STRUCTURE OF BENARD CONVECTION CELLS, PHYLLOTAXIS AND CRYSTALLOGRAPHY IN CYLINDRICAL SYMMETRY.

Crystallography is considered in two spatial dimensions, in the presence of cylindrical symmetry. Defects (non-hexagonal cells) are imposed by the symmetry and glide circles are necessary to dissipate a weak, steady shear associated with the earth's rotation. Glide circles occur naturally in phyllotaxis (leaf or floret arrangement), and the structure of daisies represents the first stage of the construction of Benard patterns. Both types of structures can be generated by an elementary algorithm, which constitutes a physical application of number theory. The structures are self-similar and locally homogeneous. They are generated by a single, irrational number lambda . Homogeneity imposes lambda to be a Noble number and explains the pervasiveness of Fibonacci numbers in phyllotaxis. Once the ideal structure is constructed, melting can be simulated and analysed.