ONE DAY MEETING - Geometry in Science

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• Professor Sir Michael Berry FRS; Professor Chris Calladine FRS ; Professor Gabor Domokos; Professor Jan Koenderink; Professor Gabriel Paternain; Professor Denis Weaire FRS
• Friday 13 January 2012, 09:00-17:30
• Cambridge University Engineering Department, LR0.

If you have a question about this talk, please contact Beverley Larner.

Organised by Professor Jim Woodhouse

Professor Chris Calladine FRS Department of Engineering, University of Cambridge

“Some spiral structures in biology” Biological structures, unlike those designed and constructed by human structural engineers, are built by a process of self-assembly. The internal machinery of biological cells produces molecular building-blocks in accordance with information encoded in the DNA , and these then find their appointed place in the assembly under construction. The simplest possible self-assembled structure is a uniform helix, or spiral: repeated addition of identical building-blocks in a regular pattern make a uniform helix, which is the simplest space-curve. In this talk I shall discuss three particular spiral structures built from molecular components. 1. The α-helix, an important motif in protein structures; and in particular the way in which two α-helices can be programmed to assemble into a “coiled coil” arrangement. 2. The DNA double-helix. How it can switch between the classical “A” and “B” geometries; and how this kind of change enables us to understand sequence-dependent curvature and flexibility of the molecule – which is important in the recognition of DNA sequences by contacting proteins. 3. Bacterial flagellar filaments – the corkscrew-like organelles which, when rotated by their motors, enable bacteria such as E.coli to swim in their watery environment and navigate towards nutrients. Here the building-block, much larger than those of examples 1 and 2, is a protein molecule which contains a “switch” feature; and this enables the filament to change from a left-handed to a right-handed corkscrew when it is driven in reverse. These examples illustrate the power of evolution to develop highly sophisticated variants of the simplest kind of self-assembled biological structures. Geometry provides an indispensable tool in elucidating the subtle structural phenomena seen in these helices.

Professor Denis Weaire FRS Trinity College Dublin

“The geometry of foam packings.” The structure of a foam was first described in detail by Plateau in the nineteenth century, and his geometrical/topological rules have guided us ever since. The foam may be monodisperse or polydisperse, ordered or disordered, wet or dry (referring to the liquid fraction), two or three dimensional, and may be confined (eg in a cylinder) or essentially infinite. Kelvin posed the question of the minimum energy structure for a monodisperse 3D dry foam, and this has been debated ever since. In the opposite limit, that of a wet foam, the bubbles are spherical and hard-sphere structures are formed. We review a variety of problems having to do with structure and stability, including the computer simulations and experiments that have been brought to bear on them.

Professor Sir Michael Berry FRS Department of Physics, University of Bristol

“The singularities of light: intensity, phase, polarization”

Geometry dominates modern optics, in which we understand light through its singularities. These are different at different levels of description. At the coarsest level, where light is described in terms of the rays of geometrical optics, the singularities are caustics: focal lines and surfaces – the envelopes of ray families. These singularities of bright light are classified by the mathematics of catastrophe theory. Wave optics smooths these singularities and decorates them with rich interference patterns, widely applicable, for example to rainbows, ship wakes and quantum scattering. Wave optics introduces a new quantity, namely phase, which has its own singularities. These are optical vortices, a.k.a nodes or wavefront dislocations. Geometrically these singularities of dark light are lines in space, or points in the plane. They occur in all types of quantum or classical waves. Currently, optical phase singularities are being used to rotate small particles (optical spanners) and as a possible way to detect extra-solar planets. On a finer scale, where the vector nature of light cannot be ignored, the new phenomenon is polarization. This possesses its own singularities, also geometrical, describing lines where the polarization is purely circular. As well as representing interesting physics at each level, these optical and wave geometries illustrate the idea of asymptotically emergent phenomena.

