We study the properties of Kokotsakis polyhedra of orthodiagonal anti-involutive type. Stachel conjectured that a certain resultant connected to a polynomial system describing flexion of a Kokotsakis polyhedron must be reducible. Izmestiev  showed that a polyhedron of the orthodiagonal anti-involutive type is the only possible candidate to disprove Stachel’s conjecture. We show that the corresponding resultant is reducible, thereby confirming the conjecture. We do it in two ways: by factorization of the corresponding resultant and providing a simple geometric proof. We describe the space of parameters for which such a polyhedron exists and show that this space is non-empty. We show that a Kokotsakis polyhedron of orthodiagonal anti-involutive type is flexible and give explicit parametrizations in elementary functions and in elliptic functions of its flexion.
|Journal||Mechanism and Machine Theory|
|Early online date||26 Dec 2019|
|Publication status||Published - 1 Apr 2020|
- Kokotsakis polyhedron
- Spherical linkage
- Stachel's conjecture
- flexible polyhedron