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

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