(13,8,5)$ to make up the thirteen cells of layer $\mu=3$.

For smaller spheres $n = 22-42$, the polar cap and circle $[(5,6,6),(5,5,5,5,5)]$ constitute the eight-cell layer $\mu = 1$ going inwards, with the five pentagons $s=4-8$ as defect inclusions. Going outwards, layer $\mu = 1$ ($(5,6,6)$) has only the three cells $s=1,2,3$, the five pentagons of the polar circle joining with the first three hexagons of grain $ (8,5,3)$ to make up the eight cells of layer $\mu=2$.

Now that the phyllotactic structure is understood in its details, we can discuss the action of an elementary topological transformation applied at vulnerable spots (a $T1$ on a grain boundary or the disappearance of the first cell). It turns out that the structure remains invariant under these specific transformations.

Core, disappearance of the first cell and invariance of the phyllotactic structure
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In equation (1), the function $\rho(s)$ specifies amongst others the area per cell (parameter $a$) and also the position of the first point ($s=0$ or $s=1$) with respect to the spiral’s origin $\rho=0$, to be discussed now. The radial law of the generative spiral follows from demanding florets of equal sizes.

We denote by the operator $\bf D$, the action 