$\pi$, $e$, $\phi$, and sunflowers

Question

While reading some internet materials on design, I came across this picture and comment:

I found it a little bit surprising. I knew that the real sunflower follows golden ratio in some way (but I don't know the details). But how come the patterns for $\pi$ and $e$ are still "periodic", but different? What affects "sparseness" of the pattern? Is there any tool to display the sunflower pattern for different values of the "constant" that generates it?

Are the claims in the picture and in its comment for real at all?

What is the math background beyond all this?

Is the resulting shape different for rational and irrational numbers? What about transcendent numbers?

NOTE: I just saw there is a mistake (most likely a typo) in the picture: $1.681$ should be $1.618$, right?

Answer 1

How are the patterns for a number $a$ generated? Plot the points with polar coordinates $r=r_0\cdot c^n$, $\theta = \theta_0+n\cdot a\cdot 2\pi$ for $n=1,2,\ldots$, where $c$ is a suitable constant just slightly below $1$ (or above $1$ if wyou want the pattern to grow from inside to outside)

Even though it is not transcendent, $\phi$ is the "most irrational number", and that's what ultimately causes the pattern to be most efficient (and pleasing). The statement "most irrational" can be made more precise with the use of continued fractions. Patterns for rational $a=\frac uv$ would look like a straight "windmill" with $v$ wings. The pattern for $\pi$ looks almost like a straight windmill with $7$ wings. That's from the famous approximation $\pi\approx\frac{22}7$.

Answer 2

This is pure biomathematical misinformed popularization. These famous constants have no more reason to appear than any other real value, and even if they did, this could be approximately true to the first or second decimal and unverifiable for the next.

The pseudo-fitness of $\phi$ as shown on the figure is just due to a smaller value of the constant, resulting in better compactness. And there is no other "periodicity" than that caused by the revolution (actually, there is no periodicity at all, these are spirals).

Besides that, there is a rationale for the golden ratio to appear in nature, due to its relation to the Fibonacci numbers ($F_n=F_{n-1}+F_{n-2}$): when a physical process involves this recurrence (like in reproduction of rabbits, unlike the pattern of sunflowers), the golden ratio is there in the limit. It also appears with rotational symmetry of order 5 (which, by the way, is known to be impossible as a symmetry order in crystals). Note that any recurrence with integer coefficients can only generate algebraic numbers.

$\pi$ can appear if you roll out the skin of a cylinder of revolution.

I can't think of a simple process that would generate $e$. (Exponential curves arise in many situations in the frame of linear systems, but the basis has no reason to be specifically $e$.)

Last but not least, there is no real relation between aesthetics and the golden ratio, this is folklore. No one would notice a difference between $\phi$ proportions and, say $1.6$ or $\sqrt3$ or $e-1$.

Answer 3

These kind of patterns comes from biological activities as cell division. The spiral form come from a method of growth and development and the pattern on the curved surface of the base of the composite seems to be the first surface-filling spiral pattern.

The seeds creates in the center and grows bigger as the surface grows wider in a time varying surface filling pattern.

It's masterful geometry.