icosahedron and dodecahedron
The icosahedron and the dodecahedron are two of the 5 Platonic solids.
These solids form pairs and the icosahedron and the dodecahedron are one of them.
The remarkable thing about them is that you can keep on placing the solids into one another.
The midpoints of the 20 side surfaces of the icosahedron can be connected to form a dodecahedron and at their turn the midpoints of the 12 side surfaces of a dodecahedron form an icosahedron.
So you might as well say that you can connect the midpoints of the 12 pentagonal side surfaces to form 3 golden rectangles.
And talking about pentagons is not at all a stranger.
So you can find in polygons were angles (or mulples) of occur as well in solids.