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Spiral of Theodorus [image Wikimedia Commons].
The square-root spiral is attributed to Theodorus, a tutor of Plato. It comprises a sequence of right-angled triangles, placed edge to edge, all having a common point and having hypotenuse lengths equal to the roots of the natural numbers.
The spiral is built from right-angled triangles. At the centre is an isosceles triangle of unit side and hypotenuse 
Rene Descartes
There are numerous branches of mathematics, from arithmetic, geometry and algebra at an elementary level to more advanced fields like number theory, topology and complex analysis. Each branch has its own distinct set of axioms, or fundamental assumptions, from which theorems are derived by logical processes. While each branch has its own flavour, character and methods, there are also strong overlaps and interdependencies. Several attempts have been made to construct a grand unified theory that embraces the entire field of maths [TM220 or search for “thatsmaths” at irishtimes.com].
In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by curves and volumes with curved surfaces were mostly beyond the scope of Euclid. Only a few volumes — for example, the sphere, cylinder and cone — could be measured using classical methods.
If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is 
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’
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
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.
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
where 
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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 <<
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Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’
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 
This enabled him to deduce the remarkable result
which he described as an unexpected and elegant formula.
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 
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’
The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, 
The non-trivial zeros of 
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!
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.
In 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].
Continue reading ‘Cornelius Lanczos – Inspired by Hamilton’s Quaternions’
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.
The Home page of OEIS: https://oeis.org
Continue reading ‘The Online Encyclopedia of Integer Sequences’
In its initial stages, the Covid-19 pandemic grew at an exponential rate. What does this mean? The number of infected people in a country is growing exponentially if it increases by a fixed multiple R each day: if N people are infected today, then R times N are infected tomorrow. The size of the growth-rate R determines how rapidly the virus is spreading. An example should make this clear [TM185 or search for “thatsmaths” at irishtimes.com].
“Flattening the curve” [image from ECDC].
We take a fresh look at the vector differential operators grad, div and curl. There are many vector identities relating these. In particular, there are two combinations that always yield zero results:
Question: Is there a connection between these identities?
Vector analysis can be daunting for students. The theory can appear abstract, and operators like Grad, Div and Curl seem to be introduced without any obvious motivation. Concrete examples can make things easier to understand. Weather maps, easily obtained on the web, provide real-life applications of vector operators.
Fig. 1. An idealized scalar field representing the mean sea-level atmospheric pressure over the North Atlantic area.
Continue reading ‘Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis’
Mathematicians have traditionally dealt with convergent series and shunned divergent ones. But, long ago, astronomers found that divergent expansions yield valuable results. If these so-called asymptotic expansions are truncated, the error is bounded by the first term omitted. Thus, by stopping just before the smallest term, excellent approximations may be obtained.
Introduction
The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.
Originally, Carl Gustav Jacobi defined the elliptic functions 
He called 
Continue reading ‘Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”’
Every branch of mathematics has key results that are so
important that they are dubbed fundamental theorems.
The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is then used to deduce the consequences of those axioms with each major result designated as a theorem.
As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the fundamental theorem of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.
Augustin-Louis Cauchy (1789–1857)
If you love mathematics and have never studied complex function theory, then you are missing something wonderful. It is one of the most beautiful branches of maths, with many amazing results. Don’t be put off by the name: complex does not mean complicated. With elementary calculus and a basic knowledge of imaginary numbers, a whole world of wonder is within your grasp.
In the early nineteenth century, Augustin-Louis Cauchy (1789–1857) constructed the foundations of what became a major new branch of mathematics, the theory of functions of a complex variable.
Given a function 
A rational function with five real zeros and a pole at x = 1.
Bernard Bolzano (1781-1848)
Bernard Bolzano, born in Prague in 1781, was a Bohemian mathematician with Italian origins. Bolzano made several profound advances in mathematics that were not well publicized. As a result, his mathematical work was overlooked, often for many decades after his death. For example, his construction of a function that is continuous on an interval but nowhere differentiable, did not become known. Thus, the credit still goes to Karl Weierstrass, who found such a function about 30 years later. Boyer and Merzbach described Bolzano as “a voice crying in the wilderness,” since so many of his results had to be rediscovered by other workers.
Continue reading ‘Bernard Bolzano, a Voice Crying in the Wilderness’
The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory.
[Image Wikimedia Commons]
Continue reading ‘Really, 0.999999… is equal to 1. Surreally, this is not so!’
In an earlier post, we discussed Grandi’s series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series
This is a divergent series: the sequence of partial sums is 
Is the Light On or Off?
Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states “on” and “off” by
Solving PDEs by a Roundabout Route
Joseph Fourier (1768-1830)
Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only in frequency [TM136 or search for “thatsmaths” at irishtimes.com].
