Static and Dynamical Phyllotaxis in a Magnetic Cactus

While the statics of many simple physical systems reproduce the striking number-theoretical patterns found in the phyllotaxis of living beings, their dynamics reveal unusual excitations: multiple classical rotons and a large family of interconverting topological solitons. As we introduce those, we also demonstrate experimentally for the first time Levitov’s celebrated model for phyllotaxis. Applications at different scales and in different areas of physics are proposed and discussed.

Figure 1

A specimen of Mammillaria elongata displaying a helical morphology ubiquitous to nature, a magnetic cactus of dipoles on stacked bearings, and a schematic of a wrapped Bravais lattice showing the angular offset (screw angle) $\Omega$ and the axial separation $a$ between particles.

Figure 2

Measured angular offsets between successive magnets for magnetic cacti with short (left) and long (right) magnets, plotted versus axial index. Flat regions are perfect spirals, and steps are kinks between domains of different spiralling angles. Dotted lines give the phyllotactic angles ${\Omega }_1$, ${\Omega }_2$, ${\Omega }_3$, and $2\pi -{\Omega }_2$ defined in the text. Dashed lines are minima of the magnetic lattice energy (insets) calculated by interpolating the measured pairwise magnet-magnet interaction. The magnetic cactus recapitulates botany.

Figure 3

Top: Lattice energy $V(\Omega )$ (in units of $ϵ=p^2{R}^{-3}$) versus screw angle for successively halving values of $a/R$ starting from 0.5, showing more highly degenerate curves. Bottom: Energy curve for $a/R=0.15$, with six distinct minima. The highest rank observed, $J=6$, matches Eq. (1).

Figure 4

The phonon dispersion relations for the six phyllotactic lattices of Fig. 3, with multiple rotons and maxons. Corresponding parastichy numbers ($j_1$ and $j_2$) (defined in the text) and the three most strongly interacting neighbors [${\tilde{j}}_1$, ${\tilde{j}}_2$, and ${\tilde{j}}_3$] (${\omega }_1{\tilde{j}}_1>{\omega }_2{\tilde{j}}_2>{\omega }_3{\tilde{j}}_3$) are given. The simple estimate for the number of rotons and maxons $2{\tilde{j}}_1-1$ holds for all but $(2,5)$. Each spectrum is offset by $5{\tau }^{-1}$ for clarity, where $\tau ={ϵ}^{-1/2}{I}^{1/2}$ is the unit of time [10].

Figure 5

Two phyllotactic solitons approach, collide, and transform. Top: The screw angle $\Omega$ plotted versus space and time for a collision of two phyllotactic solitons. Plateaux of different height correspond to spirals of different angle. Middle: The potential energy $V$ drops as the lower-energy domains advance, converting into rotational kinetic energy of the newborn domain (see online animation $S5$). Deviations starting around $t=2$ arise from precursor elastic waves propagating in front of the solitons that ensure local angular momentum conservation [15]. Bottom: The lattice energy $V(\Omega )$ corresponds to the physically realized magnetic cactus of Fig. 2 (bottom), with minima at ${\Omega }_1/2\pi =0.23$, ${\Omega }_2/2\pi =0.28$, and ${\Omega }_3/2\pi =0.38$. For clarity, weak high-wave-vector phonons were smoothed by spatial averaging, here and in Fig. 6.

Figure 6

The topological solitons can decay, merge, and change identity. An initial unstable domain boundary decays into two solitons. The left-hand soliton bounces off the free boundary, maintains its shape, propagates rightward faster than the other, collides with it, merges, and transforms into a third topological soliton with a different characteristic speed. $S5$ provides an animation. The density is $a/R=0.15$.