John Nash, who was the subject of the book and film A Beautiful Mind, won the Abel Prize recently. But his journey home from the award ceremony in Norway ended in tragedy [see this week’s That’s Maths column (TM069): search for “thatsmaths” at irishtimes.com]. We learn at school how to solve polynomial equations of first … Continue reading The Tragic Demise of a Beautiful Mind
Tag: Algorithms
Modelling the Markets
Mathematics now plays a fundamental role in modelling market movements [see this week’s That’s Maths column (TM067) or search for “thatsmaths” at irishtimes.com]. The state of the stock market displayed on a trader's screen is history. Big changes can occur in the fraction of a second that it takes for information to reach the screen. … Continue reading Modelling the Markets
The Steiner Minimal Tree
Steiner's minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner's problem allows for additional points – now called Steiner points – to be added … Continue reading The Steiner Minimal Tree
Plateau’s Problem and Double Bubbles
Bubbles floating in the air strive to achieve a spherical form. Large bubbles may oscillate widely about this ideal whereas small bubbles quickly achieve their equilibrium shape. The sphere is optimal: it encloses maximum volume for any surface of a given area. This was stated by Archimedes, but he did not have the mathematical techniques … Continue reading Plateau’s Problem and Double Bubbles
Barcodes and QR Codes: Zebra stripes and Leopard spots
Barcodes and QR codes are described in this week’s That’s Maths column in The Irish Times (TM060, or search for “thatsmaths” at irishtimes.com). Virtually everything that you buy in your local supermarket has a curious little zebra-like pattern the size of a postage stamp printed on it. Barcodes, originally devised about forty years ago to … Continue reading Barcodes and QR Codes: Zebra stripes and Leopard spots
Information Theory
That’s Maths in The Irish Times this week (TM059, or Search for “thatsmaths” at irishtimes.com) is about data compression and its uses in modern technology. The arrival of mobile phones was followed rapidly by "txtese", an abbreviation of language to enable messages to be written and transmitted rapidly using SMS (Short Message Service). The simplest … Continue reading Information Theory
The Year of George Boole
This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen's College Cork. Mathematician and logician George Boole died just 150 years ago, on 8 December 1864, following a drenching as he was walking between his home and Queen's … Continue reading The Year of George Boole
Sunflowers and Fibonacci: Models of Efficiency
The article in this week’s That’s Maths column in The Irish Times ( TM046 ) is about the maths behind the efficient packing of sunflowers and many other plants Strolling along Baggot Street in Dublin recently, I noticed a plaque at the entrance to the Ibec head office. It showed a circular pattern of dots, … Continue reading Sunflowers and Fibonacci: Models of Efficiency
The Chaos Game
The term "Chaos Game" was coined by Michael Barnsley [1], who developed this ingenious technique for generating mathematical objects called fractals. We have discussed a particular fractal set on this blog: see Cantor's Ternary Set. The Chaos Game is a simple algorithm that identifies one point in the plane at each stage. The sets of … Continue reading The Chaos Game
Speed Cubing & Group Theory
The article in this week’s That’s Maths column in the Irish Times ( TM038 ) is about Rubik's Cube and the Group Theory that underlies its solution. The Rubik's cube craze ran through the world like wildfire in the 1980s. This simple mechanical puzzle is made from small pieces, called “cubies”, in a 3x3x3 structure … Continue reading Speed Cubing & Group Theory
French Curves and Bézier Splines
A French curve is a template, normally plastic, used for manually drawing smooth curves. These simple drafting instruments provided innocent if puerile merriment to generations of engineering students, but they have now been rendered obsolete by computer aided design (CAD) packages, which enable us to construct complicated curves and surfaces using mathematical functions called Bézier … Continue reading French Curves and Bézier Splines
Pythagorean triples
The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It can be written as an equation, a2 + b2 = c2, where c is the length of the hypotenuse, and a and b are the lengths of … Continue reading Pythagorean triples
The Simpler the Better
This week’s That’s Maths in The Irish Times ( TM030 ) is about Linear Programming (LP) and about how it saves millions of Euros every day through optimising efficiency. A Berkeley graduate student, George Dantzig, was late for class. He scribbled down two problems written on the blackboard and handed in solutions a few days … Continue reading The Simpler the Better
The Ups and Downs of Hailstone Numbers
Hailstones, in the process of formation, make repeated excursions up and down within a cumulonimbus cloud until finally they fall to the ground. We look at sequences of numbers that oscillate in a similarly erratic manner until they finally reach the value 1. They are called hailstone numbers. The Collatz Conjecture There are many simply-stated … Continue reading The Ups and Downs of Hailstone Numbers
CT Scans and the Radon Transform
Last December, Dublin's Tallaght Hosptal acquired a new CT scanner, a Toshiba Aquilon Prime, the first of its type in the country. The state-of-the-art scanner is housed in a room with a 'sky ceiling' that allows patients to enjoy an attractive outdoor image during the scanning process. This equipment, which cost €600,000 will undoubtedly result … Continue reading CT Scans and the Radon Transform
Singularly Valuable SVD
In many fields of mathematics there is a result of central importance, called the "Fundamental Theorem" of that field. Thus, the fundamental theorem of arithmetic is the unique prime factorization theorem, stating that any integer greater than 1 is either prime itself or is the product of prime numbers, unique apart from their order. The … Continue reading Singularly Valuable SVD
Computer Maths
Will computers ever be able to do mathematical research? Automatic computers have amazing power to analyze huge data bases and carry out extensive searches far beyond human capabilities. They can assist mathematicians in checking cases and evaluating functions at lightning speed, and they have been essential in producing proofs that depend on exhaustive searches. The … Continue reading Computer Maths
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
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
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
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
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