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Rhombic Hexecontahedron in a Rhombohedron

enlarge rhombohedron R

enlarge small RH

show big RH

show 20 small RHs

show 30 RTs

Find a point that divides the diagonal of a large rhombohedron

in proportion to the golden ratio. Construct a small rhombohedron

around this point so that one of its vertices coincides with the center of

. Construct a rhombic hexecontahedron

so that one of its constituent 20 rhombohedra coincides with

. If you enlarge the

, then you will see two small dots on each face of

, which are actually vertices of the

. (Drag to rotate the graphic to see the dots.) Therefore,

2×6=12

vertices (out of 20) of the

are on the six faces of

. When 20 rhombohedra with

s inside them are assembled to form a large

, then the

s meet at their vertices, and a rhombic triacontahedron connected face-to-face can be inserted between them.

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