### From Wave Equations to Modern Telecoms

Mathematics has an amazing capacity to help us to understand the physical world. Just consider the profound implications of Einstein’s simple equation ${E = m c^2}$. Another example is the wave equation derived by Scottish mathematical physicist James Clerk Maxwell. Our modern world would not exist without the knowledge encapsulated in Maxwell’s equations. Continue reading ‘From Wave Equations to Modern Telecoms’

### Curvature and the Osculating Circle

Curvature is of critical importance in numerous contexts. An example is shown in the figure above, a map of the Silverstone Formula 1 racetrack. The sharp bends (high curvature) force drivers to reduct speed drastically.

### The Cosmology of the Divine Comedy

Divina Commedia: Online at Columbia University.

If you think poetry and maths are poles apart, think again. Around the sixth century, Indian poet and mathematician Virahanka codified the structure of Sanskrit poetry, formulating rules for the patterns of long and short syllables. In this process, a sequence emerged in which each term is the sum of the preceding two. This is precisely the sequence studied centuries later by Leonardo Bonacci of Pisa, which we now call the Fibonacci sequence [TM241 or search for “thatsmaths” at irishtimes.com].

### Adding a Point to Make a Space Compact

Stereographic projection between ${\mathbb{R}}$ and ${S^1}$. There is no point on the real line corresponding to the “North Pole” in ${S^1}$. Can we add another point to ${\mathbb{R}}$?

The real line is an example of a locally compact Hausdorff space. In a Hausdorff space, two distinct points have disjoint neighbourhoods. As the old joke says, “any two points can be housed off from each other”. We will define local compactness below. The one-point compactification is a way of embedding a locally compact Hausdorff space in a compact space. In particular, it is a way to “make the real line compact”.

### Summing the Fibonacci Sequence

Left: Fibonacci, or Leonardo of Pisa. Right: Italian postage stamp issued on the 850th anniversary of his birth.

The Fibonacci sequence must be familiar to anyone reading this. We define it by means of a second-order recurrence relation,

$\displaystyle F_{n+1} = F_{n-1} + F_n \,. \ \ \ \ \ (1)$

and two initial values, ${F_0 = 0}$ and ${F_1 = 1}$. This immediately yields the well-known sequence

$\displaystyle \{F_n\} = \{ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots \} \,.$

### 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.

### Closeness in the 2-Adic Metric

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’

### What’s the Next Number?

We are all familiar with simple mathematical puzzles that give a short sequence and ask “What is the next number in the sequence”. Simple examples would be

$\displaystyle \begin{array}{rcl} && 1, 3, 5, 7, 9, 11, \dots \\ && 1, 4, 9, 16, 25, \dots \\ && 1, 1, 2, 3, 5, 8, \dots \,, \end{array}$

the sequence of odd numbers, the sequence of squares and the Fibonacci sequence.

### The Rich Legacy of Indian Mathematics

Jeremy Irons and Dev Patel in The Man Who Knew Infinity, a film about the life of Srinivasa Ramanujan.

For more than three thousand years, mathematics has played an important role in Indian culture. Sometimes it was studied for practical reasons and sometimes for pure intellectual delight. The earliest traces of mathematics are found in the Indus Valley, around 3000 BC. There is clear evidence of a structured system of weights and measures and samples of decimal-based numeration [TM239 or search for “thatsmaths” at irishtimes.com].

### Mercury’s Mercurial Orbit

The tiny deviation of the orbit of Mercury from a pure ellipse might seem to be of no consequence. Yet the minute precession of this planet was one of the factors leading to a revolution in our world view. Attempts to explain the anomaly in the context of Newtonian mechanics were unsatisfactory. It was only with the emergence of general relativity that we were able to understand the observed phenomenon. Continue reading ‘Mercury’s Mercurial Orbit’

### The Power of the 2-gon: Extrapolation to Evaluate Pi

Lewis Fry Richardson

Richardson’s extrapolation procedure yields a significant increase in the accuracy of numerical solutions of differential equations. We consider his elegant illustration of the technique, the evaluation of ${\pi}$, and show how the estimates improve dramatically with higher order extrapolation.

[This post is a condensed version of a paper in Mathematics Today (Lynch, 2003).]

