defects (non-hexagonal cells) appear, notably on the polar circles (Fig.\[f6\]) that separate topologically the cylinder from the two hemispherical polar caps.

Projection on a tangent cone is different because the generative spiral on the cone cannot have constant pitch and keep a constant density of points. $h(s)$ must vary linearly with $s$.

Grain boundaries
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In plane phyllotaxis with a Fermat spiral and in spherical phyllotaxis, cells are of equal area. But they are not all hexagonal: there are annular crystalline grains of hexagonal cells (traversed by three visible reticular lines in the form of spirals, the parastichies), separated by grain boundaries that are circles of dislocations ($d$: dipole pentagon/heptagon) and square-shaped topological hexagons ($t$: squares with two truncated adjacent vertices), in a quasiperiodic sequence $d t d d t d t \dots $. The two main parastichies cross at right angle through the grain boundaries. [@rivier2].

The dislocations in the grain boundaries induce a shear strain in the grains. It is actually a Poisson shear, associated with radial compression between two circles of fixed, but different length. Thus, elastic and plastic shear can be readily absorbed by a polycrystalline phyllotactic structure described by several successive grain boundaries with successive 