Domokos, Gábor (BME Szilárdságtani és Tartószerkezeti Tanszék, Cornell University)

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

[1][2]. 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 [9] can be uniquely identified by an integer. Beyond physically existing equilibria we can

also count imaginary ones, corresponding to arbitrarily fine, equidistant polyhedral approximations [8],

providing information about curvatures.

When looking at various shapes in Nature, ranging from coastal pebbles [3],[7] to asteroids [6] , from extant

[4] to long-extinct turtles [5], 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 [10]. We are also

confronted by some puzzles: shapes corresponding to some special integer combinations appear to be

missing from Nature.

[1] Varkonyi, P.L., Domokos G., Static equilibria of rigid bodies: dice, pebbles and the Poincaré-Hopf

Theorem, J. Nonlinear Science 16 (2006), 255-281.

[2] Varkonyi, P.L., Domokos G., Mono-monostatic bodies: the answer to Arnold’s question. The

Mathematical Intelligencer 28 (4) 34-38 (2006)

[3] Domokos, G., Sipos, A.Á., Szabó, T. and Várkonyi, PL. , Pebbles, shapes, and equilibria, Mathematical

Geosciences, 42 (1), 29–47. (2010)

[4] Domokos, G., Várkonyi, PL., Geometry and self-righting of turtles. Proc Roy. Soc. B ( Biol Sci.) Jan 7,

2008; 275 (1630): 11–17. (2008)

[5] Benson, RJ, Domokos, G, Várkonyi, PL, Reisz RR., Shell geometry and habitat determination in extinct

and extant turtles (Reptilia: Testudinata). Paleobiology 37 (4) 547-562 (2011)

[6] Domokos, G., A. Á. Sipos, G. M. Szabó, and P. L. Várkonyi (2009), Formation of sharp edges and planar

areas of asteroids by polyhedral abrasion, Astrophys. J., 699, L13, doi:10.1088/0004-637X/699/1/L13.

[7] Domokos, G., Gibbons, G.W., The evolution of pebble size and shape in space and time. Proc. Roy. Soc. A

468, 2146, pp 3059-3079 (2012)

[8] Domokos, G., Lángi, Z., Szabó, T., On the equilibria of finely discretized curves and surfaces. Monatshefte

für Mathematik 168 (3-4) 321-345

[9] Domokos, G., Lángi,Z., Szabó T., The genealogy of convex solids. Preprint http://arxiv.org/abs/1204.5494

[10] Domokos, G. , Jerolmack, D.J., Sipos, A.Á., Török, Á. How river rocks round: resolving thes shape-size

paradox. PLOS One DOI: 10.1371/journal.pone.0088657 (2014)

Date: Sep. 23, Tuesday 4:15pm

Place: BME, Main Building „K”, 1st Floor, Room 50

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