Featured Book of the Month: Proofs From The Book

ProofsFromTheBOOK April’s Book of the Month is Proofs From The Book by Martin Aigner and Günter Zeigler.
(The first edition came out in 1998 — note the new editions)
From the preface to the first edition.

Paul Erdös liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdös also said that you need not believe in God but, as a mathematician, you should believe in The Book. A few years ago, we suggested to him to write up a first (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, filling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdös’ 85th birthday. With Paul’s unfortunate death in the summer of 1997, he is not listed as a co-author. Instead this book is dedicated to his memory.
We have no definition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations.

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How Google works: Markov chains and eigenvalues

Originating author is Christiane Rousseau.
From its very beginning, Google became “the” search engine. This comes from the supremacy of its ranking algorithm: the PageRank algorithm. Indeed, with the enormous quantity of pages on the World-Wide-Web, many searches end up with thousands or millions of results. If these are not properly ordered, then the search may not be of any help, since no one can explore millions of entries. Continue reading

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Dimension

Originating author is Christiane Rousseau.
How do we measure the size of a geometric object? For subsets of the plane we often use perimeter, length, area, diameter, etc. These are not sufficient to describe fractals. The fractal objects are very complex geometric objects, and we must find a way to quantify their complexity. For this purpose the mathematicians have introduced the concept of dimension. Dimension provides a measure of the complexity of a fractal. The notion of dimension is a generalization and formalization of our intuitive notion of dimension when we speak of 1D, 2D or 3D. We will discuss some ways to describe fractal objects by working on two examples: the Sierpinski carpet and the von Koch flake (see figures on the left). Continue reading

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Fair voting: the quest for gold

Originating authors are Gabriel Rosenberg and Mark Iwen.
It is a little known fact that two gold medals were awarded for the same pairs figure skating competition in the 2002 Winter Olympics. These two medals were ultimately a result of contentious voting which initially resulted in the clear crowd favorites not winning the gold medal. The outrage over this decision was so great that the International Olympic Committee (IOC) eventually had to award a second gold metal to the second place figure skating pair in order to settle the scandal. As a secondary result, the voting system for deciding which figure skaters deserve which medals was changed (NB: Prior to 2003 judges individually scored participants and used these results to rank the athletes. These ranks (not scores) were then combined to award overall prizes).

Imagine that you are on the IOC in 2003 and have been tasked with developing a better voting system for judging figure skating competitions in the future. What voting system would you choose for ranking figure skaters? How would you make sure that the voting system was fair? Not surprisingly, mathematics can help us answer these questions! Continue reading

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Goodstein Sequences: The Power of a Detour via Infinity

Originating authors are Michèle Artigue, Ferdinando Arzarello and Susanna Epp.
Studying the evolution of a natural phenomenon often leads to studying numerical sequences , especially their long-term behavior and whether they eventually converge. Polynomial, exponential, and logarithmic sequences are frequently encountered in secondary school, but certain other sequences with very simple definitions exhibit much more complex behavior. Examples include the chaotic sequences that arise in the study of dynamical systems (see [1]) and the Syracuse sequence (or 3n + 1 sequence), introduced by Luther Collatz in 1937. The Syracuse sequence has challenged mathematicians for decades. Despite the huge number of values that have been computed, it is currently unknown whether the sequence is infinite or is finite and always ends in 1 (see [2]).

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Matrices and Digital Images

Originating authors are Dirce Uesu Pesco and Humberto José Bortolossi.
The images you see on internet pages and the photos you take with your mobile phone are examples of digital images. It is possible to represent this kind of image using matrices. For example, the small image of Felix the Cat (on the left) can be represented by a 35 \times 35 matrix whose elements are the numbers 0 and 1. These numbers specify the color of each pixel (a pixel is the smallest graphical element of a matricial image, which can take only one color at a time): the number 0 indicates black, and the number 1 indicates white. Digital images using only two colors are called binary images or boolean images.

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Banach’s microscope to find a fixed point

Originating author is Christiane Rousseau.
In this vignette, we will show how we start from a small game to discover one of the most powerful theorems of mathematics, namely the Banach fixed point theorem. This theorem has fantastic applications inside and outside mathematics. In Section 3 we will discuss the fascinating application to image compression.

But, let us start with our game and look at the famous lid of a box of The Laughing Cow.

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What is the way of packing oranges? — Kepler’s conjecture on the packing of spheres

Originating author is Christiane Rousseau.
What is the densest packing of spheres? Kepler conjectured that it was the one you observe with oranges at the fruit shop, and which is called the face-centered cubic lattice (Figure 1). At the International Congress of Mathematicians in 1900, David Hilbert gave a very famous lecture in which he stated 23 problems that would have deep significance for the advance of mathematical science in the 20-th century. The problem of the densest packing of spheres, also called Kepler’s conjecture, is part of Hilbert’s 18-th problem. Kepler’s conjecture was only proved in 1998 by Thomas Hales, and the details of the proof were published in 2006.

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Higher Dimensions

Figure 1: Illustration of a Calabi-Yau-manifold (Important for the description of higher dimensional models in superstring-theory).

Originating authors are Markus Ruppert and Hans-Georg Weigand.
1. Looking for the next dimension
Does our world really have more than three dimensions? If so, do objects in higher dimension have a relation to the world around us? Is it possible to get a perception of these objects or do they withdraw any representation? The Theory of Relativity uses four dimensions to explain the concept of space-time, six dimensions are necessary to describe the bending of space-time and different string theories even use representations in up to 26 dimensions (e.g. L. Botelho, R. Botelho, 1999). Another current domain of application for higher dimensional objects and their three-dimensional representations is the study of non-periodic structures in modern crystallography. Within the concept of quasicrystals projections of higher dimensional point-sets (such as the integer-lattice in dimension 5) to three dimensional space are supposed to be good models for non-periodic crystalline structures (see section 5 below).

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Benford’s law: learning to fraud or to detect frauds?

Originating author is Christiane Rousseau.
It is very risky to change too many numbers in some fi nancial statements if one does not know some mathematics. Indeed, most often the numbers appearing in fi nancial statements follow some strange mathematical rule, called Benford’s law, or law of the fi rst signi ficant digit. If one forgets to follow the rule, then the numbers will fail some statistical tests and are likely to be scrutinized with care. Benford’s law claims that if you collect numbers at random and calculate the frequencies of their fi rst signifi cant digits, the numbers with fi rst signifi cant digit 1 should appear around 30% of the time, while the numbers with fi rst signifi cant digit 9 appear only 4.5% of the time. This rule is observed in many other sets of numbers, like powers of 2 or Fibonacci numbers.

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