Figure 1
(a) Cactus stickers are labeled according to their distance from the center. The divergence angle
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 (
) 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
,
,
, the deformation consisting of three equal-amplitude states with the wave vectors
of the standard phyllotactic wave vector sequence (SPWS) evaluated at
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
of average radius
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
,
and restricted to the space consisting of the wave vectors
of the SPWS evaluated at
, 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
,
, to the space consisting of the wave vectors
of the SPWS evaluated at
. All modes have equal amplitude.
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