Dodecahedron Measures

These facts are enough to reconstruct many of the measures and features of a dodecahedron with edge length one:
(1) The vertices of a cube with edge length coincide with those of the dodecahedron.
(2) Eight edges of a dodecahedron coincide with the faces of a cube with edge length
(3) The same eight edges are also the edges of three mutually perpendicular rectangles with side ratio .
(4) The dodecahedron fits into a golden rhombus. In other words, the angle between two adjacent faces and the angle between two nonadjacent faces of a dodecahedron correspond to the angles of a golden rhombus.
(5) The distance between two opposite faces (i.e., the diameter of the inscribed sphere) equals the spacing of two parallel sides of a golden rhombus.
(6) Eight unit cubes connected vertex to vertex to the inscribed cube fill the space together with dodecahedra and bilunabirotundas.


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