Ocean waves, asteroids, the human heart and a host of other dynamical systems exhibit transitions from periodic behavior to turbulence, unpredictability, fibrillation and chaos.

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.

Czech translation by A. Savicevic here

Hungarian translation by E. Pavlet here Polish translation by M. Murawski here