Related papers: Dissections of a metal rectangle

In the March 2025 issue of Pour la Science, Jean-Paul Delahaye described a wonderful solution to the following problem: How many ways can you divide a 3 by 2n rectangle into two connected, congruent pieces? We show that this problem can be…

Leaves and flowers frequently have a characteristic rippling pattern at their edges. Recent experiments found similar patterns in torn plastic. These patterns can be reproduced by imposing metrics upon thin sheets. The goal of this paper is…

In this article we prove the impossibility of some disentanglement puzzles, first building mathematical models that reflect the essential characteristics of these puzzles.

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other…

We prove that any diffeomorphism of the sphere S^n to itself can be decomposed into bi-Lipschitz mappings of small isometric distortion and which move points a small amount in the spherical metric.

Our first experience of dimension typically comes in the intuitive Euclidean sense: a line is one dimensional, a plane is two-dimensional, and a volume is three-dimensional. However, following the work of Mandelbrot \cite{mandelbrot},…

We decompose the regular quandle representation of a dihedral quandle $\mathcal{R}_n$ into irreducible quandle subrepresentations.

Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits…

Fractals are ubiquitous natural emergences that have gained increased attention in engineering applications, thanks to recent technological advancements enabling the fabrication of structures spanning across many spatial scales. We show how…

A two-dimensional energy-based model of fragmentation of rapidly expanding cylinder under plane strain conditions is proposed. The model allows us to estimate the average fragment length and the number of fragments produced by ductile…

In this Letter we show that a single rectangular hole exhibits transmission resonances that appear near the cutoff wavelength of the hole waveguide. For light polarized with the electric field pointing along the short axis, it is shown that…

This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.

We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking…

We utilise the two principles of decoupling introduced in [arXiv:2407.16108] to prove decoupling for two types of surfaces exhibiting radial symmetry. The first type are surfaces of revolution in $\mathbb R^n$ generated by smooth surfaces…

When cracks form in a thin contracting layer, they sequentially break the layer into smaller and smaller pieces. A rectilinear crack pattern encodes information about the order of crack formation, as later cracks tend to intersect with…

The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…

The surface reconstruction problem from sets of planar parallel slices representing cross sections through 3D objects is presented. The final result of surface reconstruction is always based on the correct estimation of the structure of the…

A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly inte\-resting, since there are…

We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner's Lemma plays a central role.

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of…