Monday, February 11, 2019

another mathematical rabbit hole

     I've been having fun playing with a new area of mathematics.  Up to now my explorations have had to do with building polyhedra.  But I do look occasionally at Kate McKinnon's peyote stitch based work done in seed beads. Under the heading Contemporary Geometric Beadwork, she's produced several books, as well as lectures and lots of other stuff.  I admit I don't have the books.  Since I don't do peyote stitch and I no longer do seed bead based work, I thought lots of it wouldn't apply to my work.  But I do look at the pictures of what she and her group do, and find it quite interesting (as an aside--since the people who read this blog are mostly beaders with a mathematical bent, if anyone knows which of the books in the Contemporary Geometric Beadwork series would have a good coverage of the theories for someone who doesn't actually do peyote stitch with seed beads, I'd love a recommendation). I've often thought I should play around with the hyperbolic plane that she uses. She calls it a warped square, and it's made of 4 triangles.  If they were right triangles it would be a flat square but since the "hypotenuse" is longer than it would be in a right triangle there's more perimeter and the structure isn't flat.  It has always seemed to me that my way of making triangles has much more flexibility than the peyote stitch way.  It seems to me the shape of her triangles is determined by the shape of the seed beads--the relationship between the height and the diameter of the bead used sets the shape of the triangle.  In that way I have much more flexibility. I can  cut any length tubes and get whatever triangle I want.  Of course it's also much faster, but on the other hand I can't do the great stuff with colors and patterns that she can.
   Above is a single warped square made into an earring, and another earring with 2 of them.  I said my way of working  gives me more flexibility, but on the other hand my "stock" lengths are sort of set to be able to build right triangles. For example, I cut lengths of 10, 14, 20, 28 and 40 mm. Each one is more or less the one before it times the sq root of 2. So 10/10/14 makes a  right triangle ( or would except that the sq root of 2 isn't 1.4 but 1.4xxxx to infinity), as does 14/14/20 or 20/20/28 etc.  I also do 17, 24 and 34mm lengths, which fill in the gap and work the same way.  But to jump to 14/14/24 makes a very warped square,
(similar to what's in the earrings, which use triangles of 10/10/17) and more than I wanted.  So I decided to try making squares where just one of the outer edges was 24 and the rest were 20s.Here's a double row of these squares.

I liked it, and thought it could make a good bracelet ( you can see on the left end an additional structure I added where I could put a clasp). But there was 1 problem.  There are 2 versions of the square.  If you arrange the "plane" so that the tube on the bottom is horizontal, the tube on top can either go from high on the left and low on the right or vice versa.  Since the triangles are theoretically rigid you can't change it once it's made. In this way it's different from the peyote triangles which are inherently flexible. In the top pic I arranged the squares so that each is the mirror image of the ones beside it, which is the way it would have to be for a bracelet.  But since the hyperbolic-ness ( hyperbolicity? no idea what the proper word is) is fairly minimal, if you push on one of the high points in the center of the structure you can flip it, and then it turns into pic 2.  You can just as easily flip it back, but a potential customer wouldn't know that and I didn't want to get bogged down in long explanations. So I changed it, making essentially warped rectangles, where the outside edges were 2 20s and 2 24s.  That made the bumpiness more extreme and hence more stable. 
   I also tried a structure where the warped squares didn't go back and forth between the 2 versions, but were all the same version.  That made the warping spiral.  That's the last picture, but you can't see what's happening very well. In order to make it work, I'd need to replace 1 tube in each square with a gold or colored one so that you could see it spiral around.
  There's another way to make a hyperbolic plane. Instead of adding length to the tubes that make the perimeter of the structure you can add an extra triangle, i.e, where 4 right triangles would make a flat square, 5 of them would make a hyperbolic pentagon.  That's the sort of structure that will get progressively more wavy in the way of a lettuce leaf.  But doing it with 5 right triangles was too extreme.  Each pentagon was very non-flat. And if the first row had n structures the next would have 2n, and then 4n etc.  Again too  extreme and fast a progression.  I did it several years ago (you can see it in a post on hyperbolic planes from that time) using equilateral triangles, and where 6 would make a flat hexagon, 7 makes a hyperbolic heptagon. And each row will have 4 units for every 3 units in the one before, so the progression is still fast, but not so very fast. I really liked the one I did in colored glass beads back then, but had some trouble with the glass cutting the thread. I'd like to try one in silver tubes, possibly bright silver, but haven't gotten it done yet. 
  


