Orthodiagonal anti-involutive Kokotsakis polyhedra

Abstract

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 [3] 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 parameterizations in elementary functions and in elliptic functions of its flexion. P 3 (u, w 1 ) = 0, P 4 (u, w 2 ) = 0. They studied this system with respect to different factorizations of the resultant R 12 (w 1 , w 2 ) of P 1 and P 2 as polynomials in z, and they classified all classes of flexible Kokotsakis polyhedra with quadrilateral base for reducible R 12 . Stachel conjectured that there is no flexible Kokotsakis polyhedron whose system has the irreducible resultant.

Generalization of the Orthodiagonal Involutive Type of Kokotsakis Flexible Polyhedra

“…A spherical quadrilateral is said to be orthodiagonal, if its diagonals are orthogonal. The orthodiagonality is equivalent to the following identity Lemma 2.4 (Lemma 6.3, [40]). The diagonals of a spherical quadrilateral with side lengths α, β, γ, δ (in this cyclic order) are orthogonal if and only if its side lengths satisfy the relation…”