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Packing problems, phyllotaxis and Fibonacci numbers

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GFSW01 - Form and deformation in solid and fluid mechanics

We study the optimal packing of hard spheres in an infinitely long cylinder. Our simulations have yielded dozens of periodic, mechanically stable, structures as the ratio of the cylinder (D) to sphere (d) diameter is varied [1, 2, 3, 4]. Up to D/d=2.715 the densest structures are composed entirely of spheres which are in contact with the cylinder. The density reaches a maximum at discrete values of D/d when a maximum number of contacts are established. These maximal contact packings are of the classic “phyllotactic” type, familiar in biology. However, between these points we observe another type of packing, termed line-slip. We review some relevant experiments with small bubbles and show that such line-slip arrangements can also be found in soft sphere packings under pressure. This allows us to compute the phase diagram of columnar structures of soft spheres under pressure, of which the main feature is the appearance and disappearance of line slips, the shearing of adjacent spirals, as pressure is increased [5].

We provide an analytical understanding of these helical structures by recourse to a yet simpler problem: the packing of disks on a cylinder [1, 2, 4]. We show that maximal contact packings correspond to the perfect wrapping of a honeycomb arrangement of disks around a cylindrical tube. While line-slip packings are inhomogeneous deformations of the honeycomb lattice modified to wrap around the cylinder (and have fewer contacts per sphere). Finally, we note that such disk packings are of relevance to the spiral arrangements found in stems and flowers, when labelled in a natural way, which are generally represented by some triplet of successive numbers from the Fibonacci series (1,1,2,3,5,8,13…). This has been an object of wonder for more than a century. We review some of this history and offer yet another straw in the wind to the never-ending debate [6].

This talk is part of the Isaac Newton Institute Seminar Series series.

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