In simple mathematical models, the onset of chaos is heralded by a cascade of period doublings. Computer experiments reveal that fine details of the cascade are the same for many different systems. The cascade is governed by new constants of nature; for example, the rate of period doubling is always 4.66920...
Familar constants like pi and the cube root of 2 are related to geometric figures, like the circle and the cube. Because of their symmetry, cubes can be stacked to fill or tile ordinary flat space.
More exotic constants come from shapes like the dodecahedron (a twelve-sided solid), which can be used to tile negatively curved space. At the edge of a curved space (like the border of Escher's `Heaven and Hell', or of the pentagon pattern above), the tiling becomes chaotic. Mathematically this chaos is related to rigidity of the tiles themselves, and hence to the uniqueness of the constants they determine.
Some of my recent research aims to use this deep link between chaos and rigidity to provide a geometric understanding of universal constants in dynamics.