Wednesday, January 23, 2019

new piece after a hiatus

   It's been quite a while since I've posted anything here. First there were 2 months on a canal boat in England. Then I came home at the end of October to the aftermath of hurricane Michael.  By the time I got back to Panama City the roads were mostly clear, and we had power and water.  But our house, though liveable, has holes in 1 wall which are only covered with plastic.  Also until a week or so ago we didn't have internet, except for slow data on our phones.  Finally, my computer crashed, taking with it all my jewelry pics as well as the copy of Photoshop that I had used to edit them.  So I've had to learn to use Gimp, which is open-source.  I'm pretty sketchy at it, but I finally got a picture of a piece I like.
   I decided to think back over the last 4 or 5 months, pick my favorite piece from that period, and post a picture of it.This is it.  It's done mostly in RAW ( I guess it's technically CRAW, but since that's the only kind of RAW I ever do I don't usually specify it).  But I wanted to break it up, make it less regular than I usually do in my RAW pieces, and I was quite pleased with the way it came out.  I hope you like it. As I get more pictures "gimped" I'll try to post some.  It feels good to get back to normal in one more way.
 

Friday, August 17, 2018

some new bracelets, and a technical advance

Just finished some new bracelets, and I like them a lot.  It also provided a start of a solution to a technical problem I've had with bracelets. 
  Up until recently, I have sort of avoided bracelets for several reasons.  One is that I don't wear them myself.  Like most everyone else, I start out by making something for myself, and I've just never been a bracelet wearer.  The second was the lack of a good clasp.I've written about this before, but I've finally found 2 clasps that work for me and don't use up too much of the bracelet length.
    It still left the major problem with bracelets--they have a very narrow range of usable lengths.  The difference between a 7" bracelet and an 8" one is pretty big.  When you are making modular structures, as I usually am, if you come out too short, you can't just add another module, or you'll be much too big.  Over time I've found out that 10 modules using 20mm lengths or 6 modules using 28mm lengths make a workable length when a short clasp is added.
    Now about the bracelets shown here--it started with the idea of a common Brancusi structure which is column with a square cross section that alternates small and bigger waists.  He did that a often and I wanted to reproduce in in a RAW structure.  My first attempt failed, because since my structures aren't rigid the way a wooden column is, they tend to straighten out on one side or another unless you exaggerate the in-and-out-ness quite a bit.  As you can see, the top bracelet is more exaggerated than the bottom one, but both of them work pretty well.  What I discovered, though, and thought was pretty cool, was that because of the zigzaging in and out on the inside of the bracelet, which is there when not stressed, but can go away if you push on it, the bracelet fits comfortably on a small wrist, but will also accommodate a larger wrist by straightening out the zigzag.  The outside distorts to allow that, but it looks fine either way.  Pretty cool. 

Wednesday, August 1, 2018

new work and galleries

I thought I'd do a post of what I've been up to lately.  Another piece where individual structures are interlinked to form a chain. For each link, I made an octahedron out of gold filled tubes, then added 8 long oxidized silver tubes to form another octahedron on the outside of it.  It took a while to get it right because so many tubes were ending at the same place.  Eventually I added a small round silver bead to separate the outer oct from the inner one, and that seemed to make it work.
The other thing I've been working on this summer is getting my work into galleries.  I traveled around and talked to places, and I've gotten a few new ones, so if you're in any of these areas, I hope you'll stop by and take a look:
Albuquerque--Mariposa Gallery, 3500 Central Ave, mariposa-gallery.com
New Orleans--Ariodante Gallery, 535 Julia St, ariodantegallery.com
Chicago--Pistachios, 55 E Grand Ave, pistachiosonline.com

