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The Golden Ratio in Arrangements of Octahedron, Cube, Tetrahedron, and Sphere

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octahedron size
1.
show:
sphere
tet. 1
tet. 2
The edge of an octahedron of length 1 and the edge of a smaller octahedron of length
1/ϕ≈0.61803…
determine a straight line, one endpoint of which is at the face center of a matching cube, while its other endpoint is on a sphere circumscribing the cube.
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