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")

Golden ratio in stereometry

Is there really such a thing as icosahedron?

Euclid found this problem difficult enough to be placed near the end of the Elements, and few of his readers ever mastered his solution. A beautiful direct construction was given by Luca Pacioli, a friend of Leonardo da Vinci, in his book De divina proportione (1509).

from Mathematics and its History by John Stillwell

The model consists of three golden-ratio rectangles passing one through another cyclically. Besides the central slits, a topological obstruction requires a temporary cut in one rectangle, which is then taped over. The convex hull of the union is an icosahedron, its vertices being the {3\cdot 4=12} vertices of the rectangles. Indeed, if the rectangles have dimensions {2\varphi\times 2}, then the vertices are {(\pm \varphi,\pm 1,0)}, {(0,\pm \varphi,\pm 1)}, {(\pm 1,0,\pm \varphi)}. To prove that the faces are regular triangles, it suffices to check that {\varphi^2 + (1-\varphi)^2+1^2 =4}, which quickly turns into {\varphi^2=\varphi+1}.

Proof of the existence of icosahedron
Proof of the existence of icosahedron

I used the Fibonacci approximation {\varphi \approx F_{n+1}/F_n} to draw the rectangles (specifically, their dimensions are {89\times 55}). The central slit in rectangle of size {F_{n+1}\times F_{n}} should begin at distance {F_{n-1}/2} from the shorter side.

The group of rotational symmetries of the paper model is smaller than the icosahedral group {A_5}: it has order {12} and acts freely on the vertices. Come to think of it, the group is {A_4}.


The second model is my favorite Catalan solid, rhombic triacontahedron. It is formed by 30 golden-ratio rhombi. I folded it from a net created by Robert Webb. The net looks pretty cool itself:

Robert Webb’s net

Assembly took a lot more scotch tape (and patience) than the first model.

Rhombic triacontahedron
Rhombic triacontahedron