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.

Canal trip

 I don't usually write blog posts about non-artsy things, but just this once I couldn't resist.  My husband and I are leaving in a couple of days for a month exploring the canals of northern England, and maybe a bit of Wales.   A friend and fellow jewelry artist, Lauran Sundin, ( and by the way you should check out her beautiful jewelry at lauransundin.com) owns the boat pictured here.  She and her husband spend several months on it each year, traveling the English canals. That's her in the pictures. My husband and I are long-time boat people.  We spent a year on a small sailboat (in our youth), 

bringing it down the Mississippi River from the Great Lakes.  Anyway, Lauran and her husband Steve offered us the use of the boat this summer, so we're off on another boating adventure.  As you can see from the pictures, though, this one will be really pastoral and quiet, mostly visiting small towns, and traveling the countryside and going up and down the hills ye means of teeny locks that you crank open and shut by hand.  That's one in the second picture  We'll start out near Manchester, which is where we're flying to.    I'm taking lots of silver tubes with me, and I
plan to experiment and play with designs, but

not to expect to actually produce a lot of jewelry on the boat.

Rolling Bridge Necklace

One structure that I totally love is the Rolling Bridge in Paddington Basin, London.  If you don't know it here's a video link www.youtube.com/watch?v=0aIl0bzyQD0.  It's a pedestrian bridge designed by Thomas Heatherwick, and it's just the most elegant structure.  I was telling a friend about it, and suddenly realized that it was a great inspiration for a necklace.  This was also a perfect idea because my husband and I are leaving in a week or so for a month in England.  We'll be up north around Manchester, so I won't actually get to see

the rolling bridge, but it still seemed just right to have a necklace inspired by a British structure.
  This one is  all octahedrons.  Like the bridge, it has 8 sections.  It's not quite finished ( I need to make more chain links, and I've ordered a hook clasp) but when it's done the clasp will be right where the chain meets the "bridge," That way you can hook it to just 1 end of the bridge and it will look like picture 1, or to both ends and it will be rolled up like picture 2.  The only way to get it to be totally straight like the bridge is to hang it upside down, like picture 3, but it's actually a bit long for that (around 7"), so that's mostly theoretical.
  I didn't initially plan on using the seed beads, but I was forgetting that if the individual octs are allowed to pivot any way they want they'll tend to hang with the longer, and heavier, beads down.  The bead

chains keep that from happening.  I'm used to using much bigger seed beads, but they would have made the strings of beads much too bulky.  Fortunately, FMG had once sent me a container of #15 beads in just the right iridescent gray, as one of their free gifts along with an order.  I almost didn't keep them, thinking  "what would I ever do with these tiny things?"  But it's hard to throw away free beads, so they were still in my stash, and worked out just right.
  That reminds me--if anyone knows of interesting galleries, artist, sbead stores etc. in the Manchester...Liverpool-ish area, I'd love to know about them.  Actually, our mobility will be pretty limited, as we'll be on a narrow boat on the canals up there, but you never know. 

Playing with an idea

Still playing with my chains of squarish octahedrons.  The last one I did, and posted about, was not as deep (that is, not as 3 dimensional, not standing up as far off the body) as most of my work.  The tubes to give it depth were 10 mm long.  Here I used more or less the same structure, but with tubes of 15 mm. They changed the geometry enough that once again the octs zigzag a bit, but that was fine here.
   It had occurred to me that you could also use those tubes as hinges and join rows of octs just at those hinges.  So that was what I did here.
   Then I started thinking of other beads I could use for the "waist" beads.  A while ago I had gotten strings and strings of marble beads on sale, and hadn't done a lot with them.  So I thought I'd try those.
  I had 2 sizes, 35mm long and 30 mm long.  I tried the larger ones first and got the structure you see here.  The ends of the marble beads were pretty wide, so I added a bronze colored bead to give a narrower end.  It came out quite zigzaggy, which I liked.  But the 35mm beads were awfully heavy, a

and I was afraid if I made a whole necklace like that it would be uncomfortable.
   I tried using the 30mm marble beads, and to keep more or less the same proportions, I changed the silver tubes from 25mm to 20mm.  The next picture shows that one.  I liked it, but somehow didn't decide to do a whole necklace like that.  But I thought that  single one made a great pendant, so I just finished off the one, and I'm putting that on etsy.










  What I finally ended up with was using the smaller (30mm) marble beads, and 25mm silver tubes.  That made each oct more

square, so I changed the 2 tubes on each outside face  to 20mm ones, and got just enough curve.  It's not quite finished, but here's a preview.  Each 2-oct structure is 2" long, so I'll have 9 of them hinged together.
  The fun part of all this is to take an idea and watch it evolve, and see what you can do with it.

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.

