Circles, polygons and the Kepler-Bouwkamp constant

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is {\cos \pi/3 = 0.5}. More generally, for an N-gon the ratio is easily shown to be {\cos \pi/N}. Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the Mysterium Cosmographicum here).

Kepler was unable to construct an accurate model using polygons, but he noted that, if successive polygons with an increasing number of sides were inscribed within circles, the ratio did not diminish indefinitely but appeared to tend towards some limiting value. Likewise, if the polygons are circumscribed, forming successively larger circles (see Figure below), the ratio tends towards the inverse of this limit. It is only relatively recently that the limit, now known as the Kepler-Bouwkamp constant, has been established. 

Continue reading ‘Circles, polygons and the Kepler-Bouwkamp constant’

Was Space Weather the cause of the Titanic Disaster?

Space weather, first studied in the 1950’s, has grown in importance with recent technological advances. It concerns the influence on the Earth’s magnetic field and upper atmosphere of events on the Sun. Such disturbances can enhance the solar wind, which interacts with the magnetosphere, with grave consequences for navigation. Space weather affects the satellites of the Global Positioning System, causing serious navigation problems [TM208 or search for “thatsmaths” at irishtimes.com].

Solar disturbances disrupt the Earth’s magnetic field [Image: ESA].
Continue reading ‘Was Space Weather the cause of the Titanic Disaster?’

The Dimension of a Point that isn’t there

A slice of Swiss cheese has one-dimensional holes;
a block of Swiss cheese has two-dimensional holes.

What is the dimension of a point? From classical geometry we have the definition “A point is that which has no parts” — also sprach Euclid. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on.

Continue reading ‘The Dimension of a Point that isn’t there’

Making the Best of Waiting in Line

Queueing system with several queues, one for each serving point [Wikimedia Commons].

Queueing is a bore and waiting to be served is one of life’s unavoidable irritants. Whether we are hanging onto a phone, waiting for response from a web server or seeking a medical procedure, we have little choice but to join the queue and wait. It may surprise readers that there is a well-developed mathematical theory of queues. It covers several stages of the process, from patterns of arrival, through moving gradually towards the front, being served and departing  [TM207 or search for “thatsmaths” at irishtimes.com].

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Differential Forms and Stokes’ Theorem

Elie Cartan (1869–1951).

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds in {n} dimensions.

The fundamental theorem of calculus on manifolds is called Stokes’ Theorem. It is a generalization of the theorem in three dimensions. In essence, it says that the change on the boundary of a region of a manifold is the sum of the changes within the region. We will discuss the basis for the theorem and then the ideas of exterior calculus that allow it to be generalized. Finally, we will use exterior calculus to write Maxwell’s equations in a remarkably compact form.

Continue reading ‘Differential Forms and Stokes’ Theorem’

Goldbach’s Conjecture: if it’s Unprovable, it must be True

The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of logic. The axiomatic method has dominated mathematics ever since [TM206 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Goldbach’s Conjecture: if it’s Unprovable, it must be True’

Mamikon’s Theorem and the area under a cycloid arch

The cycloid, the locus of a point on the rim of a rolling disk.

The Cycloid

The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are

\displaystyle x = r (\theta - \sin\theta)\,, \qquad y = r (1 - \cos\theta ) \ \ \ \ \ (1)

where {\theta} is the angle through which the disk has rotated. The centre of the disk is at {(x_0,y_0) = (r\theta, r)}.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
Full details and links to suppliers at
http://logicpress.ie/2020-3/

>>  Review in The Irish Times  <<

* * * * *

 

Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’

Machine Learning and Climate Change Prediction

Current climate prediction models are markedly defective, even in reproducing the changes that have already occurred. Given the great importance of climate change, we must identify the causes of model errors and reduce the uncertainty of climate predictions [TM205 or search for “thatsmaths” at irishtimes.com].

Schematic diagram of some key physical processes in the climate system.

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Apples and Lemons in a Doughnut

A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let {r} be the radius of the circle and {R} the distance from the axis to the centre of the circle, with {R>r}.

Generating a ring torus by rotating a circle of radius {r} about an axis at distance {R>r} from its centre.

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Complexity: are easily-checked problems also easily solved?

From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm. An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a cake (output). For a computer algorithm, the inputs are the known numerical quantities and the output is the required solution [TM204 or search for “thatsmaths” at irishtimes.com].

Al Khwarizmi, Persian polymath (c. 780 – 850) [image, courtesy of Prof. Irfan Shahid].

