Archive for the 'Occasional' Category

Topological Calculus: away with those nasty epsilons and deltas

Continuous functions[figure from Olver (2022a).

A new approach to calculus has recently been developed by Peter Olver of the University of Minnesota. He calls it “Continuous Calculus” but indicates that the name “Topological Calculus” is also appropriate. He has provided an extensive set of notes, which are available online (Olver, 2022a)].

Continue reading ‘Topological Calculus: away with those nasty epsilons and deltas’

The 3-sphere: Extrinsic and Intrinsic Forms

Figure 1. An extract from Einstein’s 1917 paper on cosmology.

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 {\mathbb{S}^1} in the plane {\mathbb{R}^2} and the sphere {\mathbb{S}^2} in three-space {\mathbb{R}^3}, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in {\mathbb{R}^4} but can also be envisaged as a non-Euclidean manifold in {\mathbb{R}^3}.

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

Dynamic Equations for Weather and Climate

“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 {(\lambda,\varphi, r)} are convenient. Here {\lambda} is the longitude and {\varphi} the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):

Fig 1. The momentum equations, as in Lorenz (1967). The metric terms are boxed.

Continue reading ‘Dynamic Equations for Weather and Climate’

Curl Curl Curl

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.

Continue reading ‘Curl Curl Curl’

X+Y and the Special Triangle

Asa Butterfield as Nathan Ellis in X+Y.

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.

Continue reading ‘X+Y and the Special Triangle’

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

The Approximating Functions

It is simple to define a mapping from the unit interval {I := [0,1]} into the unit square {Q:=[0,1]\times[0,1]}. Georg Cantor found a one-to-one map from {I} onto {Q}, 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 {I} onto {Q}, that is, a curve that fills the entire unit square. Shortly afterwards, David Hilbert found an even simpler space-filling curve, which we discussed in Part I of this post.

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

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

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 {(x,y)}, or its polar coordinates {(\rho,\theta)}. In space, we may specify the location by giving three numbers {(x,y,z)}.

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

Poincare’s Square and Unbounded Gomoku

Poincare’s hyperbolic disk model.

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.

Continue reading ‘Poincare’s Square and Unbounded Gomoku’

Fields Medals presented at IMC 2022

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].

Clockwise from top left: Maryna Viazovska, James Maynard, June Huh and Hugo Duminil-Copin. Image Credits: Mattero Fieni, Ryan Cowan, Lance Murphy.

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.

Continue reading ‘Fields Medals presented at IMC 2022’

Goldbach’s Conjecture and Goldbach’s Variation

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 ({4\times 10^{18}}) 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’

The Size of Sets and the Length of Sets

Schematic diagram of {\omega^2}. Each line corresponds to an ordinal {\omega\cdot m + n} where {m} and {n} are natural numbers [image Wikimedia Commons].

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

Continue reading ‘The Size of Sets and the Length of Sets’

Can We Control the Weather?

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.

Continue reading ‘Can We Control the Weather?’

The Arithmetic Triangle is Analytical too

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 {e} in the triangle (Brothers, 2012(a),(b)). This is indeed surprising. The number {e} 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’

Swingin’-Springin’-Twistin’-Motion

{Left: Swinging spring (three d.o.f.). Right: the Wilberforce spring (two d.o.f.).

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].

Continue reading ‘Swingin’-Springin’-Twistin’-Motion’

Parity of the Real Numbers: Part I

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 {q = m / n} (in reduced form):

  • q Odd: {m} odd and {n} odd.
  • q Even: {m} even and {n} odd.
  • q None: {m} odd and {n} even.

or, in symbolic form,

\displaystyle \mbox{Odd} = \frac{odd}{odd} \,, \qquad \mbox{Even} = \frac{even}{odd} \,, \qquad \mbox{None} = \frac{odd}{even}

Here, {None} stands for “Neither Odd Nor Even”.

Continue reading ‘Parity of the Real Numbers: Part I’

Fairy Lights on the Farey Tree

Fairy Lights on the Farey Tree. Parity types are coloured as follows: Even: Blue; Odd: Green; None: Red.

The rational numbers {\mathbb{Q}} are dense in the real numbers {\mathbb{R}}. The cardinality of rational numbers in the interval {(0,1)} is {\boldsymbol{\aleph}_0}. We cannot list them in ascending order, because there is no least rational number greater than {0}.

Continue reading ‘Fairy Lights on the Farey Tree’

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.

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

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.

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

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.

Continue reading ‘A Finite but Unbounded Universe’

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.

Continue reading ‘Following the Money around the Eurozone’

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.

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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.

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

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}.

Continue reading ‘The Chromatic Number of the Plane’

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.

Continue reading ‘Why Waffle when One Wordle Do?’

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.

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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 \}}.

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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.

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

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.

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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.

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

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.

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

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.”

Continue reading ‘Chiral and Achiral Knots’

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.

Continue reading ‘Cantor’s Theorem and the Unending Hierarchy of Infinities’

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.

Continue reading ‘How to Write a Convincing Mathematical Paper’

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}.

Continue reading ‘The Square Root Spiral of Theodorus’

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.

 

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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.

Continue reading ‘Embedding: Reconstructing Solutions from a Delay Map’

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)}.

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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.
Continue reading ‘Carnival of Mathematics’

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.

Continue reading ‘Simple Models of Atmospheric Vortices’

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.

Continue reading ‘Finding Fixed Points’

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.

Continue reading ‘Approximating the Circumference of an Ellipse’

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?

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Al Biruni and the Size of the Earth

Abu Rayhan al-Biruni (AD 973–1048)

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.

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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.

Continue reading ‘A Model for Elliptic 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?

Continue reading ‘Can You Believe Your Eyes?’

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.

Continue reading ‘The Size of Things’


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