That’s Maths in this week's Irish Times ( TM013 ) is about the branch of mathematics called topology, and treats the map of the London Underground network as a topological map. Topology is the area of mathematics dealing with basic properties of space, such as continuity and connectivity. It is a powerful unifying framework for … Continue reading Topology Underground
The Power Tower
Look at the function defined by an `infinite tower' of exponents: $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ It would seem that for x>1 this must blow up. But, amazingly, this is not so. In fact, the function has finite values for positive x up to $latex {x=\exp(1/e)\approx 1.445}&fg=000000$. We call this function the power tower … Continue reading The Power Tower
Archimedes uncovered
The That’s Maths column in this week's Irish Times ( TM012 ) describes the analysis of the ancient codex known as the Archimedes Palimpsest. Archimedes of Syracuse Archimedes (Ἀρχιμήδης, 287-212 BC) was a brilliant physicist, engineer and astronomer, and the greatest mathematician of antiquity. He is famed for founding hydrostatics, for formulating the law of … Continue reading Archimedes uncovered
Pons Asinorum
The fifth proposition in Book I of Euclid's Elements states that the two base angles of an isosceles triangle are equal (in the figure below, angles B and C). For centuries, this result has been known as Pons Asinorum, or the Bridge of Asses, apparently a metaphor for a problem that separates bright sparks from … Continue reading Pons Asinorum
Santa’s TSP Algorithm
This week's That's Maths column ( TM011 ) discusses the challenge faced by Santa Claus: he has about a billion homes to visit in one night, so he needs to be smart in picking his route. The challenge he faces is called the Travelling Salesman Problem, or TSP. Although he won't reveal his secret, Santa … Continue reading Santa’s TSP Algorithm
Sharing a Pint
Four friends, exhausted after a long hike, stagger into a pub to slake their thirst. But, pooling their funds, they have enough money for only one pint. Annie drinks first, until the surface of the beer is half way down the side (Fig. 1(A)). Then Barry drinks until the surface touches the bottom corner (Fig. … Continue reading Sharing a Pint
Ramanujan’s Lost Notebook
In the Irish Times column this week ( TM010 ), we tell how a collection of papers of Srinivasa Ramanujan turned up in the Wren Library in Cambridge and set the mathematical world ablaze. Srinivasa Ramanujan (1887—1920) Ramanujan was one of the greatest mathematical geniuses ever to emerge from India. Born into a poor Brahmin … Continue reading Ramanujan’s Lost Notebook
Where Circles are Square
Imagine a world where circles are square and π is equal to 4. Strange as it seems, we live in such a world: urban geometry is determined by the pattern of streets in a typical city grid and distance "as the crow flies" is not the distance that we have to travel from place to … Continue reading Where Circles are Square
The Root of Infinity: It’s Surreal!
Can we make any sense of quantities like ``the square root of infinity"? Using the framework of surreal numbers, we can. In Part 1, we develop the background for constructing the surreals. In Part 2, the surreals are assembled and their amazing properties described. Part 1: Brunswick Schnitzel The number system has been built up … Continue reading The Root of Infinity: It’s Surreal!
Where in the World?
Here's a conundrum: You buy a watch in Anchorage, Alaska (61°N). It keeps excellent time. Then you move to Singapore, on the Equator. Does the watch go fast or slow? For the answer to this puzzle, read on. The Global Positioning System In the Irish Times column this week ( TM009 ), we look at … Continue reading Where in the World?
