Phyllotaxis Growth. Simulations in C++ ImgGui.

Phyllotaxis is a plant with exceptional leaf arrangement that was researched by biologists and mathematicians. According to the article, we can derive two mathematical models (by H. Vogel and by J.N. Ridley) describing plant growth and shape. For the following article used the model proposed by Vogel.
The general arrangement of the leaves on the land stem is opposite (alternate), spiral, or whorled.

Typical leaves arrangements (by author)

Phyllotaxis arrangement is called “two-ranked leaf arrangement” since the opposite or alternate leaf arrangement is arranged in two vertical columns on opposite sides of the stem.
The area of phyllotaxis is dominated by intriguing mathematical relationships. One of them is the “remarkable fact that the numbers of spirals which can be traced through a phyllotactic pattern are predominantly integers of the Fibonacci sequence”. (see details in the article).

The introduction to the C++ ImGui and information on how to compile and build programs you will find in one of my previous articles.

The source code for all presented simulations you will find on my GitHub.

Mathematical model

For the simulations, I applied the Vogel model which can be formulated as follows,

where:
n is a pure number of the leaf, counting outward from the center.
φ is the angle between a reference direction and the position of the n-th leaf.
α constant divergence angle equals 137.3, 137.5, or 137.6 degrees.
r is the distance between the center of the figure (plant) and the center of the nth leaf (in simulations the center of the circle).
c is the constant calling factor.

Here I simulated the growth for a constant angle equal 137.3.

The below patterns differ only by the value of the divergence angle α.

α = 137.3 (by author)

α = 137.5 (by author)

α = 137.6 (by author)

Thank you for reading.