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