### Phyllotaxy- A Serendipitous Surprise

Simply put, Phyllotaxy is the arrangement of leaves on a plant stem. But this is not to be dismissed as a naive curiosity; Phyllotaxy is one of the most easily available and fascinating naturally occurring mathematical phenomena. It was a sunny afternoon on the ${4^{th}}$ of March 2012, that I chanced upon something very beautiful. Observing the collection of plants at my home, I got curious about a certain tree. It had a symmetric looking canopy, but noticing the branches individually, I could not guess any pattern whatsoever. Luckily, I was in a mood to experiment, and decided to spend the day trying to find out more. At the beginning of my experiment, I had no idea what Phyllotaxy was and thought that the branch positioning would be random, but after analyzing my results, I found something truly unexpected.

Below are the images of that tree:

1. Experiment

Though there were many interesting properties of the tree that could be measured, I restricted myself to measuring only the angle at which the branches of the tree emerged. If a cylindrical coordinate system is imagined to be set up with the longitudinal axis passing through the centre of the trunk, the azimuthal angle is what I measured.

I was lucky as the tree left marks wherever branches had emerged in the past, which made it easier to take more measurements. I drew a reference line along the trunk using a straight ruler as a guide. To measure the azimuthal angle of each branch, I measured the diameter of the trunk using a screw gauge and the distance of the mark from the reference line (along the circumference of the trunk, at the mark) using a measuring tape. Then, the required angle is:

$\displaystyle \frac{distance\;from\;reference\;line}{circumference}\times 360^o$

2. Observations

 Branch No. diameter(mm) circumference(cm) distance from reference line(cm) angle(degrees) 1 15.6 4.8984 0 0 2 15.5 4.867 1.9 140.5383 3 16.88 5.30032 4.2 285.2658 4 16.95 5.3223 0.7 47.34795 5 16.75 5.2595 2.3 157.4294 6 16.65 5.2281 5.2 358.0651 7 15.5 4.867 1.6 118.3481 8 15.65 4.9141 3.7 271.0568 9 15.69 4.92666 0.7 51.15027 10 15.35 4.8199 2.5 186.7259 11 15.23 4.78222 4.7 353.8106 12 15.6 4.8984 1.5 110.2401 13 15.4 4.8356 3.7 275.457 14 15.05 4.7257 0.5 38.0896 15 14.8 4.6472 2.3 178.1718 16 14.22 4.46508 4.2 338.6278 17 14.8 4.6472 1.3 100.7058 18 14.7 4.6158 3.3 257.3768 19 14.35 4.5059 0.4 31.9581 20 14.1 4.4274 2 162.6237 21 14.1 4.4274 4.4 357.7721 22 14.1 4.4274 1 81.31183

Even after tabulating the observations, I could not detect any pattern in the sequence of angles.

3. Plots

In hope of getting some insight, I decided to make a plot of the azimuthal angle vs. the branch no. . But the real turning point was when I joined consecutive points by line segments. Below are both the graphs.

4. Conclusion

From the figure above, it is evident that every fifth branch emerges at approximately the same angle. Moreover, the sequence of angles at which the intermediary branches emerge also repeats. Below is a computer generated image showing the emergence of branches around the main trunk (The upper vertical is ${0^o}$).

Actual photographs supporting the conclusion:

After the experiment, I searched about my conclusions on the internet and found that this was called Phyllotaxy, and people as early as the Ancient Egyptians had investigated about it. Since then, numerous models based on packing efficiency, energy minimization, uneven distribution of Biochemical agents etc.., have been proposed by men like Leonardo Da Vinci, Johannes Kepler, Auguste Bravais, Louis Bravais, Hofmeister, Airy, Adler, Douady, and Couder.

5.1. Classification

Distichous Phyllotaxy: Botanical elements (leaves, branches etc..) grow one by one, each at 180 degrees from the previous one.

Whorled Phyllotaxy: Two or more Botanical elements grow at the same node on the stem. Elements in a node are evenly spread around the stem, midway between those in the previous node.

Spiral Phyllotaxy: Botanical elements grow one by one, each at a constant divergence angle d from the previous one. This is the most common pattern, and most often the divergence angle d is close to the Golden Angle, which is about 137.5 degrees. Eg.: Alternate distichous leaves will have an angle of 1/2 of a full rotation. In beech and hazel the angle is 1/3, in oak and apricot it is 2/5, in sunflowers, poplar, and pear, it is 3/8, and in willow and almond the angle is 5/13 of a full rotation. The numerator and denominator normally consist of a Fibonacci number and its second successor.

Periodic Orbits: The rarest type of Phyllotaxy, is when the divergence angle is not constant but repeats periodically a finite sequence of values. On inspecting the other plants at my home, I found all the above Phyllotaxies. I was extremely lucky that the tree I had chosen had an almost periodic orbit Phyllotaxy.

5. Afterthoughts

After reading through some relevant literature, I could not help but try fitting the observations with that of a spiral Phyllotaxy (being the most common Phyllotaxy). The average of the difference between consecutive angles, i.e. the average divergence angle comes out to be ${141^o}$, which is quite close to the golden angle (${137.5^o}$). It can be seen that the graphs below match quite nicely.

As observed

Divergence angle= 141 degrees

Divergence angle= 137.5 degrees

Perhaps Spiral Phyllotaxys give way to Phyllotaxys with almost periodic orbits as the tree grows. Also, to really discuss about the Phyllotaxy one has to determine at the early stages when the branches originate from groups of cells called primordia. But still, this simple experiment was quite informative for the analysis of pattern generation with respect to branches.  Though many mathematical models which explain this phenomenon do exist, I wish I can find a general article on the recent advances in this direction and whether a clear reason for such patterns exists as of now. Also, what are the evolutionary advantages of Fibonacci Phyllotaxy?