Continue reading ‘Complexity: are easily-checked problems also easily solved?’

Euler’s Product: the Golden Key

The Golden Key

The Basel problem was solved by Leonhard Euler in 1734 [see previous post]. His line of reasoning was ingenious, with some daring leaps of logic. The Basel series is a particular case of the much more general zeta function, which is at the core of the Riemann hypothesis, the most important unsolved problem in mathematics.

Euler treated the Taylor series for {\sin x} as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result

\displaystyle \frac{\sin x}{x} = \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n}}{(2n+1)!} = \prod_{n=1}^{\infty} \left( 1 - \frac{x^{2} }{(n\pi)^2} \right) \nonumber \ \ \ \ \ (1)

This enabled him to deduce the remarkable result

\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \left( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots \right) = \frac{\pi^2}{6}

which he described as an unexpected and elegant formula.

Continue reading ‘Euler’s Product: the Golden Key’

Euler: a mathematician without equal and an overall nice guy

Mathematicians are an odd bunch. Isaac Newton was decidedly unpleasant, secretive and resentful while Carl Friedrich Gauss, according to several biographies, was cold and austere, more likely to criticize than to praise. It is frequently claimed that a disproportionate number of mathematicians exhibit signs of autism and have significant difficulties with social interaction and everyday communication [TM203 or search for “thatsmaths” at irishtimes.com].

It is true that some of the greatest fit this stereotype, but the incomparable Leonhard Euler is a refreshing counter-example. He was described by his contemporaries as a generous man, kind and loving to his 13 children and maintaining his good-natured disposition even after he became completely blind. He is comforting proof that a neurotic personality is not essential for mathematical prowess.

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The Basel Problem: Euler’s Bravura Performance

The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to {\ln 2}. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,

\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \ ?

It is not immediately clear that this series converges, but this can be proved without much difficulty, as was first shown by Jakob Bernoulli in 1689. The sum is approximately 1.645 which has no obvious interpretation.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch has just appeared.
Full details and links to suppliers at
http://logicpress.ie/2020-3/

* * * * *

Continue reading ‘The Basel Problem: Euler’s Bravura Performance’

We are living at the bottom of an ocean

Anyone who lives by the sea is familiar with the regular ebb and flow of the tides. But we all live at the bottom of an ocean of air. The atmosphere, like the ocean, is a fluid envelop surrounding the Earth, and is subject to the influence of the Sun and Moon. While sea tides have been known for more than two thousand years, the discovery of tides in the atmosphere had to await the invention of the barometer  [TM202 or search for “thatsmaths” at irishtimes.com].

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Derangements and Continued Fractions for e

We show in this post that an elegant continued fraction for {e} can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position.

Continue reading ‘Derangements and Continued Fractions for e’

Arrangements and Derangements

Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer — about 37% whatever the number of students — emerges from the theory of derangements.

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On what Weekday is Christmas? Use the Doomsday Rule

An old nursery rhyme begins “Monday’s child is fair of face / Tuesday’s child is full of grace”. Perhaps character and personality were determined by the weekday of birth. More likely, the rhyme was to help children learn the days of the week. But how can we determine the day on which we were born without the aid of computers or calendars? Is there an algorithm – a recipe or rule – giving the answer? [TM201 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘On what Weekday is Christmas? Use the Doomsday Rule’

Will RH be Proved by a Physicist?

The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, {\Re(s) = 1/2}. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:

The non-trivial zeros of {\zeta(s)} are the
eigenvalues of a hermitian operator.

Is there a Riemann operator? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?

This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!

Continue reading ‘Will RH be Proved by a Physicist?’

Decorating Christmas Trees with the Four Colour Theorem

When decorating our Christmas trees, we aim to achieve an aesthetic balance. Let’s suppose that there is a plenitude of baubles, but that their colour range is limited. We could cover the tree with bright shiny balls, but to have two baubles of the same colour touching might be considered garish. How many colours are required to avoid such a catastrophe? [TM200 or search for “thatsmaths” at irishtimes.com].

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Laczkovich Squares the Circle

The phrase `squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.

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Ireland’s Mapping Grid in Harmony with GPS

The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential  [TM199 or search for “thatsmaths” at irishtimes.com].

Transverse Mercator projection with central meridian at Greenwich.

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Aleph, Beth, Continuum

Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are “the same size”. For two finite sets, if there is a one-to-one correspondence — or bijection — between them, they have the same number of elements. Cantor extended this equivalence to infinite sets.