Asymmetrical silver tetrahedrons

Seems to me that I have a way of announcing in a blog post that I plan to do A and then immediately doing B.  Like I'll say I want to get away from symmetry and the next thing I know I've done 3 or 4 symmetrical pieces. Or vice versa. Just a few posts ago I said I wanted to make pieces using sort of stock lengths of tubing instead of cutting custom lengths for an individual piece, because if I get the geometry wrong, I'm left with odd length pieces that don't work and that could  produce a lot of waste. So what did I do next but make the most custom length piece I've made so far.  I'm sure there's a CAD-CAM program out there that could have figured out the proper lengths for me, but, of course, I don't have one.  For this one I drew the inside oval and the outside asymmetrical curve on graph paper, and then drew the triangles it would take to join them.  This gave me an approximation of the lengths I would need, but only that.  Beyond that it was pretty seat-of-the-pants. I've been mostly oxidizing my silver so I don't have to think about tarnish, but for this one I had no choice but to use bright silver, because I was cutting 2 or 3 lengths at a time, and there's no sensible way to oxidize the tubes that way. (If I were still working with copper, I could oxidize as I go, because copper doesn't need hot water to oxidize, but for silver you need very hot water, close to boiling.  I guess I could keep a liver of sulphur bath hot if I used a hot plate, but I just use a pot with a coil immersed in the water, so I can only heat plain water, and when it starts to cool I throw the bath out.) Also if I oxidized a tube and then had to shorten it a bit,I'd be left with 1 bright edge, and I'd have to oxidize it again.  Anyway, I like this one in bright silver anyway, so it worked out alright.  the longest tubes in it, by the way are 62 mm, or around 2.5" long.

New Shape

I've been working again on creating different torus shapes based on the tube structures I learned from the Beaded Molecules people.  Since they're based on 6-bead circles, which serve as hexagons, they lend themselves to circles, triangles, diamonds.  shapes with right angles come harder.  I did a post last September where I came up with several shapes.  One of them, which I did in dark brownish red, was vaguely right angled, because I used 4 9-bead circles in the middle to get the curve.  It didn't sit flat, though, and just generally wasn't satisfactory.  I was playing around the other day and came up with a much better structure.  It still uses 4 9-bead circles in the middle (they're the darker green ones) but they're arranged differently, and I like the shape much better.  As a more general thing, it's a way to make a clean right angle turn in a tube structure.

Charms for Baltimore ACC show

    This will probably be my last post before heading for Baltimore and the ACC show there.  I've been working on charms for the show.  I think I mentioned that they have a special promotion for the jewelers who want to participate by making a "mini-collection" of charms and pieces incorporating charms.  I've made several pieces, and these are the 2 I like best.  The first is a charm bracelet with charms that combine my oxidized silver tubes with gemstones.  #2 is a necklace, just using the tubes. I've also made several loose charms, because I think one of the most interesting things about the promotion is that the show people will create and sell pieces (charm bracelets, necklaces, whatever) that incorporate charms from several artists.  I think that sounds interesting.
      If any of you are in the Baltimore area next weekend (Feb 20-22) come check out the show.  It's at the Baltimore Convention Center, down town.  I'm in booth 501, and I'd love to show you my work.

Octahedron shapes.

   I've been playing around with the different shapes I can get in octahedrons.  I started thinking about how I don't get a firm shape using RAW with tube beads because a square isn't rigid the way a triangle is.  But on the other hand right angles are just so "freindly" to work with.  We're surrounded by them, and we know how they work.  So I started imagining  octahedrons (octs)that were more or less square in overall shape. I wanted the piece to be not too deep, i.e. not sticking up too far off the body (as lots of my pieces do).  So with a relatively flat oct I could just use Pythagorus to figure out how long to make the "waist" beads--here the gold beads-- so that you'd have a right triangle, and that would make a square shaped oct.  The cross-wise and length-wise tubes are 20mm long, so a 28mm gold tube would be about right.  Actually, though, I decided to use 25mm cold tubes instead, so the octs aren't quite square, but zigzag a bit.  Partly that was because I thought a bit of a zigzag was more interesting, and partly because I prefer using more or less stock lengths rather than cutting tubes just for a single design.  If I use stock lengths (5, 10, 15, 20 and 25mm), I have more flexibility in the design.  If I cut custom pieces ahead of time, I have to have it all figured out before I do it, and I like to sort of make it up as I go along, even if I have the general idea figured out at the start.  Also there would be lots of times that I would cut lots of, say 28mm tubes and then discover that to get the right curve, I really needed them to be 27.5 or 29mm or whatever. 
   Here, when I got to the back and needed to make it curve, I had to use shorter tubes on the inside triangles than on the outside ones.  The ideal length for the inside tubes would have been around 18mm, so I just alternated between 15mm tubes in 1 oct and 20mm ones in the next.  One of these days I'd like to skip the curve and do a piece that would look sort of like a tic-tac-toe grid.  

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.