Lizzy & Mark: Week 15
One thing I managed to do over the holidays was to modify the "cubical gears" 3D print design (see previous posts for YouTube links to that) to more accurately represent a the rotations of filaments in a 3D tetrahedral twist fold ... I don't have a decent video of the modification, unfortunately (further videos will forthcome) but as designed, three filaments turn in one direction (looking down on the central node down the barrel of each filament) and the fourth filament turns oppositely. The fantastic teaching lab group here in the Durham physics dept has been very helpful with the 3D printing.
The 3D gears are a curious system: each gear links to its neighbor(s), causing them to turn oppositely. Gears turning in the same direction can be achieved by putting a gear in between. The gears turn about points on a sphere, like in this spherical Voronoi tessellation. And the gears can be thought of as nodes on a graph.
In fact, for the gears to turn successfully, this graph must be "bipartite", meaning it can be colored with only two colors, without bordering gears being colored the same way. The two sets of nodes being those turning clockwise when looking down from the outside, and those turning counter-clockwise. The famous "four color theorem" refers to a similar property, of arbitrary regions of a 2D map: it's always possibly to color them with 4 colors, which took a very long time and a computer to prove.
Coincidentally, streams of dark matter in the universe (contiguous regions of the primordial universe that are bordered by caustics or folds) also form a bipartite graph: those An arbitrary 3D set of solids has no limit to the number of colors required to color it, but this bipartiteness property greatly restricts the arrangement of streams in this case, reducing the coloring number from infinity to two.
That's all for now. but this Friday I'll be visiting the Northumbria Paper Studio in Newcastle, England,
Hopefully we'll collaborate with them somehow!
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Lizzy Storm is an artist and owner of Lizzy Storm Designs based in Atlanta, Georgia.
Mark Neyrinck is an award-winning astrophysicist and cosmologist, and a postdoctoral researcher at Durham University, United Kingdom.