## Archive for the 'Occasional' Category

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

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

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

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

### A Finite but Unbounded Universe

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

### Following the Money around the Eurozone

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

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

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

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

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

### Infinitesimals: vanishingly small but not quite zero

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

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

### The Chromatic Number of the Plane

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

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

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

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

### Hyperreals and Nonstandard Analysis

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

### Why Waffle when One Wordle Do?

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

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

### Where is the Sun?

Ecliptic plane [Wikimedia Commons].

The position of the Sun in the sky depends on where we are and on the time of day. Due to the Earth’s rotation, the Sun appears to cross the celestial sphere each day along a path called the ecliptic. The observer’s position on Earth is given by the geographic latitude and longitude. The path of the Sun depends on the latitude and the date, while the time when the Sun crosses the local meridian is determined by the longitude.

### Gaussian Primes

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

$\displaystyle \mathbb{Z}[i] \equiv \{ m + i n : m, n \in \mathbb{Z} \} \,.$

The set ${\mathbb{Z}[i]}$ forms a two-dimensional lattice in the complex plane. For any element ${g \in \mathbb{Z}[i]}$ we consider the four numbers ${\{g, -g, ig, -ig \}}$ as associates. The associates of ${1}$ are known as units: ${\{1, -1, i, -i \}}$.

### Letters to a German Princess: Euler’s Blockbuster Lives On

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.

### Euler’s Journey to Saint Petersburg

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.

Drawing based on a map of Europe from about 1740 (from Calinger, 2016, pg. 39). Euler’s route from Basel to Saint Petersburg is marked by the heavy dashed line.

### De Branges’s Proof of the Bieberbach Conjecture

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.

### Number Partitions: Euler’s Astonishing Insight

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 ${n}$ is a way of writing ${n}$ 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.

### Set Density: are even numbers more numerous than odd ones?

In pure set-theoretic terms, the set of even positive numbers is the same size, or cardinality, as the set of all natural numbers: both are infinite countable sets that can be put in one-to-one correspondence through the mapping ${n \rightarrow 2n}$. This was known to Galileo. However, with the usual ordering,

$\displaystyle \mathbb{N} = \{ 1, 2, 3, 4, 5, 6, \dots \} \,,$

every second number is even and, intuitively, we feel that there are half as many even numbers as natural numbers. In particular, our intuition tells us that if ${B}$ is a proper subset of ${A}$, it must be smaller than ${A}$.
Continue reading ‘Set Density: are even numbers more numerous than odd ones?’

### Chiral and Achiral Knots

An object is chiral if it differs from its mirror image. The favourite example is a hand: our right hands are reflections of our left ones. The two hands cannot be superimposed. The term chiral comes from ${\chi\epsilon\rho\iota}$, Greek for hand. If chirality is absent, we have an achiral object.

According to Wikipedia, it was William Thomson, aka Lord Kelvin, who wrote:

“I call any geometrical figure, or group of points, ‘chiral‘, and say that it has chirality if its image in a plane mirror  …  cannot be brought to coincide with itself.”

### Cantor’s Theorem and the Unending Hierarchy of Infinities

The power set of the set {x,y,z}, containing all its subsets, has 2^3=8 elements. Image from Wikimedia Commons.

In 1891, Georg Cantor published a seminal paper, U”ber eine elementare Frage der Mannigfaltigkeitslehren — On an elementary question of the theory of manifolds — in which his “diagonal argument” first appeared. He proved a general theorem which showed, in particular, that the set of real numbers is uncountable, that is, it has cardinality greater than that of the natural numbers. But his theorem is much more general, and it implies that the set of cardinals is without limit: there is no greatest order of infinity.

### How to Write a Convincing Mathematical Paper

Let ${X}$ be a Banach Space

Open any mathematical journal and read the first sentence of a paper chosen at random. You will probably find something along the following lines: “Let X be a Banach space”. That is fine if you know what a Banach space is, but meaningless if you don’t.

### The Square Root Spiral of Theodorus

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 ${\sqrt{2}}$. Another triangle, with sides ${1}$ and ${\sqrt{2}}$ and hypotenuse ${\sqrt{3}}$ is stacked upon the first. This process continues, giving hypotenuse lengths ${\sqrt{n}}$ for all ${n}$.

