to go back and fix it.  Happy New Year.
  

Live and learn

    I wanted to do something simple and rather stark with my oxidized silver tubes.  Cubic RAW was an obvious choice.  I'd never used it with the tubes, and I didn't really think it would work, but I tried it.
    Over time I've developed a sort of rule of thumb.  With round beads I don't much like structures that use triangles (i e circles of 3 beads) because too much thread shows.  Particularly I don't like to use monofilament  nylon fishing line and have 3-bead circles, because it won't pull tight over such a hard bend, and so lots of it sticks out.  I do occasionally do cuboctahedrons, which are a mix of 3-bead and 4-bead circles, but I tend to avoid them  when I can.  Similarly, with long narrow beads, and particularly with my metal tubes, I mostly only use triangles, because that's the only shape that will stay rigid with long beads.  So I have lots of tetrahedrons and octahedrons, because they're made of triangles.If you tried to make a circle out of 5 tubes, it wouldn't stay round, but could take most any shape.  When I have used 4-bead circles, I've always kept the cross section a triangle, because then, even if the squares turned into parallelograms, it still couldn't completely lose its shape, and it would remain 3 dimensional.
   However, I've come to realize that while a single cube made out of tubes is pretty apt to not stay much like a cube, a series of them will have more stability.  Also, when I was using oxidized copper tubes I used fireline thread, because there wasn't enough room for monofilament, which is fatter.  But the silver tubes, although they're the same size on the outside, seem to have a thinner wall, so there's room inside for more thread.  So here I used monofilament line to support the corners of the cubes better.  The piece moves alot, but it still holds its shape quite well.  It's big, because I wanted it to just go over the head without a clasp, so the pieces that form the long side of each rectangle are around an inch.  Anyway, I enjoyed doing it, and may do more along that line.  One thing I'd like to do just for comparison, is a simple rectangular necklace like this, but made out of a string of octahedrons instead of cubes.  Then I can see whether a much more rigid piece works as well on the body as one, like this, with more give.

More tetrahedrons

I'm playing with tetrahedrons again.  I'm discovering that being able to cut my own tubes is changing my mindset.  When you work with beads mostly you're saying "what can I do with the beads I have?"  When you can make your own you start to ask "What beads ( or in this case lengths of beads) will do what I want to do?"  I came up with a sequence of tetrahedrons that would produce a very gentle curve.  But by lengthening the tube on the outside of the curve you can make the curve sharper.  But how much to lengthen it?  I did what I could with math--Pythagorean theorem and all that.  But that doesn't completely do it because there's always some looseness in beadwork, and errors build up.  I had

decided I wanted to do a necklace that would go over your head without needing a clasp, but I didn't want it to be too huge either. And the tubing is expensive enough that I can't just keep trying different lengths till I get what I want.
   I played around with coffee stirrer straws (top picture), using 50mm straws for 25mm tubes, 40mm straws for 20mm tubes etc, and could approximate things that way.  For the most part I'd like to stick with cutting my metal tubes in lengths of 5, 10, 15, 20mm etc.  Otherwise it's just too hard to keep track of things.  But for a given piece I can always pick one element (like the tube on the very outside) and make it whatever I need to make the shape come out right.  Anyway, I'm still learning, still having fun.

sterling silver tubes

 Buying and cutting those colored aluminum tubes that I've talked about in the last few posts has started me off on a tear.  I've thought for a while about using oxidized sterling silver instead of oxidized copper, but was put off by having to cut the tubes myself.  At the same time I knew that there were things I couldn't do with the copper tube lengths I had, that I could do if I could control the length myself.  So as soon as I had worked a bit with the aluminum tubes, I found myself buying silver tubes and cutting them up too.
  As a practical matter, the price difference between copper and silver isn't so very great because the necklaces don't actually have that much metal in them. They're light as a feather.  In fact, that may be a problem in some pieces down the road.  On many of my copper pieces I used some 18mm melon shaped beads, mostly to add a little weight to make the piece hang better.  But those same beads in silver are pretty pricey because they have a lot of silver in them.  For now, the time it takes to cut and, more importantly, de-burr the cut tubes adds more to the cost of the work than the difference in metal prices.  And I think silver has more customer appeal than copper.
   The piece on top, with the tetrahelixes, was my first one with oxidized sterling.  I find it doesn't oxidize as dark as the copper does, at least using liver of sulphur.  There may be other chemicals that make it darker if I decide I want that.  For now this is OK.  I felt I needed seed beads at the ends of the aluminum tubes, so I used them at the ends of the
silver tubes too.  On the second piece, below, I used them just on the aluminum, not on the silver.  As I was cutting the silver, I found I was seduced by how pretty the bright silver was, so for the second piece, I left it bright (also the dark blue and purple show up better against the bright silver).  However, as a practical matter, I don't want to spend a lot of time trying to clean silver, so I think I'll mostly darken it.  I've looked at a lot of Flora Book's work with silver tubes, and it's so beautiful it tempts me to do more with bright silver. I'll just have to see as I go along.
  On the bottom piece you can see the advantage of cutting my own tubes.  It's a simple chain of octahedrons.  But a chain of octahedrons would normally form a straight line.  In order to get the curve you need for a necklace I had to make the triangle on the outside edge longer than the triangle on the inside edge.
Here's where some trig would have come in handy in figuring out just how much longer, but I managed to figure it out with "lesser" math, and it came out right.

