Irrational sunflowers

A neat way to visualize a real number is to make a sunflower out of it. This is an arrangement of points with polar angles and polar radii (so that the concentric disks around the origin get the number of points proportional to their area). The prototypical sunflower has , the golden ratio. This is about the most uniform arrangement of points within a disk that one can get.

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

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

The number has stronger spirals: seven of them, due to approximation.

Of course, if was actually , the arrangement would have rays instead of spirals:

What if we used more points? The previous pictures have 500 points; here is with . The new pattern has 113 rays: .

Apéry’s constant, after beginning with five spirals, refuses to form rays or spirals again even with 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")