[4] 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.

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.

[1] As with the simpler Schönhardt polyhedron, the interior of Jessen's icosahedron cannot be triangulated into tetrahedra without adding new vertices.

It provides a counterexample to a question of Michel Demazure asking whether star-shaped polyhedra with triangular faces can be made convex by sliding their vertices along rays from this central point.

However, Adrien Douady proved that, for a family of shapes that includes Jessen's icosahedron, this sliding motion cannot result in a convex polyhedron.

[11] Jessen's icosahedron is not a flexible polyhedron: if it is constructed with rigid panels for its faces, connected by hinges, it cannot change shape.

This means that there exists a continuous motion of its vertices that, while not actually preserving the edge lengths and face shapes of the polyhedron, does so to a first-order approximation.

[4] Replacing the long concave-dihedral edges of Jessen's icosahedron by rigid struts, and the shorter convex-dihedral edges by cables or wires, produces the tensegrity icosahedron, the structure which has also been called the "six-bar tensegrity"[6] and the "expanded octahedron".

The convex shapes in this family range from the octahedron itself through the regular icosahedron to the cuboctahedron, with its square faces subdivided into two right triangles in a flat plane.

Extending the range of the parameter past the proportion that gives the cuboctahedron produces non-convex shapes, including Jessen's icosahedron.

These polyhedra are combinatorially distinct, and have chiral dihedral symmetry groups of arbitrarily large order.