### Dropping Pebbles down a Mine-shaft

Trajectory of a body falling at the Equator during a period of 10 seconds.

If you drop a pebble down a mine-shaft, it will not fall vertically, but will be deflected slightly to the East by the Coriolis force, an effect of the Earth’s rotation. We can solve the equations to calculate the amount of deflection; for a ten-second drop, the pebble falls about 500 metres (air resistance is neglected) and is deflected eastward by about 25 cm. The figure on the left shows the trajectory in the vertical xz-plane (scales are not the same).

We derive the equations after making some simplifying assumptions. We assume the mine-shaft is at the Equator; we assume the meridional or north-south motion is zero; we neglect variations in the gravitational force; we neglect the sphericity of the Earth; we neglect air resistance. We can still get accurate estimates provided the elapsed time is short. However, carrying the analysis to the extreme, we obtain results that are completely unrealistic. The equations predict that the pebble will reach a minimum altitude and then rise up again to its initial height a great distance east of its initial position. Then this up-and-down motion will recur indefinitely.

### From Sub-atomic to Cosmic Strings

The two great pillars of modern physics are quantum mechanics and general relativity. These theories describe small-scale and large-scale phenomena, respectively. While quantum mechanics predicts the shape of a hydrogen atom, general relativity explains the properties of the visible universe on the largest scales.

### Finding the Area of a Field

It is a tricky matter to find the area of a field that has irregular or meandering boundaries. The standard method is to divide the field into triangular parts. If the boundaries are linear, this is simple. If they twist and turn, then a large number of triangles may be required.

Heron of Alexandria. Triangle of sides a, b and c and altitude h.

### CND Functions: Curves that are Continuous but Nowhere Differentiable

Approximation ${W_{12}(x)}$ to the Weierstrass CND function.

A function ${f(x)}$ that is differentiable at a point ${x}$ is continuous there, and if differentiable on an interval ${[a, b]}$, is continuous on that interval. However, the converse is not necessarily true: the continuity of a function at a point or on an interval does not guarantee that it is differentiable at the point or on the interval.

### Topological Calculus: away with those nasty epsilons and deltas

Continuous functions[figure from Olver (2022a).

A new approach to calculus has recently been developed by Peter Olver of the University of Minnesota. He calls it “Continuous Calculus” but indicates that the name “Topological Calculus” is also appropriate. He has provided an extensive set of notes, which are available online (Olver, 2022a)].

### The 3-sphere: Extrinsic and Intrinsic Forms

Figure 1. An extract from Einstein’s 1917 paper on cosmology.

The circle in two dimensions and the sphere in three are just two members of an infinite family of hyper-surfaces. By analogy with the circle ${\mathbb{S}^1}$ in the plane ${\mathbb{R}^2}$ and the sphere ${\mathbb{S}^2}$ in three-space ${\mathbb{R}^3}$, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in ${\mathbb{R}^4}$ but can also be envisaged as a non-Euclidean manifold in ${\mathbb{R}^3}$.

### Making Sound Pictures to Identify Bird Songs

Top: Audio signal with three chirps. Bottom: Time-Frequency spectrogram of signal.

A trained musician can look at a musical score and imagine the sound of an entire orchestra. The score is a visual representation of the sounds. In an analogous way, we can represent birdsong by an image, and analysis of the image can tell us the species of bird singing. This is what happens with Merlin Bird ID. In a recent episode of Mooney Goes Wild, Niall Hatch of Birdwatch Ireland interviewed Drew Weber of the Cornell Lab of Ornithology, a developer of Merlin Bird ID. This phone app enables a large number of birds to be identified [TM237 or search for “thatsmaths” at irishtimes.com].

### Dynamic Equations for Weather and Climate

“I could have done it in a much more complicated way”,
said the Red Queen, immensely proud. — Lewis Carroll.

Books on dynamic meteorology and oceanography usually have a full chapter devoted to the basic dynamical equations. Since the Earth’s fluid envelop is approximately a thin spherical shell, spherical coordinates ${(\lambda,\varphi, r)}$ are convenient. Here ${\lambda}$ is the longitude and ${\varphi}$ the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):

Fig 1. The momentum equations, as in Lorenz (1967). The metric terms are boxed.