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Weather Forecasts get Better and Better

Weather forecasts are getting better. Fifty years ago, predictions beyond one day ahead were of dubious utility. Now, forecasts out to a week ahead are generally reliable  [TM198 or search for “thatsmaths” at irishtimes.com].

Anomaly correlation of ECMWF 500 hPa height forecasts over three decades [Image from ECMWF].

Careful measurements of forecast accuracy have shown that the range for a fixed level of skill has been increasing by one day every decade. Thus, today’s one-week forecasts are about as good as a typical three-day forecast was in 1980. How has this happened? And will this remarkable progress continue?

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The p-Adic Numbers (Part 2)

Kurt Hensel (1861-1941)

Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.

Hensel is best known for his introduction of the p-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, p-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s p-adics were first described in 1897, and much more completely in his books, Theorie der algebraischen Zahlen, published in 1908 and Zahlentheorie published in 1913.

Continue reading ‘The p-Adic Numbers (Part 2)’

The p-Adic Numbers (Part I)

Image from Cover of Katok, 2007.

The motto of the Pythagoreans was “All is Number”. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers {\mathbb{N}}, and ratios of these, the positive rational numbers {\mathbb{Q}^{+}}. It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.

If we arrange the rational numbers on a line, there are gaps everywhere. We can fill these gaps by introducing additional numbers, which are the limits of sequences of rational numbers. This process of completion gives us the real numbers \mathbb{R}, which include rationals, irrationals like {\sqrt{2}} and transcendental numbers like {\pi}.

Continue reading ‘The p-Adic Numbers (Part I)’

Terence Tao to deliver the Hamilton Lecture

Pick a number; if it is even, divide it by 2; if odd, triple it and add 1. Now repeat the process, each time halving or else tripling and adding 1. Here is a surprise: no matter what number you pick, you will eventually arrive at 1. Let’s try 6: it is even, so we halve it to get 3, which is odd so we triple and add 1 to get 10. Thereafter, we have 5, 16, 8, 4, 2 and 1. From then on, the value cycles from 1 to 4 to 2 and back to 1 again, forever. Numerical checks have shown that all numbers up to one hundred million million million reach the 1–4–2–1 cycle  [TM197 or search for “thatsmaths” at irishtimes.com].

Fields Medalist Professor Terence Tao.

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From Impossible Shapes to the Nobel Prize

Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.

Impossible triangle sculpture in Perth, Western Australia [image Wikimedia Commons].

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.

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Mathematics and the Nature of Physical Reality

Applied mathematics is the use of maths to address questions and solve problems outside maths itself. Counting money, designing rockets and vaccines, analysing internet traffic and predicting the weather all involve maths. But why does this work? Why is maths so successful in describing physical reality? How is it that the world can be understood mathematically? [TM196, or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Mathematics and the Nature of Physical Reality’

Doughnuts and Dumplings are Distinct: Homopoty-101

As everyone knows, a torus is different from a sphere. Topology is the study of properties that remain unchanged under continuous distortions. A square can be deformed into a circle or a sphere into an ellipsoid, whether flat like an orange or long like a lemon or banana.

Continue reading ‘Doughnuts and Dumplings are Distinct: Homopoty-101’

Will mathematicians be replaced by computers?

There are ongoing rapid advances in the power and versatility of AI or artificial intelligence. Computers are now producing results in several fields that are far beyond human capability. The trend is unstoppable, and is having profound effects in many areas of our lives. Will mathematicians be replaced by computers?  [TM195 or search for “thatsmaths” at irishtimes.com].

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TeX: A Boon for Mathematicians

Donald E Knuth, designer of the \TeX mathematical typesetting system.

Mathematicians owe a great debt of gratitude to Donald Knuth. A renowned American computer scientist and mathematician, Knuth is an emeritus professor at Stanford University. He is author of many books, including the multi-volume work, The Art of Computer Programming.

Knuth is the author of the powerful and versatile mathematical typesetting system called TeX. The original version, designed and written by Knuth, was released in 1978.

TeX is a powerful system for typesetting mathematical formulae. It is ideal both for simple mathematical notes with few formulas and for more complex documents and books involving subtle and sophisticated mathematical typography. TeX is used by almost all research mathematicians. It is also popular in computer science, engineering, physics, statistics, and and many other sciences.

