Using the same three golden rectangles at right-angles to each other, we can also make an octahedron. If we put a square as shown around each rectangle, the squares will also be at right angles to each other and form the edges of an octahedron.
Now if we join the "golden-section points" forming the corners of our three rectangles (and now on both the edges of an octahedron and also forming the vertices of an icoshedron as we saw above), we can see how to fit an icosahedron into an octahedron - and the process involves golden sections!
Here are some more Platonic-solids-within-Platonic-solids:
A Tetrahedron in a Cube
Select one corner of a cube and join it to the opposite corner on each face.
An Octahedron in a Tetrahedron
Join the mid-point of each edge to any other edge mid-point where the connecting line lies on one face of the tetrahedron.
An Octahedron in a Cube
Join the mid-points of faces: if two faces are next to each other at a corner, then their mid-points can be joined.
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