### Curl Curl Curl

Many of us have struggled with the vector differential operators, grad, div and curl. There are several ways to represent vectors and several expressions for these operators, not always easy to remember. We take another look at some of their properties here.

### X+Y and the Special Triangle

Asa Butterfield as Nathan Ellis in X+Y.

How can mathematicians grapple with abstruse concepts that are, for the majority of people, beyond comprehension? What mental processes enable a small proportion of people to produce mathematical work of remarkable creativity? In particular, is there a connection between mathematical creativity and autism? We revisit a book and a film that address these questions.

### The Navigational Skills of the Marshall Islanders

Marshallese canoe sailing on Majuro Lagoon. Image from: www.canoesmarshallislands.com

For thousands of years, the Marshall Islanders of Micronesia have been finding their way around a broadly dispersed group of low-lying islands, navigating apparently without effort from one atoll to another one far beyond the horizon. They had no maps or magnetic compass, no clocks, no weather forecasts and certainly no GPS or SatNav equipment  [TM236 or search for “thatsmaths” at irishtimes.com].

### Space-Filling Curves, Part II: Computing the Limit Function

The Approximating Functions

It is simple to define a mapping from the unit interval ${I := [0,1]}$ into the unit square ${Q:=[0,1]\times[0,1]}$. Georg Cantor found a one-to-one map from ${I}$ onto ${Q}$, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor’s map was not continuous, but Giuseppe Peano found a continuous surjection from ${I}$ onto ${Q}$, that is, a curve that fills the entire unit square. Shortly afterwards, David Hilbert found an even simpler space-filling curve, which we discussed in Part I of this post.

### Space-Filling Curves, Part I: “I see it, but I don’t believe it”

We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be located by giving its Cartesian coordinates ${(x,y)}$, or its polar coordinates ${(\rho,\theta)}$. In space, we may specify the location by giving three numbers ${(x,y,z)}$.

### Poincare’s Square and Unbounded Gomoku

Poincare’s hyperbolic disk model.

Henri Poincar’e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved outward from the centre, everything got smaller in such a way that it would take an infinite time to reach the boundary.

### Fields Medals presented at IMC 2022

Every four years, at the International Congress of Mathematicians, the Fields Medal is awarded to two, three, or four young mathematicians. To be eligible, the awardees must be under forty years of age. For the chosen few, who came from England, France, Korea and Ukraine, the award, often described as the Nobel Prize of Mathematics, is the crowning achievement of their careers [TM235 or search for “thatsmaths” at irishtimes.com].

Clockwise from top left: Maryna Viazovska, James Maynard, June Huh and Hugo Duminil-Copin. Image Credits: Mattero Fieni, Ryan Cowan, Lance Murphy.

The congress, which ran from 6th to 14th July, was originally to take place in St Petersburg. When events made that impossible, the action shifted to Helsinki and the conference presentations were moved online. The International Mathematical Union generously allowed participants to register at no cost.

### Goldbach’s Conjecture and Goldbach’s Variation

Goldbach’s Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude.

The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to Leonhard Euler. The truth of the conjecture for all even numbers up to four million million million (${4\times 10^{18}}$) has been demonstrated. There is essentially no doubt about its validity, but no proof has been found.

### The Size of Sets and the Length of Sets

Schematic diagram of ${\omega^2}$. Each line corresponds to an ordinal ${\omega\cdot m + n}$ where ${m}$ and ${n}$ are natural numbers [image Wikimedia Commons].

Cardinals and Ordinals

The cardinal number of a set is an indicator of the size of the set. It depends only on the elements of the set. Sets with the same cardinal number — or cardinality — are said to be equinumerate or (with unfortunate terminology) to be the same size. For finite sets there are no problems. Two sets, each having the same number ${n}$ of elements, both have cardinality ${n}$. But an infinite set has the definitive property that it can be put in one-to-one correspondence with a proper subset of itself.

### Can We Control the Weather?

Atmospheric motions are chaotic: a minute perturbation can lead to major changes in the subsequent evolution of the flow. How do we know this? There is just one atmosphere and, if we perturb it, we can never know how it might have evolved if left alone.

