$5$ taken up by the innermost [grain]{} boundary (polar circle) and $+1$ by the pole ($s=1$, a pentagonal cell). The core (polar cap) is the open disk of cells inside the innermost ring of defects (open circles in Fig.\[exfig9\]).

Since the successive grain boundaries are circles of given length, they are better separated on a substrate with a more positive Gaussian curvature, and more crowded on a more negative curvature substrate. Indeed, as we shall see later, spheres $n = 43-75$ have the simplest structure: only one grain $(13,8,5)$ separated from two polar caps of three cells each $(5,6,6)$ and $(6,6,5)$ by grain boundaries (polar circles) of thirteen cells each,\
$[(6,6,6,6,6),(6,6,6),(5,5,5,5,5)]$ and $[(5,5,5,5,5),(6,6,6),(6,6,6,6,6)]$. (See Fig.\[f6\]a). In smaller spheres $n = 16-42$, the grain is $(8,5,3)$ and the polar circle fills up the polar cap $[(5,6,6),(5,5,5,5,5)]$ with its $f_{6}= 8$ cells. The pole is the common vertex of cells $s=1,2,3$. Larger spheres are discussed in Appendix 1.

Phyllotaxis as a foam
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The phyllotactic structure has global features, such as the circular grain boundaries, visible in Fig.\[fg2\], but also local features, the organization into parastichies, whorls of neighbor cells. Both features are invariant under the local, elementary topological transformations $T1$ and 