Saturday, June 2, 2018

new collar with gold

I love this piece!  A few posts ago I talked about using some long beads on the outside of one of these collars and how it made the outside edge scallop. I had done it with stone and marble beads and didn't like the mix, but mentioned I might do it with gold tubes. Just finished this one and I really like it. I'm doing more these days with the gold (actually gold-filled) tubes as accents. and I think the contrast with the dark silver really works. For my own use, I mostly stick to the plain oxidized silver pieces, as they're less dressy, as is my lifestyle. But I do like making these silver and gold pieces. They tend to be simpler, and less funky, because when I piece has lots of different shapes and structures, then I think the contrasting color is  distraction.  But it makes a more classic piece pop.
By the way, after writing the earlier blog post about this structure, I forgot to read it before making this piece, and so had to redo it  after getting around 1/3 of the way around.  In the post the dark tubes were 28mm and the stone tubes on the outside edges were 20-35-20.  In this one I started out using all 28s except for the long 35mm ones on the outside, so no 20s on the outside.  That made the outside too long relative to the inside, so it curved was too tightly to fit your neck into it. So I changed the inside tubes to 31mm.  It's still a pretty tight circle.  If I wanted it a bit longer, instead of round, I'd make2 of the inside tubes in each side 35s instead of 31s.  That would add a bit more length, but mostly it would make the curve shallower at that point. There are 12 inside tubes, so I'd change 3,4, 9 and 10. Actually I might just change 3 and 10 (talking to myself here) because you really want to do it when the line of the necklace is 90 degrees from the center point, so you're making it just longer, not wider.

Monday, April 2, 2018

playing with tube lengths


This is a post that, more than most, is just me talking to myself to remember something, because it's about a piece I started to make, but don't like too much and plan to take apart.  I've always liked the piece pictured first.  It was made with 25mm tubes and quartz beads of around 20mm.  I recently bought some malachite beads that are a sort of pinkish tan, and wanted to use them on the outside edges, but I wanted to use 28 mm silver tubes everywhere else. This was partly to make the piece a bit bigger, and partly because I 'm low on 25mm tubes just now. But I found that that combination of lengths made a curve that was way too shallow (obviously, the tightness of the curve is just a matter of how much longer the outside edge is than the inside edge).  The outside edges consist of 3 beads in a sort of a straight line and then and then a shift to a new angle. So I went back and put a long(35mm) marble bead in the middle of each set of 3, in place of one of the pink ones.  I liked the way it made the set of 3 curve, so you get an interesting outline, as you can see in the bottom picture.  But I didn't like the 2 colors.  Too jumpy.  If I'd used all 28s on the inside I was headed toward a piece that was about 21" on the inside and 26" on the outside.  No Pythagorus here, I just laid it on top of a salad plate and the outside curve was pretty close to the outside curve of the plate.  A saucer (21") fit the inside. That seems pretty big, so I needed to shorten the inside edge some more.  I tried substituting a 20mm on the inside, but you have to do it in pairs, and 2 20mm tubes would have made the curve too tight (there's one 20 in the sample).  It looked like the curve with all 28s would have led to a piece with 9 units (maybe 8 and a clasp).  If I'd used all 25s on the inside that might have worked.  Or, to get a more oval, less round shape, 28s with 4 25s, to sort of make 4 "corners".  You could do it with gold tubes on those outside edges, and it would be pretty interesting too.

Sunday, April 1, 2018

Another neckwire piece

I liked playing around with shapes on a neckwire a few weeks ago, so I thought I'd try another.  I think the rectangles with gold zigzagging down them worked well.  makes for a pretty wearable piece, as well as an attractive one.