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:
- 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.
Tension integrity transformations
The Jessen's icosahedron is the geometry of the tension integrity icosahedon, a physical structure comprised of 6 semi-rigid compression rods (comprising the concave edges) and 24 tension cables (comprising the other edges). Realized in this form, the Jessen's icosahedron's property of continuous but not infinitesimal rigidity (its equilibrium shakiness) is dramatically manifest. Only the rods must be rigid, to resist compression and keep the cables taught and the whole structure drum-tight, despite the fact that the rods do not touch each other: they float in a net of tension cables.
Provided the 6 compression rods are spring-like and can be compressed slightly (down to a length of √2 times the edge length), the 6 concave pairs of isosceles triangles can be folded open (simultaneously) into square faces, transforming the Jessen's icosahedron into a cuboctahedron. Or they can be folded closed (which also compresses the 6 rods to length √2), transforming the Jessen's icosahedron into an octahedron. Released, the compressed rods spring back to their resting length, returning the Jessen's icosahedron to its original shape.
These two ways of compressing the concave edges amount to twisting the whole structure away from its resting shape in one of two opposing directions. The vertices move slightly all together in either outward or inward spirals around their radii, and the long radius (center to vertex) of the entire structure grows from its resting length (the icosahedral radius) to a maximum (the cuboctahedral radius of 1 edge length) when fully open, or shrinks to a minimum (the octahedral radius of √2/2 edge lengths) when fully closed. As the isosceles faces fold outward or inward, they distort (because their long edge shortens). The compression rods move slightly as well. They are 3 pairs of parallel rods, one pair lying in each of the three orthogonal planes, exactly one edge length apart from each other (at rest). When compressed, they move either apart (outward) or toward each other (inward), but they remaining parallel and in their original plane throughout. The equilateral faces displace also, but do not change shape during the twisting; they merely rotate around their centers and travel outward or inward slightly: opposite faces rotate in opposite directions around an axis joining them through their centers. Therefore the physical structure can even be skinned with rigid equilateral panels, and with foldable or stretchable isosceles panels, without impeding its ability to traverse the entire range of motion.
Finally, if the compression rods are not springs but nearly rigid, the structure still exhibits its equilibrium springiness, even though the isosceles faces are not able to fold fully open or fully closed. To whatever extent the rods can be compressed, even minutely, a very evident small range of inward and outward motion (springy radial expansion and contraction) remains.
- 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.
- Kenner, Hugh (1976). Geodesic Math and How to Use It. University of California Press. ISBN 978-0520029248. 2003 reprint ISBN 0520239318. The mathematics and dynamics of the tensegrity icosahedron.