Jessen's icosahedron

Jessen's icosahedron or Jessen's orthogonal icosahedron is a concave isogonal icosahedron with pyritohedral symmetry. It has the same face, vertex, and edge count as the regular icosahedron. All edges have a dihedral angle of 90 degrees, but the six longer edges are concave.

The vertex coordinates are given by the twelve points $$(\pm 2,\pm 1,0)$$, $$(0,\pm 2,\pm 1)$$, and $$(\pm 1,0,\pm 2)$$.

Jessen's icosahedron is a well-known example of a "shaky polyhedron." That is, if the faces are made of a rigid material and the edges are hinges, the solid is flexible, but only infinitesimally so.

If the longer edges are realized with rigid bars and the shorter edges strings, the figure supports its own weight, forming a tensegrity structure known as the six-bar spherical tensegrity. Some research has been done on applying this structure in robotics.

A shape similar to Jessen's icosahedron can be produced by symmetrically "indenting" six edges of a regular icosahedron, but the longer-to-shorter edge ratio of the resulting figure is different (equal to the golden ratio in this case). This shape is sometimes mistakenly labeled as Jessen's icosahedron, but it is no longer shaky and does not form a tensegrity structure.