Astrophysical symmetries

Astrophysical objects, ranging from meteorites to the entire universe, can be classified into about a dozen characteristic morphologies, at least as seen by a blurry eye. Some patterns exist over an enormously wide range of distance scales, apparently as a result of similar underlying physics. Bipolar ejection from protostars, binary systems, and active galaxies is perhaps the clearest example. The oral presentation included about 130 astronomical images which cannot be reproduced here.

Symmetries in bacterial motility

Descriptions are given of three kinds of symmetries encountered in studies of bacterial locomotion, and of the ways in which they are circumvented or broken. A bacterium swims at very low Reynolds number: it cannot propel itself using reciprocal motion (by moving through a sequence of shapes, first forward and then in reverse); cyclic motion is required. A common solution is rotation of a helical filament, either right- or left-handed. The flagellar rotary motor that drives each filament generates the same torque whether spinning clockwise or counterclockwise. This symmetry is broken by coupling to the filament. Finally, bacterial populations, grown in a nutrient medium from an inoculum placed at a single point, usually move outward in symmetric circular rings. Under certain conditions, the cells excrete a chemoattractant, and the rings break up into discrete aggregates that can display remarkable geometric order.

Symmetries throughout organic evolution

The biological realm has inherited symmetries from the physicochemical realm, but with the increasing complexity at higher phenomenological levels of life, some inherited symmetries are broken while novel symmetries appear. These symmetries are of two types, structural and operational. Biological novelties result from breaking operational symmetries. They are followed by acquisition of regularity and stability, in a recurrent process throughout complexity levels.

New perspectives on forbidden symmetries, quasicrystals, and Penrose tilings

Quasicrystals are solids with quasiperiodic atomic structures and symmetries forbidden to ordinary periodic crystals—e.g., 5-fold symmetry axes. A powerful model for understanding their structure and properties has been the two-dimensional Penrose tiling. Recently discovered properties of Penrose tilings suggest a simple picture of the structure of quasicrystals and shed new light on why they form. The results show that quasicrystals can be constructed from a single repeating cluster of atoms and that the rigid matching rules of Penrose tilings can be replaced by more physically plausible cluster energetics. The new concepts make the conditions for forming quasicrystals appear to be closely related to the conditions for forming periodic crystals.

Symmetries in geology and geophysics

Symmetries have played an important role in a variety of problems in geology and geophysics. A large fraction of studies in mineralogy are devoted to the symmetry properties of crystals. In this paper, however, the emphasis will be on scale-invariant (fractal) symmetries. The earth’s topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.