Phyllotactic Patterns on Plants

We demonstrate how phyllotaxis (the arrangement of leaves on plants) and the deformation configurations seen on plant surfaces may be understood as the energy-minimizing buckling pattern of a compressed shell (the plant’s tunica) on an elastic foundation. The key new idea is that the strain energy is minimized by configurations consisting of special triads of almost periodic deformations. We reproduce a wide spectrum of plant patterns, all with the divergence angles observed in nature, and show how the occurrences of Fibonacci-like sequences and the golden angle are natural consequences.

Figure 1

(a) Cactus stickers are labeled according to their distance from the center. The divergence angle $D=2\pi d$ is noted as are the families of 3,8 (5) spirals emanating clockwise (anticlockwise) from the center of the plant. Also note the irregular hexagons, one of whose vertices are marked in white. (b) This cactus also exhibits a 3,5-spiral arrangement, but the dominant pattern is the 8 ($=3+5$) radial ridges. (c) The maxima of the succulent surface are arranged in pairs that alternate in angle; this is the alternating 2-whorl (decussate) pattern. (d) Irregular hexagons are prominent on this cactus, whose stickers are arranged along 12 radial lines. (e) A theoretical reproduction of the cactus in (a). For the parameters $\Gamma =4$, $\chi =1$, $\frac{R_{\alpha }}{R_r}=1$, the deformation consisting of three equal-amplitude states with the wave vectors ${\mathrm{k}}_4,{\mathrm{k}}_{\mathrm{5}}\mathrm{,}{\mathrm{k}}_{\mathrm{6}}$ of the standard phyllotactic wave vector sequence (SPWS) evaluated at $g=1$ is an energy-minimizing configuration and is shown plotted on a sphere. The three families of spirals are also marked, as well as the vertices of irregular hexagons. (f) Schematic of a plant tip. The shoot apical meristem (SAM) consists of region 1 (undifferentiated tissue) and region 2 (the annular region $M$ of average radius $R$ where growth forces lead to buckling of the plant tunica). Tunica material goes from being in regions 1 to 2 to 3 as the shoot extends upward. Region 3 consists of material that has now hardened, thus setting the phyllotactic pattern developed when it was in region 2; the bumps may now develop into leaves, etc. Region 4 is the corpus of squishy material that acts as an elastic foundation to the tunica. (g) Theoretical reproduction of (c). In this energy-minimizing configuration found by minimizing the elastic energy with the parameters $\chi =2$, $\Gamma =1.7$ and restricted to the space consisting of the wave vectors ${\mathrm{k}}_0+{\mathrm{k}}_{\mathrm{2}}\mathrm{,}{\mathrm{k}}_{\mathrm{0}}\mathrm{,}{\mathrm{k}}_{\mathrm{2}}\mathrm{,}{\mathrm{k}}_{\mathrm{1}}$ of the SPWS evaluated at $g=2$, the first mode has an amplitude that is one-fifth the amplitude of the remaining modes. (h) Theoretical reproduction of cactus (d) found by restricting the energy, with $\chi =2$, $\Gamma =3$, to the space consisting of the wave vectors ${\mathrm{k}}_{\mathrm{1}}\mathrm{,}{\mathrm{k}}_{\mathrm{2}}\mathrm{,}{\mathrm{k}}_{\mathrm{3}}$ of the SPWS evaluated at $g=6$. All modes have equal amplitude.