Archive for December, 2022

Spiric curves and phase portraits

Left: Conic sections. Right: Spiric sections [images Wikipedia Commons].

We are very familiar with the conic sections, the curves formed from the intersection of a plane with a cone. There is another family of curves, the Spiric sections, formed by the intersections of a torus by planes parallel to its axis. Like the conics, they come in various forms, depending upon the distance of the plane from the axis of the torus (see Figure above). We examine how spiric curves may be found in the phase-space of a dynamical system.

When is 144 closer to 8 than to 143?

The usual definition of the norm of a real number ${x}$ is its modulus or absolute value ${|x|}$. We measure the “distance” between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric ${\rho(x,y) = |x-y|}$ and, using it, we can define the usual topology on the real numbers ${\mathbb{R}}$.

The standard arrangement of the real numbers on a line automatically ensures that numbers with small Euclidean difference between them are geometrically close to each other. It may come as a surprise that there are other ways to define norms and distances, which provide other topologies, leading us to a radically different concept of closeness, and to completely new number systems, the p-adic numbers.

Convergence of mathematics and physics

The connexions between mathematics and physics are manifold, and each enriches the other. But the relationship between the disciplines fluctuates between intimate harmony and cool indifference. Numerous examples show how mathematics, developed for its inherent interest in beauty, later played a central role in physical theory.

A well-known case is the multi-dimensional geometry formulated by Bernhard Riemann in the mid 19th century, which was exactly what Albert Einstein needed 50 years later for his relativity theory [TM240 or search for “thatsmaths” at irishtimes.com].

Curvature and Geodesics on a Torus

Geodesics on a torus [image from Jantzen, Robert T., 2021].

We take a look at the curvature on a torus, and the various forms that geodesics can have. These are compared to the geodesics on a “flat torus”.
Continue reading ‘Curvature and Geodesics on a Torus’

Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein

In last week’s post we looked at aspects of puzzles of the form “What is the next number”. We are presented with a short list of numbers, for example ${1, 3, 5, 7, 9}$ and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as the next number.

Borwein integrals evaluated by Mathematica. The first seven integrals are all equal to ${\pi}$. The eighth is a tiny bit less than this.

In this article we consider a sequence of seven ones: ${1, 1, 1, 1, 1, 1 ,1}$. Most people would agree that the next number in the sequence is ${1}$. We will show how the number ${1 - 1.47\times10^{-11} \approx 0.999\,999\,999\,985}$ could be the “correct” answer. Continue reading ‘Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein’