Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry

Sadoc, J.-F.; Rivier, N.; Charvolin, J.

doi:10.1107/S0108767312018910

Figure 10
Strip cut in a square lattice (a). This strip is a good approximation of the ring of defects between points s = 101 and s = 155 containing 55 points. The colour of the points corresponds to their type of Voronoi cells. Point s = 101 is the first green point down, then points are numbered, increasing by 21 going up or decreasing by 34 going right. This strip can be divided into three strips of heptagons, hexagons and pentagons. (b, c) Continuation of the grain towards the other boundary on the same square grid $[\{(x,y):x-1\leftrightarrow s+34, y-1\leftrightarrow s+21\}]$ which can be considered as folded on a cylinder with axis perpendicular to the strip. This grain (in grey) is limited by an inner border (b) and an outer border (c). Blue points (pentagons) in (b) are those of (a), orange points belong to the large grain. The grain is bordered in (b) by a Fibonacci chain defined by the lower part of the grey domain. There are 21 steps in it containing 21 blue points and 34 orange points (hexagons of the large grain). The other border of the grain [in (c)] is geometrically identical, but it contains 34 green points (heptagons of the next ring of defects) and 21 red points (hexagons of this ring of defects). This next ring of defects from s = 290 to s = 378 contains 89 points. The beginning of this ring is expanded in (d), with two Fibonacci chains, one which closes the grain from s = 156 to s = 289 and the other which begins the next grain, from s = 379 to s = 800. The two Fibonacci chains could be coded with long (purple) and short (blue) segments forming a `staircase'. Steps of the first chain appear with $[\pi/2]$ angles, but for the other angles are $[\pi/4]$ . In fact the strip should be deflated (stretched along by a rational approximant of $[\tau]$ and compressed across by an approximant of $[1/\tau]$ , that is a Poisson shear). Introducing progressively the Poisson shear, instead of a part of a cylinder the substrate of the grain and its two boundaries are a part of a surface with a negative constant Gaussian curvature around the same axis. This Poisson shear changes the rhombic cell, darker in (d), into a square. Refer also to Figs. 11

and 12

FOUNDATIONS
ADVANCES

ISSN: 2053-2733

Volume 68| Part 4| June 2012| Pages 470-483

doi:10.1107/S0108767312018910

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