Found Mathematics on the Playa

I've been sorting through my photos from Burning Man, and I found this set of photos of the dried playa ground.  The playa is composed of a very fine particle dirt or dust that is wet mud in the winter and dry, flat earth in the late summer.  As the lake bed dries, cracks are formed in the mud.  The resulting cracks nicely demonstrate the idea of the self similarity of fractals in nature.

The idea is that as you zoom in, you continue to see more and more detail, and the over all shapes at every level always look the same, at least in theory.

Zoom, zoom, zoom.

Of course, this is the real world and not theoretical mathematics, so that perfect self similarity breaks down if we zoom in far enough.  That reminds me of a riddle.  What's the difference between theory and practice?  In theory they're the same, but in practice, they're different.  This is like that.

Anyway, the above photos were taking is the shade of bright daylight.  Below is what the Playa looks like at night with blue and green laser lights shining on it. You can still see the cracks, but instead look at the patterns in the lights.  

This pattern is quite complex since the lights are projected with more than one periodic pattern superimposed on top of one another.   See how the blue forms one repeating pattern, and the larger green clusters of dots form another, but one doesn't quite match up with the other.  And maybe you can even make out a repeating pattern in the smaller green dots.  Pattern on top of pattern on top of pattern...  This pleases me. 

Infinity Truncated Rhombic Dodecahedron Beaded Bead

This beaded bead has a really long, and very descriptive name.  The weaving technique is Infinity Weave.  I use this name for the weave because there are an infinite number of polyhedra that can be realized with Infinity Weave.  The mathematical shape of this particular beaded bead is a truncated rhombic dodecahedron. I think of it first in terms of its symmetry, which is that of a cube (or octahedron).  I imagine that on each face of the cube, there is a smaller square. This beaded bead has 6 squares holes just like this:

Square Face of the Truncated Rhombic Dodecahedron

The shape is further modified by replacing each edge of the cube with a hexagon.  Since a cube has 12 edges, so does this beaded bead have 12 hexagons.  In the photo below, see how the hole in the center has six sides?  That's the hexagon.

If you would like to read more about this beaded bead, and see photos of it from other angles, click on the photos above.