Analyzing the pattern in the artichoke heart

Co-authors Jacques Dumais and Christophe Golé have developed software to analyze the structure of inflorescence patterns, working together with their collaborators Pau Atela and Scott Hotton.  On the tiny artichoke heart, the software reveals striking and orderly transitions moving from the outer to inner regions. 

At its outer edge, the artichoke has 55 clockwise spirals and 34 counter-clockwise spirals (both Fibonacci numbers). In its middle section, there are fewer spirals: 21 and 34. And finally, close to the center, the artichoke's transition yields 21 and 13 spirals.  These can be read—by a computer or a human—in the numbers of green zigs and red zags in the "zigzag fronts" encircling the center. The red pentagons indicate the loss  of a zig or zag in the successive fronts, where a spiral terminates. 

Seeing this computer image made Chris realize how these fronts could geometrically encode the recursive rule of the Fibonacci sequence. In the case of the artichoke heart, the numbers of spirals decrease along the Fibonacci sequence. There had to be a mechanism for the number of spirals to increase during growth. Indeed this is what the next pictures show.  

Both the computer simulations and the plant photos above reveal the genesis of Fibonacci spirals via front transitions. This time, as the numbers of spirals increase,  we see triangles instead of pentagons.

These images are representations of unrolled cylindrical objects. In the computer simulation, the plant leaves are represented as disks. In the photograph, co-author Stéphane Douady has trimmed the leaves of an ornamental cabbage to reveal a similar structure. 

Both representations show a dynamical/geometric mechanism in which the growth of the plant progressively flattens the fronts, which are then forced into orderly transitions that enact the Fibonacci recursion. These transitions take a  (3, 5) front, for example, to a (5,  8). Each of the 5 triangles creates a new spiral that, added to the 3 original ones,  gives a total of 8.  The cabbage show the typical situation where, at the onset of growth, there is just one proto-leaf (cotyledon) that gets the pattern starting with (1, 1) spirals. 

Moreover, as simulations show, go too fast in the process and the triangles' preference for one side breaks down,  giving you quasi-symmetry!

How then does the microbiology set up these simple, local rules? Enter auxin, a plant growth hormone. In this case,  the concentration of auxin determines where there is enough space for a new leaf (or other plant organ) to grow, in a ring around the meristem, the growing tip of the plant. Concentration of auxin prompts the formation of a new leaf.  Then the newly formed leaf pumps auxin out of the surrounding area, preventing other leaves from forming nearby. 

This is an oversimplification, of course. The complex biochemical and biophysical mechanisms that set the stage were only discovered in the early 2000s, and are still being actively researched. 

All of the above translates into a simple geometrical rule: new leaves form at the edge of the tip of the plant when and where there is enough space. In the disk model, the rule looks slightly different: place a new disk in the lowest position above the existing disks, without any overlap.  

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