Shapes of toruses

   After spending a lot of time working with my tube beads, I spent some time this week going back to my molecular shapes.  My idea was to make several long narrow toruses (I suppose it's actually "tori", but I can't bring myself to say that) that I would join together, probably with metal tube structures. I found I couldn't make what I wanted, so I took some time to just try and figure out how shaping these structures works.
  The basic idea is that if you build a tube out of hexagons ( 6-bead circles serve as hexagons), it will be just that--a straight tube.  If you want it to curve, you add  heptagons on the inside and  pentagons on the outside of the curve.  It takes 12 heptagons on the inside and 12 pentagons on the outside to make a full  rotation and create a torus.  Actually, you can do it with 10 and 10, and that's what many of the toruses in the beaded molecules blog do, but to get it all around without having to force it you need 12.  Often you get a firmer structure by making it do something it doesn't quite "want" to do, but here I was trying to not do that.
  What I've come to realize, after making a jillion of the things, is that what you really need is just to add 12 extra beads on the inside and a corresponding number left out on the outside, and you can do that in any configuration.  The easiest, and one I had done before, is to use 6 octagons on the inside and 6 squares on the outside. The picture here is taken from one of my very first posts, and shows  a hexagonal torus done that way on the right of the piece.  It's easy to see because the octagons are green, the squares blue and the rest red-brown. The big square and triangle of the necklace were made by trying to stretch those same octagons into ever tighter angles, and show how much I didn't know what I was doing back then.  On the other hand, it's nice to read that post and see that I've actually learned something since then.
  The 6 octagon/6 square torus makes for a very blocky shape that I don't much like, so I haven't used it often.  But now I can see that there are lots of ways to add 12 extra beads on the inside of the circle, the number 12 being divisible in so many ways.

The first torus in this picture was made by making the center out of 4 circles of 9 beads.  That way each circle has 3 extra beads for a total of 12 extra. It's a bit odd because 9 is an odd number, so sometimes you're adding more to the back and sometimes to the front, so it doesn't lie quite flat.  But you get a sort of 4-cornered torus. The center is actally a short fat diamond on 1 side and a tall skinny one on the other, so it's a bit odd. The second torus uses 3 circles of 10 beads each, and so makes a triangle.  And the last one uses 2 circles of 12 beads each, and makes a sort of long skinny shape.  I can now see that if I had added some hexagons (6-bead circles) between the 12-bead ones, I'd have gotten a longer torus, which was what I was originally looking for.  But by then I was into exploring torus shapes.Actually in any of these configurations, you could make the donut hole larger by just putting 6-bead circles between the larger circles.

  Having gotten  handle on the inside of the torus, I started playing with the outside.  I liked the triangular center best, so I used a center of 3 10-bead circles in each of these.  On the outside you need to take 12 beads away from your structure of hexagons.  You can do that with 12 pentagons, or with 6 squares, or with a combination of the 2.  Using all squares ( the 1st one) makes the hardest angles.  Using some or all pentagons makes the curves gentler, which I prefer.  Also it makes a difference whether you put the smaller circles near the points of the triangle or along the flat sides.  If you could space your small circles evenly around the torus, you'd get a circular outside with a triangular hole in the center.  I couldn't quite figure out how to do that. but I came close in #2.  It has 6 5-bead circles and 3 4-bead ones.  There are 2 pentagons on each flat side and a square at each point of the triangle.  The last 2 are both done with 12 pentagons, so they have nice, gentle curves.  #3 has 4 pentagons at each point of the triangle, so it maintains it's triangular shape. #4 has 2 pentagons at each point and 2 on each flat side.
  I have no idea if anyone is still reading, or if this makes any sense at all.  But I do like being able to talk out my ideas, even if no one is listening.  And if you actually are reading this and have questions, be sure and ask.

