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.

Follow on twitter: @thatsmaths

Tags: 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.

Tags: 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].

Tags: 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’

Tags: 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’

Tags: 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.

Tags: 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.

Tags: 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’

Tags: Analysis

Tags: 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’

Tags: 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’

Tags: 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.

Tags: 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

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

Tags: 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”’

Tags: 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.

Tags: 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.

Tags: 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.

Tags: 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 .

Tags: 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’

Tags: 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.

Tags: 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.

Tags: 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?’

Tags: 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.

Tags: Analysis, Mechanics

[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.

Continue reading ‘Mamikon’s Visual Calculus and Hamilton’s Hodograph’

Tags: Analysis, History

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

Continue reading ‘Infinitesimals: vanishingly small but not quite zero’

Tags: Analysis, Combinatorics, Topology

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

.

Tags: Social attitudes, Statistics

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].

Continue reading ‘The Improbability Principle and the Seanad Election’

Tags: Analysis, Logic

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 – definition of limits and continuity, which allowed derivatives and integrals to be defined without recourse to infinitesimal quantities. Continue reading ‘Hyperreals and Nonstandard Analysis’

Tags: Numerical Weather Prediction

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].

Continue reading ‘A Prescient Vision of Modern Weather Forecasting’

Tags: Algorithms, Recreational Maths

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.

Where does new mathematics come from? The great French mathematician Henri Poincaré, a brilliant expositor of the scientific method, described how he grappled for months with an arcane problem in function theory. Exasperated by lack of progress, he went on vacation and forgot about the problem. But, as he was boarding a bus in Caen, the answer came to him in a flash. He was later able to return to his office and complete a proof of the result [TM229 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Sources and Scenes of Mathematical Inspiration’

Tags: Astronomy, Spherical Trigonometry

In his scientific best-seller, *A Brief History of Time, *Stephen Hawking remarked that every equation he included would halve sales of the book, so he put only one in it, Einstein’s equation relating mass and energy, *E *= *mc*^{2}. This cynical view is a disservice to science; we should realize that, far from being inimical, equations are our friends [TM228 or search for “thatsmaths” at irishtimes.com].

We are all familiar with splitting natural numbers into prime components. This decomposition is unique, except for the order of the factors. We can apply the idea of prime components to many more general sets of numbers.

The **Gaussian integers** are all the complex numbers with integer real and imaginary parts, that is, all numbers in the set

The set forms a two-dimensional lattice in the complex plane. For any element we consider the four numbers as *associates*. The associates of are known as units: .

Tags: Euler, History

The great Swiss mathematician Leonhard Euler produced profound and abundant mathematical works. Publication of his *Opera Omnia* began in 1911 and, with close to 100 volumes in print, it is nearing completion. Although he published several successful mathematical textbooks, the book that attracted the widest readership was not a mathematical work, but a collection of letters [TM227 or search for “thatsmaths” at irishtimes.com].

For several years, starting in 1760, Euler wrote a series of letters to Friederike Charlotte, Princess of Brandenburg-Schwedt, a niece of Frederick the Great of Prussia. The collection of 234 letters was first published in French, the language of the nobility, as *Lettres à une Princesse d’Allemagne*. This remarkably successful popularisation of science appeared in many editions, in several languages, and was widely read. Subtitled “On various subjects in physics and philosophy”, the first two of three volumes were published in 1768 by the Imperial Academy of Sciences in St. Petersburg, with the support of the empress, Catherine II.

Continue reading ‘Letters to a German Princess: Euler’s Blockbuster Lives On’

Tags: Euler, History

It all began with an invitation to Leonhard Euler to accept a chair of mathematics at the new Imperial Academy of Science in the city founded by Peter the Great. Euler’s journey from Basel to Saint Petersburg was a highly influential factor for the development of the mathematical sciences. The journey is described in detail in a full-length biography of Euler by Ronald Calinger (2016). The account below is heavily dependent on Calinger’s book.

Tags: Social attitudes

What are mathematicians really like? What are the characteristics or traits of personality typical amongst them? Mathematicians are rarely the heroes of novels, so we have little to learn from literature. A few films have featured mathematicians, but most give little insight into the personalities of their subjects [TM226 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Some Characteristics of the Mathematical Psyche’

Tags: Analysis

It is a simple matter to post a paper on **arXiv.org** claiming to prove Goldbach’s Conjecture, the Twin Primes Conjecture or any of a large number of other interesting hypotheses that are still open. However, unless the person posting the article is well known, it is likely to be completely ignored.

Mathematicians establish their claims and convince their colleagues by submitting their work to peer-reviewed journals. The work is then critically scrutinized and evaluated by mathematicians familiar with the relevant field, and is either accepted for publication, sent back for correction or revision or flatly rejected.

Continue reading ‘De Branges’s Proof of the Bieberbach Conjecture’

Tags: Euler, Number Theory

In 1740, French mathematician Philippe Naudé wrote to Leonhard Euler asking in how many ways a positive integer can be written as a sum of distinct numbers. In his investigations of this, Euler established the theory of partitions, for which he used the term *partitio numerorum.*

Many of Euler’s results in number theory involved divergent series. He was courageous in manipulating these but had remarkable insight and, almost invariably, his findings, although not rigorously established, were valid.

** Partitions**

In number theory, a *partition* of a positive integer is a way of writing as a sum of positive integers. The order of the summands is ignored: two sums that differ only in their order are considered the same partition.

Continue reading ‘Number Partitions: Euler’s Astonishing Insight’

Tags: History, Probability

Jakob Bernoulli, head of a dynasty of brilliant scholars, was one of the world’s leading mathematicians. Bernoulli’s great work, *Ars Conjectandi*, published in 1713, included a profound result that he established “after having meditated on it for twenty years”. He called it his “golden theorem”. It is known today as the law of large numbers, and it was the first limit theorem in probability, and the first attempt to apply probability outside the realm of games of chance [TM225 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Bernoulli’s Golden Theorem and the Law of Large Numbers’