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")
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 vertices of the rectangles. Indeed, if the rectangles have dimensions , then the vertices are , , . To prove that the faces are regular triangles, it suffices to check that , which quickly turns into .
I used the Fibonacci approximation to draw the rectangles (specifically, their dimensions are ). The central slit in rectangle of size should begin at distance from the shorter side.
The group of rotational symmetries of the paper model is smaller than the icosahedral group : it has order and acts freely on the vertices. Come to think of it, the group is .
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:
Assembly took a lot more scotch tape (and patience) than the first model.