This conjecture connects the rank of the group of rational points to the number of points on an elliptic curve mod p and is backed up by a wealth of experimental evidence. Elliptic curves are essential mathematical objects that appear in numerous contexts, including Wiles’ demonstration of the Fermat Conjecture, the factorization of numbers into primes, and cryptography, to name just three. Elliptic curves are described by cubic equations in two variables. Bryan John Birch Professor Sir Peter Swinnerton-Dyer Early History Problems on curves of genus 1 feature prominently in Diophantus’ Arithmetica. It is easy to see that a straight…

The knight’s tour problem is the mathematical problem of finding a knight’s tour, and probably making knight the most interesting piece on the chess board. The knight visits every square exactly once, if the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed; otherwise, it is open. The knight’s tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight’s tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path…

The Painter’s Paradox is based on the fact that Gabriel’s horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel’s horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel’s horn has a convergent series associated with volume and a divergent series associated with surface area. The dimensions of this object, which are in the heart of the above mentioned paradox, were first studied by Torricelli. To situate the paradox historically, we provide…

Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject. Besides the obvious fact that the Fourier transform is itself an oscillatory integral par excellence, one need only recall the occurrence of Bessel functions in the work of Fourier, the study of asymptotic related to these functions by Airy, Stokes, and Lipschitz, and Riemann’s use of the method of “stationary phase” in finding the asymptotic of certain Fourier transforms, all of which took place well over 100 years ago. The dyadic decomposition of a function Littlewood–Paley theory uses a decomposition of a function f into a sum of…

Modern theories describe physical forces in terms of fields, e.g. the electromagnetic field, the gravitational field, and fields that describe forces between the elementary particles. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. A transformation from one such field configuration to another is called a gauge transformation; the lack of change in the measurable quantities, despite the field being transformed, is a property called gauge invariance. For example, if you could measure the color of lead balls and discover that when you change the color, you…

A gyroid structure is a distinct morphology that is triply periodic and consists of minimal iso-surfaces containing no straight lines. The gyroid was discovered in 1970 by Alan Schoen, a NASA crystallographer interested in strong but light materials. Among its most curious properties was that, unlike other known surfaces at the time, the gyroid contains no straight lines or planar symmetry curves. In 1975, Bill Meeks discovered a 5-parameter family of embedded genus 3 triply periodic minimal surfaces. Photonic crystals were synthesized by deposition of a-Si/Al2O3 coatings onto a sacrificial polymer scaffold defined by two-photon lithography. We observed a 100%…

The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not “solids” in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. Dissection and reassembly of a figure is an important concept in classical geometry. It was…

Chaos theory is a branch of mathematics focusing on the study of chaos states of dynamical systems whose apparently random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. When employing mathematical theorems, one should remain careful about whether their hypotheses are valid within the frame of the questions considered. Among such hypotheses in the domain of dynamics, a central one is the continuity of time and space (that an infinity of points exists between two points). This hypothesis, for example, may be invalid In the cognitive neurosciences of perception, where a finite time threshold often needs to be considered. The…

In topology, knot theory is the study of mathematical knots. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (in topology, a circle isn’t bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. Although people have been making use of knots since the dawn of our existence, the actual mathematical study of knots is…

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is “simply connected,” but that the surface of the doughnut is not. Poincaré, almost…