We know, from simple nonlinear models that exhibit chaos, that the flow is very sensitive to the starting conditions. We can run “identical twin” experiments, where the initial conditions for two runs are almost identical, and watch how the two solutions diverge. This — and an abundance of other evidence — leads us to the conclusion that the atmosphere behaves in a similar way.

### The Arithmetic Triangle is Analytical too

Pascal’s triangle is one of the most famous of all mathematical diagrams. It is simple to construct and rich in mathematical patterns. There is always a chance of finding something never seen before, and the discovery of new patterns is very satisfying.

Not too long ago, Harlan Brothers found Euler’s number ${e}$ in the triangle (Brothers, 2012(a),(b)). This is indeed surprising. The number ${e}$ is ubiquitous in analysis but it is far from obvious why it should turn up in the arithmetic triangle.

### ICM 2022 — Plans Disrupted but not Derailed

In just three weeks the largest global mathematical get-together will be under way. The opening ceremony of the 2022 International Congress of Mathematicians (ICM) opens on Wednesday 6 July and continues for nine days. Prior to the ICM, the International Mathematical Union (IMU) will host its 19th General Assembly in Helsinki on 3–4 July [TM234 or search for “thatsmaths” at irishtimes.com].

### Swingin’-Springin’-Twistin’-Motion

{Left: Swinging spring (three d.o.f.). Right: the Wilberforce spring (two d.o.f.).

The swinging spring, or elastic pendulum, exhibits some fascinating dynamics. The bob is free to swing like a spherical pendulum, but also to bounce up and down due to the stretching action of the spring. The behaviour of the swinging spring has been described in a previous post on this blog [Reference 1 below].

### Parity of the Real Numbers: Part I

In some recent posts, here and here we discussed the extension of the concept of parity (Odd v. Even) from the integers to the rational numbers. We found that it is natural to consider three parity classes, determined by the parities of the numerator and denominator of a rational number ${q = m / n}$ (in reduced form):

• q Odd: ${m}$ odd and ${n}$ odd.
• q Even: ${m}$ even and ${n}$ odd.
• q None: ${m}$ odd and ${n}$ even.

or, in symbolic form,

$\displaystyle \mbox{Odd} = \frac{odd}{odd} \,, \qquad \mbox{Even} = \frac{even}{odd} \,, \qquad \mbox{None} = \frac{odd}{even}$

Here, ${None}$ stands for “Neither Odd Nor Even”.

### Fairy Lights on the Farey Tree

Fairy Lights on the Farey Tree. Parity types are coloured as follows: Even: Blue; Odd: Green; None: Red.

The rational numbers ${\mathbb{Q}}$ are dense in the real numbers ${\mathbb{R}}$. The cardinality of rational numbers in the interval ${(0,1)}$ is ${\boldsymbol{\aleph}_0}$. We cannot list them in ascending order, because there is no least rational number greater than ${0}$.

### Image Processing Emerges from the Shadows

Satellite images are of enormous importance in military contexts. A battery of mathematical and image-processing techniques allows us to extract information that can play a critical role in tactical planning and operations. The information in an image may not be immediately evident. For example, an overhead image gives no direct information about the height of buildings or industrial installations, but shadows, together with the time, date and basic trigonometry, enable heights to be determined  [TM233 or search for “thatsmaths” at irishtimes.com].

### Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes

In last week’s post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found — even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal density in ${\mathbb{Q}}$. This article is a condensation of part of a paper [Lynch & Mackey, 2022] recently posted on arXiv.

### Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes

We define an extension of parity from the integers to the rational numbers. Three parity classes are found — even, odd and none. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure.

### A Finite but Unbounded Universe

Henri Poincaré described a beautiful geometric model with some intriguing properties. He envisioned a circular disk in the Euclidean plane, where distances were distorted to give it geometric properties quite different from those of Euclid’s Elements. He supposed that the temperature varied linearly from a fixed value at the centre of the disk to absolute zero on the boundary, and that lengths varied in proportion to the temperature.

### The Whole is Greater than the Part — Or is it?

Euclid flourished about fifty years after Aristotle and was certainly familiar with Aristotle’s Logic.  Euclid’s organization of the work of earlier geometers was truly innovative. His results depended upon basic assumptions, called axioms and “common notions”. There are in total 23 definitions, five axioms and five common notions in The Elements. The axioms, or postulates, are specific assumptions that may be considered as self-evident, for example “the whole is greater than the part”  [TM232 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘The Whole is Greater than the Part — Or is it?’

