Two years ago I linked to a post by Adam Goucher, solving an old MathOverflow question by showing that it is possible to find a dodecahedron, combinatorially equivalent to a regular one, with rational coordinates, inscribed in a unit sphere. But now there are infinitely many! Some messy algebra, and then some work with elliptic curve group operations, eventually simplifies down to a parametric family with a dodecahedron for each integer right triangle. See

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