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 bookDe 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.