Professor Gabor Domokos Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics

“Natural numbers, natural shapes”

The first step towards understanding natural shapes might be their systematic description. Instead of creating a hierarchical list of names in the spirit of Linné, we try to classify shapes based on naturally assigned integers, carrying information on the number, type and interrelation of static equilibrium points. In mechanical language, these are points where the body is at rest on a horizontal surface, in mathematical language these are the singularities of the gradient flow associated with the surface. While at first sight this appears to be a rather meager source of information compared to the abundance of three-dimensional shapes, we found that often meaningful information is condensed here.

One advantage of this classification is that we count (instead of measure) and thus do not add observer-related noise to the obtained data. Counting equilibria results in several, distinct integers describing different geometrical aspects of the investigated shape. One can distinguish between stable and unstable equilibria, also, the graph (called the Morse-Smale graph) carrying the topological information about their arrangement can be uniquely identified by an integer. Beyond physically existing equilibria we can also count imaginary ones, corresponding to arbitrarily fine, equidistant polyhedral approximations, providing information about curvatures.

When looking at various shapes in Nature, ranging from coastal pebbles to asteroids, from extant to long-extinct turtles, the integers extracted by the described means appear to carry information relevant to natural history. One could also imagine the long evolution of these shapes (whether biological or mechanical) as a coding sequence. Whether or not equilibria are the ‘true code’, we do not know, however, these simple numbers certainly help to better understand evolutionary history. We are also confronted by some puzzles: shapes corresponding to some special integer combinations appear to be missing from Nature.

——————————————————————————- Some links to recent work on the subject: http://arxiv.org/abs/0904.4423 http://arxiv.org/abs/1106.0626 http://arxiv.org/abs/1104.4813 http://www.springerlink.com/content/6880172060n40l18/ http://www.akademiai.com/content/e4646nr7083g47w6/ http://www.gomboc.eu/100.pdf —————————————-

Professor Gabriel Paternain Department of Pure Mathematics and Mathematical Statistics, University of Cambridge

“Contact geometry in dynamics: the 3-body problem”

We have known for a long time how to write down the equations of motion of a satellite that moves under the influence of the gravitational fields of the Earth and the Moon, but surprisingly, we still do not fully understand the long term behaviour of the satellite since we cannot explicitly solve the equations. At the end of the 19th century, Poincar\’e noticed the presence of chaos in the system and kick-started the modern theory of dynamical systems. There have been some remarkable applications of these chaotic motions to low fuel transfer orbits, specially since the celebrated rescue of the Japanese satellite Hiten in 1991 (Belbruno). Recently a new type of geometry called contact geometry (the odd-dimensional relative of symplectic geometry) has been proposed as a tool for understanding this old problem in celestial mechanics. In the talk I will try to explain what contact geometry is and why it is relevant for the 3-body problem.

Professor Jan Koenderink Delft University of Technology and Katholieke Universiteit Leuven

“Pictorial space: A geometry of visual awareness”

The visual field is two-fold, the visual world three-fold extended. Pictorial space assumes an intermediate position, “depth” being a quale, much like color.

From an evolutionary perspective vision is an optical user interface. This has important implications, e.g., vision is not necessarily veridical, since an effective interface shields the user from irrelevant complexity. This is indeed obvious in immediate awareness, where one frequently “sees” objects or processes that reflective thought reveals as illusory. This even extends to the space-time structure of the visual field. On artificial spatiotemporal scrambling of the retinal image, visual awareness is often coherent.

Depth is a quality of “separateness from the self”, even though the eye is not in pictorial space. Since there is no natural depth origin, nor a unit of length, a convenient formal description is the full affine line. Then “pictorial space” would be a fiber bundle, the base space being represented by the Euclidean plane, the fibers the depth dimension. An analysis of the structure of “depth cues” lets one derive a group of movements and scalings that transforms configurations such that the cues are respected. One expects observers to mutually agree modulo such a transformation.

We have developed methods to operationalize (“measure” is not apt because pictorial depth configurations are only operationally defined) pictorial reliefs (surface shapes) as well as arbitrary point configurations. We find apparently very significant differences between different observers. However, such differences are explained in considerable quantitative detail through the group of movements mentioned above. Apparently “pictorial space” has a tight (non-Euclidean) structure.

This talk is part of the Cambridge Philosophical Society series.

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