Irrational sunflowers

A neat way to visualize a real number {\alpha} is to make a sunflower out of it. This is an arrangement of points with polar angles {2 \pi \alpha k} and polar radii {\sqrt{k}} (so that the concentric disks around the origin get the number of points proportional to their area). The prototypical sunflower has {\alpha=(\sqrt{5}+1)/2}, the golden ratio. This is about the most uniform arrangement of points within a disk that one can get.

Golden ratio sunflower
Golden ratio sunflower

But nothing stops us from using other numbers. The square root of 5 is not nearly as uniform, forming distinct spirals.

Square root of 5
Square root of 5

The number {e} begins with spirals, but quickly turns into something more uniform.

Euler's sunflower
Euler’s sunflower

The number {\pi} has stronger spirals: seven of them, due to {\pi\approx 22/7} approximation.

pi sunflower
pi sunflower

Of course, if {\pi} was actually {22/7}, the arrangement would have rays instead of spirals:

Rational sunflower: 22/7
Rational sunflower: 22/7

What if we used more points? The previous pictures have 500 points; here is {\pi} with {3000}. The new pattern has 113 rays: {\pi\approx 355/113}.

pi with 3000 points
pi with 3000 points

Apéry’s constant, after beginning with five spirals, refuses to form rays or spirals again even with 3000 points.

Apéry's constant, 3000 points
Apéry’s constant, 3000 points

The images were made with Scilab as follows, with an offset by 1/2 in the polar radius to prevent the center from sticking out too much.

n = 500
alpha = (sqrt(5)+1)/2
r = sqrt([1:n]-1/2)  
theta = 2*%pi*alpha*[1:n]
plot(r.*cos(theta), r.*sin(theta), '*');
set(gca(), "isoview", "on")

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