The remarkable polymath Christopher Wren died in March 1723, just 300 years ago. Sarah Hart, Professor of Geometry at Gresham College, recently presented a lecture, *The Mathematical Life of Sir Christopher Wren*; a video of her presentation in available online (see sources below). The illustration above is from the Gresham College website.

### Christopher Wren and the Cycloid

Published March 23, 2023 Occasional Leave a CommentTags: Analysis, History

### Bach and Euler chat in Frederick’s Court

Published March 16, 2023 Occasional Leave a CommentTags: History, Music

Frederick the Great of Prussia, a devoted patron of the arts, had a particular interest in music, and admired the music of Johann Sebastian Bach. In 1747, Bach visited Potsdam, where his son Carl Philipp Emanuel was the Kapellmeister in Frederick’s court. When Frederick learned of this, he summoned ‘Old Bach’ to the palace and invited him to try out his collection of pianofortes. As they went from room to room, Bach improvised a new piece of music on each instrument [TM243 or search for “thatsmaths” at irishtimes.com].

### Sets that are Elements of Themselves: Verboten

Published March 9, 2023 Occasional Leave a CommentTags: Logic, Set Theory

Can a set be an element of itself? A simple example will provide an answer to this question. Continue reading ‘Sets that are Elements of Themselves: Verboten’

### Benford’s Law Revisited

Published March 2, 2023 Occasional Leave a CommentTags: Probability, Statistics

Several researchers have observed that, in a wide variety of collections of numerical data, the leading — or most significant — decimal digits are not uniformly distributed, but conform to a logarithmic distribution. Of the nine possible values, occurs more than of the time while is found in less than of cases (see Figure above). Specifically, the probability distribution is

A more complete form of the law gives the probabilities for the second and subsequent digits. A full discussion of Benford’s Law is given in Berger and Hill (2015).

### A Puzzle: Two-step Selection of a Digit

Published February 23, 2023 Occasional ClosedTags: Number Theory

Here is a simple problem in probability.

(1) Pick a number *k* between 1 and 9. Assume all digits are equally likely.

(2) Pick a number *m* in the range from 1 to *k*.

**What is the probability distribution for the number m?**

A graph of the probability distribution is shown in the figure here.

Can you derive a formula for this probability distribution?

Can you generalise it to the range from 1 to 10^n?

Can you relate this problem to Benford’s Law [described here]?

Solution, and more on Benford’s Law, next week.

### Weather Warnings in Glorious Technicolor

Published February 16, 2023 Irish Times ClosedTags: Algorithms, Geophysics

Severe weather affects us all and we need to know when to take action to protect ourselves and our property. We have become familiar with the colourful spectrum of warnings issued by Met Éireann.

For several years, Met Éireann has issued warnings of extreme weather. These depend on the severity of the meteorological event and the level of confidence in the forecast. They are formulated using forecasts produced by computer, algorithms that determine the likely impacts of extreme weather, and the expertise of the forecasters [TM242 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Weather Warnings in Glorious Technicolor’

### Ford Circles & Farey Series

Published February 9, 2023 Occasional 1 CommentTags: Geometry, Number Theory

American mathematician Lester Randolph Ford Sr. (1886–1967) was President of the Mathematical Association of America from 1947 to 1948 and editor of the American Mathematical Monthly during World War II. He is remembered today for the system of circles named in his honour.

For any rational number in reduced form ( and coprime), a Ford circle is a circle with center at and radius . There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.

Mathematics has an amazing capacity to help us to understand the physical world. Just consider the profound implications of Einstein’s simple equation . 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 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

Published January 19, 2023 Irish Times ClosedTags: Astronomy, Cosmology, Topology

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].

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

Published January 5, 2023 Occasional ClosedTags: Arithmetic, Number Theory

### Spiric curves and phase portraits

Published December 29, 2022 Occasional ClosedTags: Mechanics, Topology

*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

Published December 22, 2022 Occasional ClosedTags: Algebra, Number Theory

*When is 144 closer to 8 than to 143?*

The usual definition of the *norm* of a real number is its modulus or absolute value . We measure the “distance” between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric and, using it, we can define the usual topology on the real numbers .

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

Published December 15, 2022 Irish Times ClosedTags: Applied Maths, 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

Published December 8, 2022 Occasional ClosedTags: Geometry, Topology

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

Published December 1, 2022 Occasional ClosedTags: Analysis, Numerical Analysis

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 and asked for the next number in the sequence. Arguments were given indicating why *any* number might be regarded as the next number.