### Following the Money around the Eurozone

Take a fistful of euro coins and examine the obverse sides; you may be surprised at the wide variety of designs. The eurozone is a monetary union of 19 member states of the European Union that have adopted the euro as their primary currency. In addition to these countries, Andorra, Monaco, San Marino and Vatican City use euro coins so, from 2015, there have been 23 countries, each with its own national coin designs. For the €1 and €2 coins, there are 23 distinct national patterns; for the smaller denominations, there are many more. Thus, there is a wide variety of designs in circulation.

National designs of Finland, France, Germany, Ireland and Netherlands.

### Mamikon’s Visual Calculus and Hamilton’s Hodograph

[This is a condensed version of an article [5] in Mathematics Today]

A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon’s Theorem states that `The area of a tangent sweep of a curve is equal to the area of its tangent cluster’.  We shall illustrate how this theorem can help to solve a range of integration problems.

### Infinitesimals: vanishingly small but not quite zero

Abraham Robinson (1918-1974)  and his book, first published in 1966.

A few weeks ago, I wrote about  Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two.

### The Chromatic Number of the Plane

To introduce the problem in the title, we begin with a quotation from the Foreword, written by Branko  Grünbaum, to the book by Alexander Soifer (2009): The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators:

If each point of the plane is to be given a color, how many colors do we need if every two points at unit distance are to receive distinct colors?

About 70 years ago it was shown that the least number of colours needed for such a colouring is one of 4, 5, 6 and 7. But which of these is the correct number? Despite efforts by many very clever people, some of whom had solved problems that appeared to be much harder, no advance has been made to narrow the gap

${4\le\chi\le 7}$.

### The Improbability Principle and the Seanad Election

A by-election for the Seanad Éireann Dublin University constituency, arising from the election of Ivana Bacik to Dáil Éireann, is in progress. There are seventeen candidates, eight men and nine women. Examining the ballot paper, I immediately noticed an imbalance: the top three candidates, and seven of the top ten, are men. The last six candidates listed are all women. Is there a conspiracy, or could such a lopsided distribution be a matter of pure chance?

To avoid bias, the names on the ballot paper are always listed in alphabetical order. We may assume that the name of a randomly chosen candidate is equally likely to appear at any of the positions on the list; with 17 candidates, there about 6% chance for each of the 17 positions; the distribution for a single candidate is uniform. However, when several candidates are grouped, the distribution is more complicated  [TM231 or search for “thatsmaths” at irishtimes.com].

### Hyperreals and Nonstandard Analysis

Following the invention of calculus, serious concerns persisted about the mathematical integrity of the method of infinitesimals. Leibniz made liberal use of infinitesimals, with great effect, but his reasoning was felt to lack rigour. The Irish bishop George Berkeley criticised the assumptions underlying calculus, and his objections were not properly addressed for several centuries. In the 1800s, Bolzano, Cauchy and Weierstrass developed the ${\varepsilon}$${\delta}$ definition of limits and continuity, which allowed derivatives and integrals to be defined without recourse to infinitesimal quantities. Continue reading ‘Hyperreals and Nonstandard Analysis’

### A Prescient Vision of Modern Weather Forecasting

Lewis Fry Richardson in 1931.

One hundred years ago, a remarkable book was published by Cambridge University Press. It was a commercial flop: although the print run was just 750 copies, it was still in print thirty years later. Yet, it held the key to forecasting the weather by scientific means. The book, Weather Prediction by Numerical Process, was written by Lewis Fry Richardson, a brilliant, eccentric mathematician. He described in detail how the mathematical equations that govern the evolution of the atmosphere could be solved by numerical means to deduce future weather conditions from a set of observations [TM230 or search for “thatsmaths” at irishtimes.com].

### Why Waffle when One Wordle Do?

A game of Wordle solved in 3 guesses (a birdie).

Hula hoops were all the rage in 1958. Yo-yos, popular before World War II, were relaunched in the 1960s. Rubik’s Cube, invented in 1974, quickly became a global craze. Sudoku, which had been around for years, was wildly popular when it started to appear in American and European newspapers in 2004.