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Suitable Names for Large Numbers

One year ago, there were just two centibillionaires, Jeff Bezos and Bill Gates. Recently, Facebook’s Mark Zuckerberg has joined the Amazon and Microsoft founders. Elon Musk, CEO of Tesla and SpaceX, is tipped to be next to join this exclusive club [TM194 or search for “thatsmaths” at irishtimes.com].

Shot from “A Suitable Boy” with Maan Kapoor (Ishaan Khatter), Mrs. Mahesh Kapoor (Geeta Agarwal) and Bhaskar (Yusuf Akhtar), covered in colours during the Holi festival [image from Instagram.  See also here].

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Jung’s Theorem: Enclosing a Set of Points

Let us imagine that we have a finite set {P} of points in the plane {\mathbb{R}^2} (Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle?  The answer is given by Jung’s Theorem.


Left: a set P of points in the real plane. Right: The span s is the maximum distance between two points of P.

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Is There Anyone Out There? The Drake Equation gives a Clue

The Drake Equation is a formula for the number of developed civilizations in our galaxy, the Milky Way. This number is determined by seven factors. Some are known with good accuracy but the values of most are quite uncertain. It is a simple equation comprising seven terms multiplied together [TM193 or search for “thatsmaths” at irishtimes.com].

Drake-Equation-Plaque-NRAO

A plaque commemorating the first appearance of the Drake Equation at a conference at the National Radio Astronomy Observatory, Green Bank, West Virginia in 1961.

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Think of a Number: What are the Odds that it is Even?

Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime?

EulerProbDist

Probability distribution {P(n)=1/(\zeta(s)n^s)} for s=1.1 (red), s=1.01 (blue) and s=1.001 (black).

Continue reading ‘Think of a Number: What are the Odds that it is Even?’

Resolution of Paradox: a Gateway to Mathematical Progress

Weyl-49-Cover

Hermann Weyl (1885-1955)

A paradox is a statement that appears to contradict itself, or that is counter-intuitive. The analysis of paradoxes has led to profound developments in mathematics and logic. One of the richest sources of paradox is the concept of infinity. Hermann Weyl, one of the most brilliant mathematicians of the twentieth century, defined mathematics as “the science of the infinite”  [TM192 or search for “thatsmaths” at irishtimes.com].

Ever since there has been time to wonder, humankind has been intrigued by the enigma of infinity. The idea has perplexed philosophers and mystified mathematicians for millennia. The Greeks were contemplating infinity from the time of Pythagoras. Head-on confrontation of infinity led to contradictory conclusions; the paradoxes of Zeno are amongst the most noted of these.

Continue reading ‘Resolution of Paradox: a Gateway to Mathematical Progress’

Berry’s Paradox and Gödel’s Incompleteness Theorem

 

Chaitin-Boolos

Left: Argentine-American mathematician
Gregory Chaitin [image from here]. Right: American philosopher and logician
George Boolos [image Wikimedia Commons].

A young librarian at the Bodleian Library in Oxford devised an intriguing paradox. He defined a number by means of a statement of the form

THE SMALLEST NATURAL NUMBER THAT CANNOT BE
DEFINED IN FEWER THAN TWENTY WORDS.

Continue reading ‘Berry’s Paradox and Gödel’s Incompleteness Theorem’

Does Numerical Integration Reflect the Truth?

Many problems in applied mathematics involve the solution of a differential equation. Simple differential equations can be solved analytically: we can find a formula expressing the solution for any value of the independent variable. But most equations are nonlinear and this approach does not work; we must solve the equation by approximate numerical means. The big question is:

Does the numerical solution resemble the true solution of the equation?

The answer is: “Not necessarily”.

There are often specific criteria that must be satisfied to ensure that the answer `crunched out’ by the computer is a reasonable approximation to reality. Although the principles of numerical stability are quite general, they are best illustrated by simple examples. We will look at some of these below.

CFL-Criterion-1

Smooth curve: True solution. Black dots: stable solution. Red dots: unstable solution (time step too large).

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Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Lanczos240In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at irishtimes.com].

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Buridan’s Ass

Jean-Buridan

Jean Buridan (c. 1300-1360).

“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.

Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.

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The Ever-growing Goals of Googology

In 1920, a kindergarten class was asked to describe the biggest number that they could imagine. One child proposed to “write down digits until you get tired”. A more concrete idea was to write a one followed by 100 zeros. This number, which scientists would express as ten to the power 100, was given the name “googol” by its inventor [TM190; or search for “thatsmaths” at irishtimes.com ].

