Twenty Tetrahedra in Icosahedral Cluster
A golden tetrahedron is the tetrahedral part of a golden rhombohedron. Twenty such golden tetrahedra can be put together to form a regular icosahedron. (If regular tetrahedra are used, there are gaps.) Twenty regular tetrahedra can be brought to meet face to face by rotating around the axes through the centers of the tetrahedra and the icosahedron.
 F. Fang, K. Irwin, J. Kovacs, and G. Sadler. "Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos((3ϕ - 1)/4)." arxiv.org/abs/1304.1771.