Fibonacci numbers. The packing efficiency problem is thereby solved: One grain boundary constitutes a perfect circular boundary for the disk into which objects are to be packed (see also [@sadoc3]). Grain boundaries can be moved (dislocation glide) by local neighbor exchanges ($T1$). In that way, the phyllotactic structure responds easily, locally and naturally to an external force that is expressed in the curvature of the substrate. A grain bounded by two boundaries of fixed length and quasicrystalline topology, is depleted as the curvature becomes less positive.

A circular ring of defects constitutes a grain boundary between grain $(f_{i-1},f_{i-2},f_{i-3})$ (hexagonal cells in phyllotaxis $(f_{i-1},f_{i-2},f_{i-3})$) and grain $(f_{i},f_{i-1},f_{i-2})$. It has a given length of $f_{i+1}$ non-hexagonal cells and uses $f_{i}$ turns of the generative spiral. The $f_{i+1}$ non-hexagonal cells are $f_{i-1}$ heptagons, $f_{i-2}$ hexagons (with the two smaller sides not opposite but next to adjacent) and $f_{i-1}$ pentagons, ordered in the quasiperiodic sequence $d t d d t d t \dots $. (Details in Fig.\[fg4\]a or in Fig.A1 of [@sadoc2]: strip cut in a square lattice). It is denoted as $[ f_{i-1},f_{i-2},f_{i-1}]$, and encoded as $[7,7, \dots,6,\dots, 5,5,\dots ]$ in the sequence $c(s)$.

On the grain boundary, the Voronoi cells are square-shaped, 