red tube necklace

Once I had my building blocks figured out, I couldn't stop beading till I had a necklace.  I think the color adds a lot in terms of defining the structure.  More to come, as I'm on a roll now.

colored tubes

   I just got my first shipment of colored (anodized aluminum) tubes, and I'm loving them.  I've been wanting to make some pieces that are quite open, as all my tube pieces are, but still have a bit more presence.  The aluminum tubes accomplish this in 2 ways.  First, obviously, they're red.  But also, they're fatter than the oxidized copper tubes.  The copper ones are 1.5 mm in diameter, and the aluminum ones are 1/16" which is almost 2mm.  Because the tubes are relatively big, I found I needed seed beads at the ends, as having 3 big raw tube ends at each point of 
a tetrahedron was unattractive.  I had used #11 beads at the ends of copper tubes in the past, but I thought that the fatter tubes might "want" a bigger bead.  So I tried it with #8s.  But I didn't quite like it so I tried #11 beads (this is all in the 1st picture) and I liked that better.  A #11 bead fits quite nicely into the hole in the red tube.
   Then I had to figure out how to combine the red tubes with the blackened copper ones.  When I cut up the 1st 1' tube, I got 14 beads, but they were just a hair shorter than the long copper ones.  So I thought I might put seed beads only on the ends of the red tubes, and not the black ones, and that would even things out.  I did that in the 1st tet in the 2nd picture, and it pretty much worked, in terms of making the sides even.  But I wasn't sure I liked it. so I did a second tet in which all the tubes had beads at the ends.  I liked that better, even though the lengths weren't quite right.  On my next 1' tube I lengthened the individual tubes just a bit and got just 13 tubes out of a foot, but the length was better. 
  Actually, I need to work on getting tubes of a consistent length.  The red ones were better than the gold filled ones I did a few weeks ago, and the last red tube I did was better than the first,  But they're still not as consistent as I'd like.  Rio Grande sells a tube cutter that you can set to a length, and that should improve both consistency and speed.  I want to play around with the lengths of the tubes, but it will be trial and error as to how a given length affects the geometry of a piece, and I can't do much trial and error without wasting beads, so that's a problem.  I imagine there are CAD type programs where you could  play with  polyhedra that have sides of various lengths, but I don't know what they would be.  The only other thing I can think of is to buy unanodized aluminum, which is really cheap ( something like $.35/ft) and play with those.  That, of course, is expensive in time.  Anyway, for now I'm having great fun with my new red tubes.

half knot necklace

    I'm still not very good at the mechanics of blogging:  I wanted to put these pictures side by side, but couldn't get that to happen.
    Anyway, as soon as I figured out how to do get my tube structures to spiral, I thought of making a piece that would look like a loosely knotted scarf.  As you can see, you can make twist the 2 ends together in 2 different ways, although I think the vertical twist (the bottom picture) would be better with 2 twists instead of just 1.   I plan to do another one, probably with the twist closer to the neck and longer tassels hanging below the twisted part.  The only problem with it is that  once it's tied, it just looks like a flexible tube, and you can't tell that the curves had to be built into it.  Oh, well.

Sampler

This is a piece I'm really proud of--even though it's not pretty, or something I can sell.  For a while now, I've been working on these tube pieces, and gradually finding what works and what doesn't.  The trouble is, I often try various ways to, for example, make a smooth curve, till I get what I want.  But some of the ones I didn't want might be a structure that's useful for something else--except that I've torn it out and after a while I can't remember how I did it.  So this is a sort of sampler.  In it I have several ways to make a 60, 120, and (most importantly) 90 degree turn.  The hardest was to make a corner, like you'd have if you were making an open cube.  The inside curve was easy, the the outside curve would either have a big knob on it, or be too flat and pull the inside out of its 90 degree-ness.  Anyway, I learned an enormous amount, and have generated lots of ideas for future pieces.