# #71 | gdn01 – Phyllotaxis and Fibonacci sequence

Hi guys! It’s me again!

So yeah, it might be a bit crazy but I find a deep correlation between plants and my project. It is really out of nowhere when I remember something very useful about the leaves arrangement and the optimization of light they can get. So… I rushed myself to make a chapter on Mars Desert Research Station, but I’ve finally written another subject in my master’s thesis. Shame on me, I guess… Anyway! Today, let’s talk about plants!

Phyllotaxis means the study of leaves (or analogs parts of the plant) arrangement.
The subject is really interesting because there are a lot of studies centuries ago and they’re still going on today. For example, Leonard da Vinci observed that the first leaf of a plant and the fifth one are aligned. One century later, the astronomer Johannes Kepler found the correlation between the leaves arrangement and the “Fibonacci sequence. It is quite simple: the sequence starts with”1″ and a next number in the sequence is the addition of the two previous numbers. If you changed the second number of the sequence by 3 and using the same addition process, you can get another sequence: “Lucas sequence“.The formula is the following:

The leaves displayed in two categories:
– Whorled, a group of leaves (n) grew after another in a sequence. Some next group of leaves grew in a symmetrically opposite position and an angle of 120° between each leaves (whorled opposite) like cannabis plant with n=3 (a), or by couple of two by two with a 90° angle between each leaves (whorled decussated) like the mint plant (b), or even an enfilade of  binomials leaves sequence disposed in 180° (whorled couplet) like a begonia plant (c).

– Spiraled, as the name said each leaf when growing up on the stem, will be displayed with a “divergence angle” that can be calculated with a Fibonacci ratio (example: 2/3, 3/8, 5/13, etc.), making the plant grown in a spiral. In fact, this angle is a ratio of the Φ number, also known as the golden angle (Φ x 360° = 137.5°). This kind of arrangement can be also seen in the stem or in a flat model like in a sunflower.

Flat model in a sunflower head get show us two things: the first one is a grid composed of two curves called parastichies (m) and anti-parastichies (n). For a Fibonacci sequence m=21 and n=34. If you want fewer curves but with the same optimized compacity, you can just use the Lucas sequence.
The curves are calculated by the formula: r = a*θ^1/2 with a, a constant, r the radius and θ is the angle. In this grid, the disk florets of the sunflower head are arranged with a maximum of compacity and each disk floret are arranged with an angle α from the next one in the curve (see fig.04 (b)). fig.04 – (a) 21/34 curves in a sunflowers head. You can observe that these numbers are also from the Fibonacci sequence, (b) the α angle between two disk florets, (c) a cyclotronic spiral based on the relationship between all the disk florets.

In a vertical/3D model (I still don’t know exactly how you can call it), a stem of a cactus can also be modeled by a cylinder. When you cut the volume on the height  “D” and put it on a table, the flat model is a rectangle with a grid composed of m and n lines. On each intersection, a circle for each organ placed of a height “r”, an equidistance “a” from each other with the formula rule: 0 < a  D/2, and a diameter “δ”.  That graph was discovered by Gerrit Van Itterson in 1903 and he found the correlation between the circle placed on the grid (like in cartesian coordinates graphs: example, m=1 and n=2 -> (1,2)) and a visualization of these coordinates in a fractal tree. fig.05 – A stem of a cactus modeled into a cylinder fig.06 – The model can be flattened and used like a grid. fig.07 – A fractal tree by the bias of Van Iterson’s graph. The red line is the Fibonacci sequence representation and the blue lines are the Lucas sequence.

What did I learn?

• The Lucas sequence
• How the Fibonacci works in a 3D model and how to calculate it
• The correlation between golden ratio and fractals
• Names of Phyllotaxis and the categories

References

Pennybacker, Matthew F., Patrick D. Shipman, et Alan C. Newell. « Phyllotaxis: Some progress, but a story far from over ». Physica D: Nonlinear Phenomena 306 (15 juin 2015): 48‑81. https://doi.org/10.1016/j.physd.2015.05.003.

Yerly, Florence, Chrystel Feller, et Christian Mazza. « Plantes, spirales et nombre : les plantes font-elles des maths? » Academic Press Fribourg, 2010. https://doi.org/10.5169/seals-308889.