That's Maths

The Mathematical Magic in Everyday Life

Peter Lynch

• Editore: Gill Books
• Anno: 2016
• ISBN:
  9780717169566
• Argomento: Mathematics
• Categoria: History & Philosophy of Mathematics

From atom bombs to rebounding slinkies, open your eyes to the mathematical magic in the everyday. Mathematics isn't just for academics and scientists, a fact meteorologist and blogger Peter Lynch has spent the past several years proving through his Irish Times newspaper column and blog, That's Maths.Here, he shows how maths is all around us, with chapters on the beautiful equations behind designing a good concert venue, predicting the stock market and modelling the atom bomb, as well as playful meditations on everything from coin-stacking to cartography. If you left school thinking maths was boring, think again!

THE FARADAY OF STATISTICS

In October 2012 a plaque was unveiled at St Patrick’s National School, Blackrock, to commemorate William Sealy Gosset, who had lived nearby for 22 years during the early twentieth century. Sir Ronald Fisher, a giant among statisticians, called Gosset ‘the Faraday of statistics’, recognising his ability to grasp general principles and apply them to problems of practical significance. Gosset’s name may not be familiar, but his work is known to anyone who has taken an introductory course in statistics. Using the pseudonym Student, he published a paper in 1908 that has been of importance ever since.

Gosset was born in Canterbury, Kent, in 1876 into an old Huguenot family. He studied chemistry and mathematics at Oxford, graduating in 1899 with first class honours in both subjects. He then joined Arthur Guinness & Son in Dublin as a chemist, and worked at the brewery in St James’s Gate for 36 years, before becoming head brewer at a new Guinness brewery at Park Royal in London.

Guinness was interested in agricultural experimentation and hired scientists who could apply their expertise to the business. Gosset was one of these, and he used statistics to solve a range of problems connected with brewing, ranging from barley production to yeast fermentation, that affected the quality of the product. One problem involved the selection of varieties of barley that produced maximum yields in given soil types and allowing for the vagaries of climate.

To extend his knowledge, Gosset spent a year at the biometric laboratory of the leading statistician Karl Pearson at University College London. Reliable statistics require an adequate sample size. Gosset soon realised that Pearson’s large-sample theory required refinement if it was to be useful for the small-sample problems arising in brewing. His fame today rests on a statistical test called Student’s t-test.

But why Student? Gosset’s main paper, ‘The Probable Error of a Mean’, was published in 1908. But to protect trade secrets, Guinness would not allow employees to publish the results of their research. They wished to keep secret from competitors the advantages gained from employing statisticians. Gosset persuaded his bosses that there was nothing in his work that would benefit competitors, and they allowed him to publish, but under an assumed name. Hence, anyone studying statistics encounters the name Student rather than that of the true author of the method.

Gosset’s work has proved fundamental to statistical inference as practised today. His great discovery was to derive the correct distribution for the sample mean. Student’s t-test arises when we estimate the average value of a randomly varying quantity from a small sample. It plays a crucial role in statistical analysis: for example, it is used to evaluate the effect of medical treatment, when we compare patients taking a new drug with a control group taking a placebo. It was also central to the development of quality control, which is vital in modern industry.



THE CHAOS GAME

The term ‘Chaos Game’ was coined by Michael Barnsley, who developed this ingenious technique for generating mathematical objects called fractals. The Chaos Game is a simple algorithm that identifies one point in the plane at each stage. The sets of points that ultimately emerge from the procedure are remarkable for their intricate structure. The relationship between the algorithm and fractal sets is not at all obvious because there is no evident connection between them. This element of surprise is one of the delights of mathematics.

THE SIERPINSKI GASKET

Let’s specify the algorithm in a simple case. We need a dice (I should say die but that feels odd) with three numbers. We can use a standard dice and identify 6 with 1, 5 with 2 and 4 with 3; that is, for face n we call it min (n, 7 – n).

1. Fix three points in the plane, C₁, C₂ and C₃. For definiteness, we take the points C₁ = (0, 0), C₂ = (1, 0) and C₃ = (0.5, √3/2), the corners of an equilateral triangle.

2. Pick any point P₀ and draw a dot. This is our starting point. At each stage, we denote the current point by P_k and call it the game point.

3. Roll the dice. If n comes up, draw a point halfway between P_k and C_n. For example, if we roll a 2, we pick the point halfway between the current point P_k and C₂. This is the new game point.

