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Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same number of vertices, edges and faces as the regular icosahedron. It was introduced by Børge Jessen in 1967 and has several interesting geometric properties:

The regular icosahedron and the Jessen's icosahedron.

It is vertex-transitive (or isogonal), meaning that it has symmetries taking any vertex to any other vertex.
It has only right dihedral angles.
It is (continuously) rigid but not infinitesimally rigid. That is, in less formal language, it is a shaky polyhedron.
As with the simpler Schönhardt polyhedron, its interior cannot be triangulated into tetrahedra without adding new vertices.
It is scissors-congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.

Although a shape resembling Jessen's icosahedron can be formed by keeping the vertices of a regular icosahedron in their original positions and replacing certain pairs of equilateral-triangle faces by pairs of isosceles triangles, the resulting polyhedron does not have right-angled dihedrals. The vertices of Jessen's icosahedron are perturbed from these positions in order to give all the dihedrals right angles.

The Fifty-Nine Icosahedra

References

B. Jessen, Orthogonal Icosahedra, Nordisk Mat. Tidskr. 15 (1967), pp. 90–96.
Peter R. Cromwell, Polyhedra, Cambridge University Press, (1997) pp. ?
M. Goldberg, Unstable Polyhedral Structures, Math. Mag. 51 (1978), pp. 165–170
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin, (1991). p. 161.