In this article we consider a sequence of seven ones: . Most people would agree that the next number in the sequence is . We will show how the number could be the “correct” answer. Continue reading ‘Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein’

### What’s the Next Number?

Published November 24, 2022 Occasional ClosedTags: Arithmetic, Numerical Analysis

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

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

### The Rich Legacy of Indian Mathematics

Published November 17, 2022 Irish Times ClosedTags: History, 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

Published November 10, 2022 Occasional ClosedTags: Astronomy, Mechanics, Relativity

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

Published November 3, 2022 Occasional ClosedTags: Algorithms, Numerical Analysis, Pi

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 , 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).]

Continue reading ‘The Power of the 2-gon: Extrapolation to Evaluate Pi’

### Dropping Pebbles down a Mine-shaft

Published October 27, 2022 Occasional ClosedTags: Geophysics, Mechanics

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

Published October 20, 2022 Irish Times , Occasional ClosedTags: Astronomy, Physics

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.

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.

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

Published October 6, 2022 Occasional ClosedTags: Analysis

A function that is differentiable at a point is continuous there, and if differentiable on an interval , 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.

Continue reading ‘CND Functions: Curves that are Continuous but Nowhere Differentiable’

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

Published September 29, 2022 Occasional ClosedTags: Analysis

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

Published September 22, 2022 Occasional ClosedTags: Relativity, Topology

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 in the plane and the sphere in three-space , we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in but can also be envisaged as a non-Euclidean manifold in .

Continue reading ‘The 3-sphere: Extrinsic and Intrinsic Forms’

### Making Sound Pictures to Identify Bird Songs

Published September 15, 2022 Irish Times ClosedTags: Algorithms, Fourier analysis

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].

Continue reading ‘Making Sound Pictures to Identify Bird Songs’

### Dynamic Equations for Weather and Climate

Published September 8, 2022 Occasional ClosedTags: Analysis, Geometry, Relativity

*“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 are convenient. Here is the longitude and the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):

Continue reading ‘Dynamic Equations for Weather and Climate’

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.

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.

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

Published August 11, 2022 Occasional ClosedTags: Analysis

It is simple to define a mapping from the unit interval into the unit square . Georg Cantor found a one-to-one map from ** onto** , 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 onto , 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.

Continue reading ‘Space-Filling Curves, Part II: Computing the Limit Function’

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

Published August 4, 2022 Occasional ClosedTags: Analysis

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 , or its polar coordinates . In space, we may specify the location by giving three numbers .

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

### Poincare’s Square and Unbounded Gomoku

Published July 28, 2022 Occasional ClosedTags: Games, Geometry, Relativity

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

Published July 21, 2022 Occasional ClosedTags: Combinatorics, Geometry, Number Theory

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].

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

Published July 14, 2022 Occasional ClosedTags: Logic, Number Theory

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 () has been demonstrated. There is essentially no doubt about its validity, but no proof has been found.

Continue reading ‘Goldbach’s Conjecture and Goldbach’s Variation’

** 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 of elements, both have cardinality . But an infinite set has the definitive property that it can be put in one-to-one correspondence with a proper subset of itself.

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

Published June 23, 2022 Occasional ClosedTags: Arithmetic, Euler

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 in the triangle (Brothers, 2012(a),(b)). This is indeed surprising. The number is ubiquitous in analysis but it is far from obvious why it should turn up in the arithmetic triangle.

Continue reading ‘The Arithmetic Triangle is Analytical too’

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].

Continue reading ‘ICM 2022 — Plans Disrupted but not Derailed’

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].

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 (in reduced form):

**q Odd:**odd and odd.**q Even:**even and odd.**q None:**odd and even.

or, in symbolic form,

Here, stands for “Neither Odd Nor Even”.

The rational numbers are dense in the real numbers . The cardinality of rational numbers in the interval is . We cannot list them in ascending order, because there is no least rational number greater than .

### Image Processing Emerges from the Shadows

Published May 19, 2022 Irish Times ClosedTags: Astronomy, Trigonometry

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].

Continue reading ‘Image Processing Emerges from the Shadows’

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

Published May 12, 2022 Occasional ClosedTags: Number Theory

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 . 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

Published May 5, 2022 Occasional ClosedTags: Number Theory

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.

Continue reading ‘Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes’

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?

Published April 21, 2022 Irish Times ClosedTags: Cantor, Geometry, Set Theory

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

Published April 14, 2022 Occasional ClosedTags: modelling, Numerical Analysis

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.