Let \(\phi = \frac{1}{2}(1 + \sqrt{5})\) be the golden ratio. Start with a cube aligned with the Cartesian axes. Inscribe a regular tetrahedron whose vertices are alternate vertices of the cube. Rotate this tetrahedron by multiples of one-fifth of a turn about the axis through \((0, 1, \phi)\) (or any other axis obtained by permuting coordinates). The resulting union of five tetrahedra has icosahedral symmetry.
Rotations of space that preserve the union of the five tetrahedra permute the set of five tetrahedra by an ``even permutation'' (a composition of an even number of swaps). Conversely, every even permutation on the set of five tetrahedra is effected by a unique rotation of space. The group of rotation symmetries of the union is isomorphic to the alternating group on five letters.
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