with respect to the nearest defect ring. The topological distance between two cells is the minimum number of interfaces one has to cross to go from (the interior of) one to the other. Nearest neighbor cells are distant of $1$. All cells of layer $\mu$ are nearest neighbors to at least one cell of layer $\mu-1$ going up ($\mu+1$ going down); most, but in general not all, are also neighbors to layer $\mu+1$ (resp. $\mu-1$). A cell without direct contact with layer $\mu+1$ (resp. $\mu-1$) is called defect inclusion.

Grain boundaries as layers
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With this terminology, a grain boundary in phyllotaxis, e.g. $[7,7,7,7,7,6,6,6,5,5,5,5,5]$, constitutes a complete layer $\mu$ going down, (south-eastwards), with its five pentagons as inclusions (Fig.\[fg4\]c). Going up (north-westwards), only the five heptagons and the three hexagons constitute layer $\mu$, whereas the five pentagons are part of layer $\mu+1$. (Fig.\[fg4\]b).

In phyllotaxis, layers have normally a Fibonacci number of cells. It is indeed the case of grain boundaries. We shall see that the layers are invariant (a statement to be qualified) with respect to the choice of the first point, and thus under disappearance of the first cell. They fill up naturally under growth (addition of further 