Concept
Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it in 1967. In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by Adrien Douady under the name six-beaked shaddock; later authors have applied variants of this name more specifically to Jessen's icosahedron. The faces of Jessen's icosahedron meet only in right angles, even though it has no orientation where they are all parallel to the coordinate planes. It is a "shaky polyhedron", meaning that (like a flexible polyhedron) it is not infinitesimally rigid. Outlining the edges of this polyhedron with struts and cables produces a widely-used tensegrity structure, also called the six-bar tensegrity, tensegrity icosahedron, or expanded octahedron. The vertices of Jessen's icosahedron may be chosen to have as their coordinates the twelve triplets given by the cyclic permutations of the coordinates . With this coordinate representation, the short edges of the icosahedron (the ones with convex angles) have length , and the long (reflex) edges have length . The faces of the icosahedron are eight congruent equilateral triangles with the short side length, and twelve congruent obtuse isosceles triangles with one long edge and two short edges. Jessen's icosahedron is vertex-transitive (or ), meaning that it has symmetries taking any vertex to any other vertex. Its dihedral angles are all right angles. One can use it as the basis for the construction of an infinite family of combinatorially distinct polyhedra with right dihedral angles, formed by gluing copies of Jessen's icosahedron together on their equilateral-triangle faces. As with the simpler Schönhardt polyhedron, the interior of Jessen's icosahedron cannot be triangulated into tetrahedra without adding new vertices. However, because its dihedral angles are rational multiples of , it has Dehn invariant equal to zero.