### The Spine of Pascal’s Triangle

We are all familiar with Pascal’s Triangle, also known as the Arithmetic Triangle (AT). Each entry in the AT is the sum of the two closest entries in the row above it. The ${k}$-th entry in row ${n}$ is the binomial coefficient ${\binom{n}{k}}$ (read ${n}$-choose-${k}$), the number of ways of selecting ${k}$ elements from a set of ${n}$ distinct elements.

### Embedding: Reconstructing Solutions from a Delay Map

M

In mechanical systems described by a set of differential equations, we normally specify a complete set of initial conditions to determine the motion. In many dynamical systems, some variables may easily be observed whilst others are hidden from view. For example, in astronomy, it is usual that angles between celestial bodies can be measured with high accuracy, while distances to these bodies are much more difficult to find and can be determined only indirectly.

### The Signum Function may be Continuous

Abstract: Continuity is defined relative to a topology. For two distinct topological spaces ${(X,\mathcal{O}_1)}$ and ${(X,\mathcal{O}_2)}$ having the same underlying set ${X}$ but different families of open sets, a function may be continuous in one but discontinuous in the other. Continue reading ‘The Signum Function may be Continuous’

### Real Derivatives from Imaginary Increments

The solution of many problems requires us to compute derivatives. Complex step differentiation is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations.

Rounding error and formula error as functions of step size ${h}$ [Image from Wikimedia Commons].

For finite difference approximations, the choice of step size ${h}$ is crucial: if ${h}$ is too large, the estimate of the derivative is poor, due to truncation error; if ${h}$ is too small, subtraction will cause large rounding errors. The finite difference formulae are ill-conditioned and, if ${h}$ is very small, they produce zero values.

Where it can be applied, complex step differentiation provides a stable and accurate method for computing ${f^\prime(x)}$.

### Carnival of Mathematics

The Aperiodical is described on its About’ page as “a meeting-place for people who already know they like maths and would like to know more”. The Aperiodical coordinates the Carnival of Mathematics (CoM), a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on mathematics from around the internet. This month, it is the turn of thatsmaths.com to host CoM.

### Simple Models of Atmospheric Vortices

Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any detail, because the effects of moisture condensation and convection dominate their behaviour.

### Finding Fixed Points

An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group ${E(n)}$ is the group of isometries of ${n}$-dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane ${\mathbb{E}^2}$, we have the group ${E(2)}$, comprising all combinations of translations, rotations and reflections of the plane.

### Approximating the Circumference of an Ellipse

The realization that the circumference of a circle is related in a simple way to the diameter came at an early stage in the development of mathematics. But who was first to prove that all circles are similar, with the ratio of circumference ${C}$ to diameter ${D}$ the same for all? Searching in Euclid’s Elements, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that ${C / D}$ is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post here.

### Gauss Predicts the Orbit of Ceres

Ceres (bottom left), the Moon and Earth, shown to scale [Image NASA].

On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without success to locate it. Without accurate knowledge of its orbit, the search seemed hopeless. How could its trajectory be determined from a few observations made from the Earth, which itself was moving around the Sun?

### Al Biruni and the Size of the Earth

Al Biruni at Persian Scholars Pavilion in Vienna.

The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni’s method was brilliant and, for its era, spectacular, the accuracy claimed must be regarded with suspicion.

Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius ${a}$. He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.

### Hanoi Graphs and Sierpinski’s Triangle

The Tower of Hanoi is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:

• Only one disk can be moved at a time.
• No disk can be placed upon a smaller one.

Tower of Hanoi [image Wikimedia Commons].

Continue reading ‘Hanoi Graphs and Sierpinski’s Triangle’

### A Model for Elliptic Geometry

For many centuries, mathematicians struggled to derive Euclid’s fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.

### Can You Believe Your Eyes?

Scene from John Ford’s Stagecoach (1939).

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained?

### The Size of Things

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.

### 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].

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.

### 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.

### 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  <<

* * * * *

### 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.

### 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.

### 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/

* * * * *

### 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.

### 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!

### 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.

### 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.

### 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.

### 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 ${\pi}$.

### 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.

### 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.