Related papers: Deformations of period lattices of flexible polyhe…
We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again…
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued…
There are (at least) two surfaces of particular interest in eternal de Sitter space. One is the timelike hypersurface constituting the lab wall of a static patch observer and the other is the future boundary of global de Sitter space. We…
The Article demonstrates the spontaneous symmetry breaking of isotropic homogeneous elastic medium in form of transition from Euclidean to Riemann-Cartan internal geometry of medium. The deformation of elastic medium without defects is…
We give an example of an infinitesimally nonrigid polyhedron in the Lobachevsky 3-space and construct an infinitesimal flex of that polyhedron such that the volume of the polyhedron isn't stationary under the flex.
The class of 2-dimensional non-integrable flat dynamical systems has a rather extensive literature with many deep results, but the methods developed for this type of problems, both the traditional approach via Teichm\"{u}ller geometry and…
We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson…
The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two polyhedra are isometric or not by using their…
Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations…
In this paper, we introduce and study a remarkable class of mechanisms formed by a $3 \times 3$ arrangement of rigid quadrilateral faces with revolute joints at the common edges. In contrast to the well-studied Kokotsakis meshes with a…
A one-parameter deformation of a periodic bar-and-joint framework is expansive when all distances between joints increase or stay the same. In dimension two, expansive behavior can be fully explained through our theory of periodic…
Using the orthogonal connectedness, we introduce the notion of orthogonal decomposability of convex polytopes and study it in the case of Platonic and Archimedean solids. While doing so, we also encounter polytopes which are not…
We show that a smooth 1-parameter family of foliations by circles of a closed 3-manifold, deforming the foliation whose leaves are the fibers of a circle bundle, is trivial, i.e. all the foliations of the family arise from circle bundles…
Soft solids and their surface deformations control the response of many natural and artificial systems. Yet, their underlying properties are vigorously debated, particularly for polymer networks. While molecular-scale theories predict no…
We consider the dynamics of a layer of an incompressible electrically conducting fluid interacting with the magnetic field in a two-dimensional horizontally periodic setting. The upper boundary is in contact with the atmosphere, and the…
In this paper we pose and prove a spacetime version of Gromov's dihedral rigidity theorem (Gromov, Li) for cubes when the dimension is 3 by studying the level sets of spacetime harmonic functions (Stern, Bray-Stern, Hirsch-Kazaras-Khuri),…
In a previous article it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex…
A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal…
We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover…
We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper…