Classification of Flexible Kokotsakis Polyhedra with Quadrangular Base

Abstract

A Kokotsakis polyhedron with quadrangular base is a neighborhood of a quadrilateral in a quad surface. Generically, a Kokotsakis polyhedron is rigid. Up to now, several flexible classes were known, but a complete classification was missing. In the present paper, we provide such a classification. The analysis is based on the fact that the dihedral angles of a Kokotsakis polyhedron are related by an Euler-Chasles correspondence. It results in a diagram of elliptic curves covering complex projective planes. A polyhedron is flexible if and only if this diagram commutes.

Orthodiagonal anti-involutive Kokotsakis polyhedra

“…We do not discuss the nature of the written equations here. A comprehensive explanations can be found in [3]. However, we give a brief explanation in Section 6, where we show that Stachel's conjecture is a simple consequence of the results from [3].…”

“…We prove that the resultant R 12 is always reducible, which confirms Stachel's conjecture. In addition, we provide a simple geometric proof of this fact in Theorem 6 that directly follows from the results of [3].…”

“…(2) P 1 (z, w 1 ) = 0, P 2 (z, w 2 ) = 0, Considering a complexified version of system (2), Izmestiev [3] classified all possible classes of flexible polyhedra disregarding reality and embeddability. In particular, from the results of [3] it follows that there is only one possible candidate for a flexible Kokotsakis polyhedron with the irreducible resultant R 12 . He called elements of this class Kokotsakis polyhedra of orthodiagonal anti-involutive type (OAI).…”

“…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.…”

Flexible Polyhedral Surfaces with Two Flat Poses

“…This is a polyhedral surface with quadrangular faces, such that each 3 × 3 complex is a flexible Kokotsakis mesh [5] of the isogonal Type 3 (according to the enumeration given in [6]): at each vertex, opposite interior angles are either equal or complementary (Lemma 1); and there is an additional equation to satisfy ( [7], p. 12). A long lasting open problem, the classification of flexible quadrangular Kokotsakis meshes [3] has recently been solved by I. Izmestiev [8]. The classification of flexible Kokotsakis meshes with a central n-gon and n ≥ 5 is still open.…”