Shackleton’s spectacular boat-trip
A little mathematics goes a long, long way; in the adventure recounted below, elementary geometry brought an intrepid band of six men 800 sea miles across the treacherous Southern Ocean, and led to the saving of 28 lives. Endurance For eight months, Ernest Shackleton's expedition ship Endurance was carried along, ice-bound, until it was finally … Continue reading Shackleton’s spectacular boat-trip
A Mersennery Quest
The theme of That's Maths (TM008) this week is prime numbers. Almost all the largest primes found in recent years are of a particular form M(n) = 2n−1. They are called Mersenne primes. The Great Internet Mersenne Prime Search (GIMPS) is aimed at finding ever more prime numbers of this form. The search for the … Continue reading A Mersennery Quest
The Popcorn Function
Continuity is not what it seems. In 1875, the German mathematician Carl Johannes Thomae defined a function P(x) with the following extraordinary property: P(x) is discontinuous if x is rational P(x) is continuous if x is irrational. A graph of this function on the interval (0,1) is shown below. The function has many names. We … Continue reading The Popcorn Function
Carving up the Globe
This week, That’s Maths (TM007) describes various ways of dividing up the sphere. This is an important problem in geometry, biology, chemistry, astronomy, meteorology and climate modelling. The problem of defining a uniform distribution of points on the sphere has challenged mathematicians for centuries. The vertices of the five Platonic solids achieve this but, in … Continue reading Carving up the Globe
Falling Slinky
If you drop a slinky from a hanging position, something very surprising happens. The bottom remains completely motionless until the top, collapsing downward coil upon coil, crashes into it. How can this be so? We all know that anything with mass is subject to gravity, and this is certainly true of the lower coils of … Continue reading Falling Slinky
Contagion
This week, That’s Maths (TM006) describes the use of mathematical models to study the spread of infections like the SARS epidemic and swine flu. Simple models such as the SIR model of Kermack and McKendrick (1927) can simulate the broad features of epidemics, but much more sophisticated models have been developed using the same approach. … Continue reading Contagion
The End of Smallpox
Daniel Bernoulli was born in 1700 into a remarkably talented family. He excelled in mathematics, but also studied and lectured in medicine, botany and physics. In 1760, he submitted a paper to the Academy of Sciences in Paris dealing with the effects of inoculation on morbidity. The practice of inoculation, the deliberate introduction of a … Continue reading The End of Smallpox
Khan Academy
This week, That's Maths (TM005) discusses the large range of maths education videos that are available free of charge from the Khan Academy website. There are about 3,200 tutorials, covering the whole range of second-level mathematics. Salman Khan's Technology, Entertainment and Design (TED) presentation is available on-line: Let’s Use Video to Reinvent Education. There are … Continue reading Khan Academy
The Beautiful Game
What is the most beautiful rectangular shape? What is the ratio of width to height that is most aesthetically pleasing? This question has been considered by art-lovers for centuries and one value appears consistently, called the golden ratio or Divine proportion. I must admit that the notion of an ideal ratio makes me uncomfortable. How … Continue reading The Beautiful Game
Packing & Stacking
In That's Maths this week (TM004), we look at the problem of packing goods of fixed size and shape in the most efficient way. Packing problems, concerned with storing objects as densely as possible in a container, have a long history, and have broad applications in engineering and industry. Johannes Kepler conjectured that the standard … Continue reading Packing & Stacking
No Maths Involved!
Whether or not you enjoy solving them, those 9x9 grids with numbers and blank cells cannot have escaped your notice. Sudoku puzzles have swept the world since exploding on the scene in 2005. They are found in newspapers everywhere, providing daily amusement to all who like a minor mathematical challenge. The objective of Sudoku is … Continue reading No Maths Involved!
Sproutology
Sprouts is a simple and delightfully subtle pencil-and-paper game for two players. The game is set up by marking a number of spots on a page. Each player makes a move by drawing a curve that joins two spots, or that loops from a spot back to itself, without crossing any lines drawn earlier, and … Continue reading Sproutology
Analemmatic Sundials
This week’s That’s Maths article, TM003, describes the analemmatic sundial on the East Pier in Dun Laoghaire. An article in Plus Magazine, by Chris Sangwin and Chris Budd, gives a description of the theory of these sundials and instructions on how to build one. A script to design an analemmatic sundial, written by Alexander R. Pruss, … Continue reading Analemmatic Sundials
Napier’s Nifty Rules
Spherical trigonometry is not in vogue. A century ago, a Tripos student might resolve a half-dozen spherical triangles before breakfast. Today, even the basics of the subject are unknown to many students of mathematics. That is a pity, because there are many elegant and surprising results in spherical trigonometry. For example, two spherical triangles that … Continue reading Napier’s Nifty Rules
Google PageRank
This week's That's Maths article, at TM002, describes how Google's PageRank software finds all those links when you enter a search word, by solving an enormous problem in linear algebra. A comprehensive description of PageRank is given in the book Google's PageRank and Beyond: The Science of Search Engine Rankings, by Amy N. Langville & … Continue reading Google PageRank
The Two Cultures
The Two Cultures Of course I've heard of Beethoven, but who is this guy Gauss? The “Two Cultures”, introduced by the British scientist and novelist C. P. Snow in an influential Rede Lecture in 1959, are still relevant. Ludwig von Beethoven and Carl Friedrich Gauss were at the height of their creativity in the early … Continue reading The Two Cultures
Irish Times Articles
"That's Maths", a new series of articles on mathematics and its importance in society, will be published in the Irish Times, starting on 19 July 2012. The initial article, at TM001, looks at the statistics of Usain Bolt's performance in the 100m and the prospects for a new record at the London Olympics. The column … Continue reading Irish Times Articles
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