A kinematic approach to Kokotsakis meshes

Abstract

A Kokotsakis mesh is a polyhedral structure consisting of an n-sided central polygon P 0 surrounded by a belt of polygons in the following way: Each side a i of P 0 is shared by an adjacent polygon P i , and the relative motion between cyclically consecutive neighbor polygons is a spherical coupler motion. Hence, each vertex of P 0 is the meeting point of four faces. In the case n = 3 the mesh is part of an octahedron. These structures with rigid faces and variable dihedral angles were first studied in the thirties of the last century. However, in the last years there was a renaissance: The question under which conditions such meshes are infinitesimally or continuously flexible gained high actuality in discrete differential geometry. The goal of this paper is to revisit the wellknown continuously flexible examples (Bricard, Graf, Sauer, Kokotsakis) from the kinematic point of view and to extend their list by a new family.

Flexible Polyhedral Surfaces with Two Flat Poses

“…Miura-ori is a very particular case of a Voss surface. 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].…”

“…Hence, each flexion of the initial pyramid with apex V 1 is compatible with a flexion of the 3 × 3 complex of quadrangles. This can be extended to the complete tessellation (in [6], it is shown that this tessellation is a particular case of the line-symmetric Type 5 of flexible Kokotsakis meshes). The product of two half-turns is a helical motion about the common perpendicular of the axes of the two half-turns.…”

On the flexibility and symmetry of overconstrained mechanisms

“…. , f 4 For n = 4, there are five types of flexible Kokotsakis meshes known until recently [44]. In the following, we use the term fold for triples of edges where at each vertex V i opposite edges are combined.…”

“…Orthogonal type or T-flat [41]: here the horizontal folds are located in parallel (say horizontal) planes and the vertical folds in vertical planes. V. Line-symmetric type [44]: a half-rotation maps the pyramid at V 1 onto that of V 4 ; another one exchanges the pyramids at V 2 and V 3 . Additionally, δ 1 + δ 2 = α 1 + β 2 must hold together with the 'Dixon angle condition' (note angle ψ at A 1 and B 2 in figure 15) sin α 1 sin γ 1 : sin β 1 sin δ 1 : (cos α 1 cos γ 1 − cos β 1 cos δ 1 ) = ± sin β 2 sin γ 2 : sin α 2 sin δ 2 : (cos α 2 cos δ 2 − cos β 2 cos γ 2 ).…”

Shapes of Miura Mesh Mechanism with Mobility One

“…Working on the classification of flexible assemblies made of four bar linkages, Satchel recognizes the mesh as a special case of the flexible meshes Kokotsakis discovered in 1932 [10], and gives the geometric description of the conditions that ensure the movement.…”