Tetrahedrons

It turns out I said something wrong in an earlier post.  I said that tetrahedrons would tile in space, i.e. fill up 3 dimensional space completely the same way cubes can.  Turns out that's wrong.  If wikipedia is correct, though, I'm in good company in my error, because they say that Aristotle believed it too. 
  It starts out with the idea that 5 tetrahedrons will make a pentagonal solid like the one pictured here.  Actually it's close enough that it works fine in beadwork, but as an actual matter of geometry the 5 tets would leave just a few degrees left in the circle.  Knowing that, I was aware, as I made this piece, that I had to tug just a bit extra to get the fifth tetrahedron to close.  Never noticed that before. 
  If the 5 tets did actually make the figure they seem to make, then 20 tets would fill an icosahedron, and you could keep adding on tets forever to fill space.  close, but no cigar.
  This probably has very little relevance in actual beadwork, since, as I said, it's close enough that you can pretty much make it work, but I had to mention it.

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.

Complexity v simplicity

I'm never sure when simple is too simple, or when adding a bit of complexity is an improvement.  Sometimes I throw something in in the middle of a repetitive sequence, just because I'm bored with the process, and that's usually a mistake.  This time I did something different.  I had finished my red and black tets necklace, which, I showed in an earlier post, and I liked it a lot.  Still, I thought it read as a very simple structure, and I thought I might add something so that it would show more complexity.  I added 5 extra tets to give the necklace a bit of "spikiness" instead of being a simple ring of tets. I think it was an improvement, but I'd love to know what others think, looking at the 2 together.. 

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.

tetrahelix

   I haven't posted in  while, but I've been playing with a spiraling structure of tetrahedrons that I mentioned a few months ago. In the interim, the beaded molecule folks talked about the same structure in their blog, so now I know it's called a tetrahelix, i.e. a helix made of tetrahedra.  I borrowed their idea of accenting the spiral-ness by using different color beads for the 3 outside edges of the helix. 
   I've been thinking for a while about using different tubes to contrast with the oxidized copper.  To do that I have to cut my own tube beads, but that opens up lots of possibilities too, because I can vary the length to alter the structures.  I suspect that soon I'm going to wish I knew some trigonometry.  My only memory of trig is looking up 4 digit decimal numbers in tables in the back of the math book.  This was, of course, in the bygone days before calculators.  But now I find myself wanting to know things like what angle I'm creating between 2 faces, or  2 beads, and I think that's the realm of trig.  One of the nice things about doing this stuff with beads is that if the angle is close to what you need, you can make it work.
   In this piece I didn't play with angles and lengths.  Learning to saw the tubing was a first step and that was enough for starters. I bought gold filled tubing and cut tubes the same length as the long copper ones I've been using.  I like the effect.  Another possibility is stainless steel tubes ( which are actually hypodermic syringe tubes), or anodized and colored aluminum.  The stainless would, of course, be much harder to cut, and I like the idea of colored aluminum, but the tubes I've found so far are much fatter than the copper ones, so I'm not sure they'll look right if used in the same piece.  So much to learn (and such fun doing it!).

evolution of a design

I've been trying to get into better shows with my beadwork, and while I'm improving, I'm still not getting into some of the shows I want to ( as if that isn't always the case).  Part of my problem is that for years I've been applying to shows, with the rugs, in the fiber category, which is generally one of the less competitive categories; now I'm applying in jewelry, which is always one of the most competitive.  Anyway, I've been using the oxidized copper tube pieces, because I think they're the most uniquely my work.  But I wanted another piece with some color, and since some of my pieces are rather irregular, I wanted one that was more controlled.  Hence the top piece.
  I like the piece, but, for a jury shot, I thought maybe I had moved too much in the direction of repetition.  I wanted a piece that was controlled, but

not quite so predictable.  Also, although I had color, still it had a lot of browns and neutrals.   I decided  I wanted to make 3 changes:  more color, copper tube pieces that were octahedra like the ones in the first piece, but irregular octahedra, and more interesting shapes than the ovoid ones in the first
piece.
     The second pic shows me playing around with shapes till I got one I liked, sort of a fat crescent.  Incorporating the other changes gave me more the sort of piece I was looking for, controlled, but just s bit unpredictable.  I still have to work on getting a better picture of the new necklace--it's really brighter than this pic makes it appear.