Breaking Complex Objects into Simple Pieces
“In a memorable session of the French Academy on the
21st of December 1807, the mathematician and engineer
Joseph Fourier announced a thesis which inaugurated a
new chapter in the history of mathematics. The claim of
Fourier appeared to the older members of the Academy,
including the great analyst Lagrange, entirely incredible.”
Introduction
Joseph Fourier (1768-1830)
The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?
We all know that division by zero is a prohibited operation, and that ratios that reduce to “zero divided by zero” are indeterminate. We probably also recall proving in elementary calculus class that
This is an essential step in deriving an expression for the derivative of
Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.
Sin t (blue) and its evolute (red).
Torricelli’s Trumpet
Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli’s Trumpet. It is the surface generated when the curve 
Continue reading ‘Torricelli’s Trumpet & the Painter’s Paradox’
Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com].
FIg. 1: Brook Taylor (1685-1731). Image from NPG.
The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his Methodus Incrementorum Directa et Inversa, published in 1715. This work contained the famous formula known today as Taylor’s formula. In 1772, Lagrange described it as “the main foundation of differential calculus” (Wikipedia: Brook Taylor). Taylor also wrote a treatise on linear perspective (see Fig. 1).
It is noteworthy that the series for 
Continue reading ‘Taylor Expansions from India’
Which is greater, 
Contour plot of x^y – y^x, positive in the yellow regions, negative in the blue ones.
It is well-known that an ellipse is the locus of all points such that the sum of their distances from two fixed points, the foci, is constant. Thus, a gardener may map out an elliptical flower-bed by driving two stakes into the ground, looping a rope around them and pulling it taut with a pointed stick, tracing out a curve on the ground.
We consider the convergence of the random harmonic series
where 
The Gudermannian is named after Christoph Gudermann (1798–1852). The Gompertz function is named after Benjamin Gompertz (1779–1865). These are two amongst several sigmoid functions. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions.
Continue reading ‘Sigmoid Functions: Gudermannian and Gompertz Curves’
We can construct a beautiful fractal set by defining an operation of iterating exponentials and applying it to the numbers in the complex plane. The operation is tetration and the fractal is called the power tower fractal or sometimes the tetration fractal. A detail of the set is shown in the figure here.
Detail of the power tower fractal.
This is a continuation of last week’s post: LINK
The complex power tower is defined by an `infinite tower’ of exponents:
The sequence of successive approximations to this function is
If the sequence 
In Part I we described an attempt to fit a logarithmic spiral to the sequence 
The function defined by an `infinite tower’ of exponents,
is called the Power Tower function. We consider the sequence of successive approximations to this function:
As 
We will construct a sequence of functions on the unit interval such that it converges uniformly to zero while the arc-lengths diverge to infinity.
Black: Frog hop. Blue: Cricket hops. Magenta: Flea hops.
Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks.
The circular functions occur throughout mathematics. Fourier showed that, under very general assumptions, an arbitrary function can be decomposed into components each of which is a circular function. The functions get their name from their use in defining a circle in parametric form: if
then 
Sine Function:
The function 
The sine function, the essence of good behaviour.
“A bicycle, certainly, but not the bicycle,” said Holmes.
In Conan-Doyle’s short story The Adventure of the Priory School Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal level of brilliance.
As front wheel moves along the positive x-axis the back wheel, initially at (0,a), follows a tractrix curve (see below).
Tap any number into your calculator. Yes, any number at all, plus or minus, big or small. Now tap the cosine button. You will get a number in the range [ -1, +1 ]. Now tap “cos” again and again, and keep tapping it repeatedly (make sure that angles are set to radians and not degrees). The result is a sequence of numbers that converge towards the value 0.739085 … .
For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s That’s Maths column in The Irish Times (TM063, or search for “thatsmaths” at irishtimes.com).
The Scottish Café, Lvov in earlier times (left), now Hotel Atlas in Lviv (right).
Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. Together they founded what would, in 1920, become the Polish Mathematical Society.
A coin and a postage stamp commemorating Stefan Banach.
Special Curves
A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space 
![{\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbf%7B%5Cgamma%7D+%3A+%5Ba%2Cb%5D%5Clongrightarrow+%5Cmathbb%7BR%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002)
Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it does not have a unique value there, the point is singular.
Slinky traces a smooth helical curve in three dimensions.
Generally, if we zoom in close to a point on a curve, the curve looks increasingly like a straight line. However, at a singularity, it may look like two lines crossing or like two lines whose slopes converge as the resolution increases. Continue reading ‘Curves with Singularities’
Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:
Conjecture: Definitions are invented. Theorems are discovered.
The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’
Next week’s post will be about a model of the future of civilization! It is based on the classical predator-prey model, which is reviewed here.
Solution for X (blue) and Y (red) for 30 time units. X(0)=0.5, Y(0)=0.2 and k=0.5.
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