TITLE: Proof of golden rectangle inside an Icosahedron
QUESTION [10 upvotes]: This is an regular Icosahedron with three golden rectangles inscribed in it. I realize that they are clearly rectangles, I was wondering if someone could prove that they are a golden rectangles, assuming all edges are length $s$.
(I know that it's true from a result and from google but I'm not sure how to show it.)
I am completely fine with a proof from its dual, the dodecahedron.

REPLY [4 votes]: Assume $s=1$.
The generalized diamete is the max distance between vertices. Consider two of them, opposite wrt the origin $A\left(0,0,\dfrac{5}{\sqrt{50-10 \sqrt{5}}}\right)$ and $B\left(0,0,-\dfrac{5}{\sqrt{50-10 \sqrt{5}}}\right)$
$AB=\sqrt{\dfrac{1}{2} \left(5+\sqrt{5}\right)}$
The side of the golden rectangle is then
$\sqrt{AB^2-1}=\sqrt{\dfrac{3}{2}+\dfrac{\sqrt{5}}{2}}=\dfrac{1}{2} \left(1+\sqrt{5}\right)$
QED
Hope this helps