Related papers: Arnold's problem on paper folding
For each integer n, an n-folding curve is obtained by folding n times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding…
This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces…
In this pedagogical note we present a short proof of the following main result of arxiv.org/abs/0911.5319, and clarify its relation to the isoperimetric problem. On the hyperbolic plane consider triangles ABC with fixed lengths of AB and…
The lifting effect of an effervescent tablet in the water is ideal for educational purposes, because it combines various mathematical and physical principles. We developed a mathematical toy model for the time evolution of the buoyancy and…
Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each…
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.
Given a finite family of squares in the plane, the packing problem asks for the maximum number $\nu$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $\tau$ of points hitting all of them. Clearly,…
This article has been written for an educational magazine whose target audience consists of students and teachers of mathematics in universities, colleges and schools. It concerns a notion of duality between rectangles. A proof is given…
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric…
Previous research has shown that students often struggle to develop an understanding of linear and quadratic relationships. Covariational reasoning has been identified as a way to support this development. This study aims to investigate how…
Let $M$ be a 3-manifold. Every knotted (embedded) surface in $M \times \R$ can be moved via an ambient isotopy in such a way that its projection into $M$ is a generic surface. A surface is generic if every point on it is either a regular,…
The paper analyzes how the enlarging of the sample affects to the mitigation of collinearity concluding that it may mitigate the consequences of collinearity related to statistical analysis but not necessarily the numerical instability. The…
In this paper we will be considering a basic geometric problem, the extension problem of classical Hamilton-Cartan variational theory to higher jet prolongations on fibered manifolds.
Origami is the art of paper folding, and it borrows its name from two Japanese words \emph{ori} and \emph{kami}. In Japanese, {ori} means folding, and the paper is called {kami}. While origami is just a hobby to most, there is a lot more to…
The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the…
In the first part of the article using a direct calculation two-dimensional motion of a particle sliding on an inclined plane is investigated for general values of friction coefficient ($\mu$). A parametric equation for the trajectory of…
A strip of square stamps can be folded in many ways such that all of the stamps are stacked in a single pile in the folded state. The stamp folding problem asks for the number of such foldings and has previously been studied extensively. We…
Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two…
A kind of fixed-point problem in the area of discrete tomography is proposed and investigated. Our chief concern in this paper is the case of square windows in the plane. Dealing with the arrays which are bounded, of polynomial growth, and…
In this paper we review nine previous proposed and solved problems of elementary 2D geometry, and we extend them either from triangles to polygons or polyhedrons, or from circles to spheres (from 2D-space to 3D-space) and make some…