OneGoogol

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The Online Encyclopedia of Integer Sequences

Suppose that, in the course of an investigation, you stumble upon a string of whole numbers. You are convinced that there must be a pattern, but you cannot find it. All you have to do is to type the string into a database called OEIS — or simply “Slone’s” — and, if the string is recognized, an entire infinite sequence is revealed. If the string belongs to several sequences, several choices are offered. OEIS is a great boon to both professional mathematicians and applied scientists in fields like physics, chemistry, astronomy and biology.

OEIS-Homepage

The Home page of OEIS: https://oeis.org

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The Geography of Europe is Mapped in our Genes

It may seem too much to expect that a person’s geographic origin can be determined from a DNA sample. But, thanks to a mathematical technique called principal component analysis, this can be done with remarkable accuracy. It works by reducing multi-dimensional data sets to just a few variables  [TM189; or search for “thatsmaths” at irishtimes.com ].

GenesGeography-2

Predicted locations for more than 1200 individuals, based on DNA markers in their genome (figure from Nature).

Continue reading ‘The Geography of Europe is Mapped in our Genes’

Dimension Reduction by PCA

We live in the age of “big data”. Voluminous data collections are mined for information using mathematical techniques. Problems in high dimensions are hard to solve — this is called “the curse of dimensionality”. Dimension reduction is essential in big data science. Many sophisticated techniques have been developed to reduce dimensions and reveal the information buried in mountains of data.

Correlated-Variables

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Pooling Expertise to Tackle Covid-19

Our lives have been severely restricted in recent months. We are assured that the constraints have been imposed following “the best scientific advice”, but what is the nature of this advice? Among the most important scientific tools used for guidance on the Covid-19 outbreak are mathematical models  [TM188; or search for “thatsmaths” at irishtimes.com ].

IEMAG-Nolan

Prof Philip Nolan, Chairman of IEMAG (Photograph: Tom Honan

Continue reading ‘Pooling Expertise to Tackle Covid-19’

The Monte-Carlo Method

Learning calculus at school, we soon find out that while differentiation is relatively easy, at least for simple functions, integration is hard. So hard indeed that, in many cases, it is impossible to find a nice function that is the integral (or anti-derivative) of a given one. Thus, given {f(x)} we can usually find {d f /d x}, whereas we may not be able to find {\int f(x)\,d x}.

Monte-Carlo-Wide-4panel

Continue reading ‘The Monte-Carlo Method’

Changing the way that we look at the world

Albrecht-Durer-26

Self-portrait by Dürer when aged 26.

Albrecht Dürer was born in Nuremberg in 1471, third of a family of eighteen children. Were he still living, he would be celebrating his 549th birthday today. Dürer’s artistic genius was clear from an early age, as evidenced by a self-portrait he painted when just thirteen [TM187; or search for “thatsmaths” at irishtimes.com ].

In 1494, Dürer visited Italy, where he travelled for a year. A novel connection between art and mathematics was emerging around that time. By using rules of perspective, artists could represent objects in three-dimensional space on a plane canvas with striking realism. Dürer was convinced that the new art must be based upon science; in particular, upon mathematics, as the most exact, logical, and graphically constructive of the sciences”.

Continue reading ‘Changing the way that we look at the world’

A New Perspective on Perspective

The development of perspective in the early Italian Renaissance opened the doors of perception just a little wider. Perspective techniques enabled artists to create strikingly realistic images. Among the most notable were Piero della Francesca and Leon Battista Alberti, who invented the method of perspective drawing.

School-of-Athens-MidRes

School of Athens, a fresco painted by Raphael in 1509-11 illustrates the power of perspective.

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John Casey: a Founder of Modern Geometry

John-Casey-01

John Casey (1820-1891).

Next Tuesday – 12th May – is the 200th anniversary of the birth of John Casey, a notable Irish geometer. Casey was born in 1820 in Kilbeheny, Co Limerick. He was educated in nearby Mitchelstown, where he showed great aptitude for mathematics and also had a gift for languages. He became a mathematics teacher, first in Tipperary Town and later in Kilkenny [TM186; or search for “thatsmaths” at irishtimes.com ].

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Order in the midst of Chaos

We open with a simple mathematical puzzle that is easily solved using only elementary reasoning. Imagine a party where some guests are friends while others are unacquainted. Then the following is always true:

No matter how many guests there are at the party, there are
always two guests with the same number of friends present.

If you wish, try proving this before reading on. The proof is outlined at the end of this post.

Complete-Graphs-6-10

Complete graphs with 6 to 10 vertices.

Continue reading ‘Order in the midst of Chaos’


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