CYCLAMEN & THE MANDELBROT SET?
Photo: © Angela Stephens
Photo: © Angela Stephens
The leaf has a cycloid shaped perimeter and darker markings within which resemble a leaf. Your Author notes that the combination of cycloid and inner leaf also bear a strong resemblance to a part of the Mandelbrot Set, a mathematically derived fractal shape, and he ponders if this is just coincidental, or whether there might be some hidden reason for this.
Photo: © Larry One Eye Shone
It is such a remarkably faithful representation of that part of the Mandelbrot set, even down to the curved veins at the back, that your Author wonders whether it is indeed performing those Mandelbrot fractal calculations. Think about it, that is not as farfetched as it seems. Certainly, cyclamens perform no mathematical computation, but the patterns on leaves are subject to growth and inhibition feedback rules that may approximate to the Mandelbrot calculation rules. Even down to the cardioid outline of the entire leaf outline, which in the Mandelbrot Set itself is an integral part of the pattern. A rather interesting speculation...
It is well known that there are inhibitory and promotory substances in plants that together orchestrate growth in particular ways. Substance A promotes substance B, but substance B inhibits substance A, a feedback arrangement that has large regulatory powers. These positive and negative feedbacks are analogous to those which occur in the BelousovZhabotinsky (BZ) chemical reaction which continually creates swirling nonrepeating and chaotic patterns until the two reagents are used up. The one requirement for oscillation to occur (rather than the two reagents reaching a steadystate resolution) is that there has to be a phase delay of greater than 90 degrees. This is easily achieved by the slow diffusion of reagents across the medium (solvent).
In the case of plants, the reagents are phytoalexins (growth hormones in plants) and phytoalexin inhibitors or antagonists, which regulate each other and thus orchestrate growth and form in plants.
It can be envisaged that the generation of the cardioid shape of the leaf if there were two growth centres. One a radial growth pattern (governed by a radially diffusing reagent). And the other a linear unidirectional component upon which the radial component rides. This growth pattern would generate a cardioid shaped leaf.
If the linear growth component was zero, a leaf with a circular outline (orbicular, as in Marsh Pennywort) is generated. If the radial component is zero, long linear leaves result (needle leaves as in Yew). If the radial component is just slightly less vigorous than the linear growth component, a leaf with a cadioid profile is produced. It should also be noted that what would otherwise be radial veins in an orbicular leaf, become automatically curved in the case of a cardioidshaped leaf  with the curvature being greatest at rightangles to the linear growth component.
And if the radial component is directional (less in the perpendicular direction than that at rightangles) and the linear component is strong, you end up with longish kidneyshaped leaves (amplexicaul, as in Rape ).
Whilst temporal variation of the two components can generate lobes and teeth on the leaves, depending upon the magnitude of the variation. Large lobes if the temporal variation in the relative variation of the two radial components was large, such as the basal leaves of Wild Radish.
By varying the magnitude of just three growth components a whole series of leaf shapes can be generated.