4. Repeat this procedure many times, drawing a new point at each step.

We seek the attractor of the mapping, so we eliminate the initial points, let’s say the first 100 points. It is difficult to anticipate the outcome of this algorithm, but the figures below reveal all. We plot the results at several stages of the procedure, with k = 500, 1000 and 2000. We can see a structure forming. It is the famed Sierpinski Gasket.

The Chaos Game for three points at the vertices of an equilateral triangle, showing the output after 500, 1000 and 2000 steps.

BARNSLEY FERNS

Michael Barnsley generalised the Chaos Game to produce a wide range of fractal sets. He based it on a mathematical structure called an Iterated Function System (IFS). Starting with a point (x_k, y_k) in the plane, the next point is f_r(x_k, y_k), where f_r is a member of the IFS, chosen with a specified probability p_r. In the simplest cases, the maps f_r are linear or affine transformations.

Probably the most famous example is the so-called Barnsley Fern. The algorithm or procedure is very similar to that described for the Sierpinski Gasket, but now four different linear maps are used instead of one, and the choices between them are made on the basis of random numbers, with prescribed probability for each map. The algorithm can be coded in a few lines of computer code. We show the results at several stages below.

Barnsley Fern generated by the Chaos Game. From left to right: 1000, 10,000 and 100,000 points.

The final result is a set of considerably beauty. The structure of the set built up by the Chaos Game bears a remarkable similarity to a real fern:

Barnsley Fern after one million iterations.

SOURCES

Barnsley, Michael (1993, 2000), Fractals Everywhere. San Diego, California: Academic Press.

Hutchinson, John (1980), ‘Fractals and self-similarity’, https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf. First published in Indiana University Mathematics Journal, 30, 713–47.

Wikipedia, ‘Chaos game’, https://en.wikipedia.org/wiki/Chaos_game

Wikipedia, ‘Barnsley fern’, https://en.wikipedia.org/wiki/Barnsley_fern



FIBONACCI NUMBERS ARE GOOD FOR BUSINESS

Outside the entrance to the head office of the Irish Business and Employers Confederation (Ibec) in Baggot Street, Dublin, there is a plaque with a logo comprising a circular pattern of dots. According to Ibec’s website, the logo brings ‘dynamism’ and hints at Ibec’s ‘member-centric ethos’. In fact, it is more interesting than this.

The logo is based on the spiral patterns found in many flowers and plants. Examining it we find 34 clockwise and 21 counter-clockwise spirals. These numbers are sequential entries in a famous number sequence called the Fibonacci sequence. This is no coincidence.

In 1202, mathematician Leonardo of Pisa, usually known as Fibonacci, published a book, Liber Abbaci, in which he described the sequence now known as the Fibonacci numbers. They are easily defined by an iterative process: starting with 0 and 1, each entry is the sum of the previous two. Thus, the sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and 144.

As the values increase, the ratios of successive Fibonacci numbers – like 8/5, 13/8 and 21/13 – tend to a definite limit, about 1.618, called the golden number. It was known to the Greeks through their study of proportions and the geometry of the pentagon. If we divide a circle into two arcs whose ratio is the golden number, the shorter arc subtends an angle of about 137.5°. This is called the golden angle.

Distinctive spiral patterns are found in many plants. For example, the hexagons on pineapples fit together in interlocking families of helical spirals. The numbers of spirals are successive Fibonacci numbers. Sunflowers, which belong to the daisy family, usually have 55, 89 or 144 petals, and spiral patterns are evident in their seeds.

Biologists described long ago how phylla (leaves, petals, seeds, etc.) are arranged on plants: this branch of botany is called phyllotaxis. But explaining why these patterns form is much more difficult than describing them, and it is only recently that real progress has been made.

The seeds of a sunflower are arranged in a manner that makes efficient use of the available space, giving maximum room for each seed to flourish and minimising wastage of space. As a new seed sprouts forth at the growth tip of a plant, it naturally tends to grow where there is most open space. Each seed is displaced from the previous one by the golden angle.

But why the golden angle? Recent research shows that the angle emerges naturally as a feature of the dynamics of plant growth. Some years ago, Dublin-born Alan Newell, at the University of Arizona, applied elasticity theory to continuum models of growing cacti shoots. But mechanics could not answer all the questions.

Recently, Newell has shown that biochemistry, mechanics and geometry all play a role in generating the observed patterns. The growing seeds exert forces on each other, triggering the production of auxin, a growth-enhancing plant hormone. The solutions generated by the model of auxin concentration are found to be very similar to the patterns found in ...