Addendum

    This is sort of an add-on to my post earlier today.  I have a couple of local art festivals coming up, so I may not do any blogging for a few weeks.  So I thought I'd take a quick picture of the straight, but spiraled string of tetrahedrons I came up with.  I think it's kind of cool. 
    By the way, thinking of festivals, if anyone will be in the Chicago area, I'll be showing my jewelry at the Highland Park (northern suburb) Festival of Fine Craft the last weekend in June.

Back to geometry

   First of--apologies for the blurry pictures; I didn't haul out the tripod as I should have.
    I've been playing with structures of tetrahedrons again.  Last September I did a post or 2 about some necklaces I made using really long bugle beads to form chains of tetrahedrons (tets).  I liked them but got concerned about the sharpness of the edges of the bugle beads.  Even adding a seed bead at the end of each bugle, I was a bit worried about the fireline fraying.  I decided not to use magnet clasps because I didn't like to be pulling the necklace apart as you do with a magnet clasp, for fear of cutting a thread. 
  Anyway, I started in again, but this time using the oxidized copper tubes I've been using lately.  If you just keep adding tets in the most logical order they make a circle.  I think ( if anyone knows for sure, I'd love corroboration or correction)  that tetrahedrons will tile infinitely in space with no gaps the same way that cubes will.  But they do it in a very curvilinear way.  For example, 20 tets form an icosahedron, which is essentially spherical.  Still, if they'll tile in 3 dimensions, there had to be a chain of tets that would curve  gradually, or even go straight.  But I was having a hard time getting a chain of tets that had a shallow enough curve.  The chain I had used with the 30mm bugles last fall, didn't work with my 22 mm tubes.  The curve was too tight, and I ended up turning it into a bracelet instead of a necklace.
     I spent lots of time coming up with the second structure.  I found a chain of 8 tetrahedrons which didn't bend much.  Then I marked the last tet with a piece of yarn and started adding tets to the chain so that that marked tet was the center of a symmetrical chain of 15 tets.  Then I moved the yarn and extended the 15 tet chain to 29.  I'm sure there are computer designing programs that would figure out something like this for you, but I'll admit I learned alot using this seat of the pants method.  And I got a chain that curved just the right amount.  I've even found a chain that goes straight.  It makes a gentle spiral as it goes, so it's pretty cool.  I don't have a picture of it yet, but I'll post one when I do.  All in all, it was a fun endeavor.

More Asawa inspired work

I'm still playing around with shapes that overlap/intersect one another.  I thought I'd try black(ish) and white so that you could see the intersection better.  The white beads are mother of pearl; the black are jungle jasper in the top one and black silkstone in the bottom one.  Neither is quite finished as I ran out of mother of pearl beads and I'm waiting for more.  In the top piece there's a continuous black tube that goes sometimes inside of and sometimes outside of a continuous white tube.  In the bottom there are 4 "pod" shapes with pointy ends that overlap, and tubes on each end that overlap the last pod.  An interesting project.  The design limitation is you have to arrange them so that the circles where the 2 shapes interlock have to all be 6-bead circles, because a smaller circle won't let a bead pass through the middle.  If I had beads that were slightly oval shaped, so that a bead would pass through a 5-bead circle, I'd have more flexibility.  Also, it makes for a pretty rigid structure.  I'd thought I could make a whole necklace of interlocked pods, but it just doesn't curve enough.  As it turns out, it would also be pretty heavy.

Ruth Asawa again

Here's my second pendant based on Ruth Asawa's interlocking shapes, spheres at the bottom and sort of fluttery cones half way up.  These are stone beads, mother of pearl for most of it, and African jade blending into blue sodalite for the colored part.  I was really pleased with it.  Doing it was an exercise in not doing too much. The long stretch of mother of pearl tube was repetitious.  When I do a long repetitious part, I have a tendency to add things, out of boredom, that are often better left out.  Several times on this piece I started to do extra things to "jazz it up" a bit, like stripes of blue beads to break up the long stretch of white, and things like that.  I ended up ripping them all out and keeping it simple, and I think it was right.  The only change I ended up making from the original 2-color design was to blend in the blue beads at the end of the green stalk, and I do like that.  Less is more.