Thursday, April 12, 2012

Hyperbolic Beading No 3

I tried beading another hyperbolic tiling.  It's called Order 4 3 3 Dual, boring name, un-boring tiling.  A picture is worth a thousand cryptic sentences, so here's an illustration of Order 4 3 3 Dual using the Poincaré disk model of hyperbolic space.

This beautiful creature is composed entirely of hexagons.  Notice the center has four hexagons meeting at a point, and if you go out a bit, you can find places where just three hexagons meet at a point.  Compare this with the red tiling below.  This tiling also has all hexagons, but it lies flat.   Here, three hexagons meet at every corner.

If you weave an across edge weave of this red tiling, you get hexagon angle weave.  Here's a tutorial where I explain how to do that weave.

Back to the hyperbolic tiling:  What makes "Order 4 3 3 Dual" hyperbolic (and not spherical or planer) are the four hexagons meeting at some of the corners.  The extra angles jammed into all of those corners is what makes it ruffle when you build it with hexagons that are all the same size.  In contrast, if you wanted to make a spherical tiling with hexagons, you could make them get infinitely smaller at a point like on the Sheikh Lotf Allah Mosque dome:
Alternately, you could add some pentagons, squares or triangles to make it work.  This image shows a pentagon near the top right, and I'm pretty sure there's another one diametrically opposite that I can't see.

Coloring:  I fiddled with the coloring a bit in my beaded version of Order 4 3 3 Dual.  In particular, I used gray beads for the blue and red tiles, and yellow beads for the yellow and green tiles.  I also made some rules for the beads sewn across the edges of the tiling depending upon whether the two adjacent tiles are gray-gray, gray-yellow, or yellow-yellow.

Chronologically, the photo below starts with figure 5.  Notice what a ruffled mess I had.  In figure 6, I did a bit to organize the ruffles to show the 4-fold symmetry, but it's still a mess. Figure 6 has the same orientation as the illustration of Order 3 3 4 Dual above.

Figures 1 through 4 show the finished piece.  It's a bit thicker than I'd like it to be for a pendant, but Figure 2 shows how easy it is to string as a pendant. 

Alas, this piece is a little wonky, more than I like.  No matter how hard I try to make it perfectly symmetric, it just wouldn't cooperate.  In my frustration, I just kept adding more crystals.  When all else fails, add more crystals, right?  Still, little wonky bits of beadwork flip this way and that, as you play with it.  I'm trying not to think about it.  I showed it to my friend Andrew, and he didn't mind the wonky bits.  So maybe it's just me.  Anyway, it's done.  Thanks for not reminding me that it's wonky.  I know.

What I learned: Why is it wonky?  I think my largest "core" beads are a bit too big for the space I put them in.  That'll do it.  Also, it's too ruffly.  I couldn't get the ruffles to all fit together neatly.  I think I need to make smaller patches of tiles.  Also, the patch of tiles I used was quite round.  I think if I made my patches more "square" (that's a hyperbolic square), it should be easier to fold all of the ruffles neatly into place.  At least that's the theory.  Hey, you know the difference between theory and practice?  In theory they're the same, but in practice they're different, sometimes.

Want more? I've been playing with beaded hyperbolas here:
hyperbolic beaded bead No. 1 
hyperbolic beaded bead No. 2


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