相关论文: On the flexibility of Siamese dipyramids
We study a class of mechanisms known as Kokotsakis polyhedra with a quadrangular base. These are $3\times3$ quadrilateral meshes whose faces are rigid bodies and joined by hinges at the common edges. In contrast to existing work, the…
For flexibility of an octahedron we find necessary metric conditions in terms of edge lengths. These conditions yield a new description of Bricard's octahedra, suitable for solving some problems in metric geometry of octahedra, in…
A surface $\Sigma$ in a 4-manifold $M$ is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism $(M,\Sigma) \to (M,\Sigma)$. We construct flexible surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$…
A regular polyhedron of type {p, q} has at least 2pq flags, and it is called tight if it has exactly 2pq flags. The values of p and q for which there exist tight orientably regular polyhedra were previously known. We determine for which…
We say that a tile is $\sigma$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $\sigma$-morphic tile exist in the plane. In this note we present a…
Integrable models are often constructed with real systems in mind. The exact solvability of the models leads to results which are unambiguous and provide the correct physical picture. In this review, we discuss the physical basis of some…
By evaporating a drop of lipid dispersion we generate the myelin morphology often seen in dissolving surfactant powders. We explain these puzzling nonequilibrium structures using a geometric argument: The bilayer repeat spacing increases…
Many compliant shell mechanisms are periodically corrugated or creased. Being thin, their preferred deformation modes are inextensional, i.e., isometric. Here, we report on a recent characterization of the isometric deformations of periodic…
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic…
In the end of the 19th century Bricard discovered a phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit non-trivial flexes. One of the most important results in this field is a theorem of…
Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs…
We show that the equations of motion associated with a complexified sigma-model action do not admit manifest dual SO(n,n) symmetry. In the process we discover new type of numbers which we called `complexoids' in order to emphasize their…
We define a notion for unfolding smooth, ruled surfaces, and prove that every smooth prismatoid (the convex hull of two smooth curves lying in parallel planes), has a nonoverlapping "volcano unfolding." These unfoldings keep the base…
The dynamics of a rigid, rotating, precessing, massive ring orbiting a point mass within the perimeter of the ring are considered. It is demonstrated that orbits dynamically stable against perturbations in three dimensions exist for a range…
Supersymmetric non-linear sigma-models are described by a field dependent Kaehler metric determining the kinetic terms. In general it is not guaranteed that this metric is always invertible. Our aim is to investigate the symmetry structure…
We construct some example of a closed nondegenerate nonflexible polyhedron $P$ in Euclidean 3-space that is the limit of a sequence of nondegenerate flexible polyhedra each of which is combinatorially equivalent to $P$. This implies that…
A widely used mathematical model for the bouncing motion of an ideally elastic ball -- referred to in previous work by the first two authors and collaborators as a {\em no-slip billiard} system -- exhibits some notable dynamical behavior…
It is conjectured that all decomposable (i.e. interior can be triangulated without adding new vertices) polyhedra with vertices in convex position are infinitesimally rigid and only recently has it been shown that this is indeed true under…
Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid.…