Featured Book of the Month: Mathematics for the Million

Lancelot HogbenJanuary’s Book of the Month is Mathematics for the Million: How to Master the Magic of Numbers by Lancelot Hogben. First published in 1936 and reprinted many times.

From the Prologue: “The customary way of writing a book about mathematics is show how each step follows logically from the one before without telling you what use there will be in taking it. This book is written to show you how each step follows historically from the step before and what use it will be to you or someone else if it is taken. The first method repels many people who are intelligent and socially alive, because intelligent people are suspicious of mere logic, and people who are socially alive regard the human brain as an instrument for social activity”

Chapters:
Mathematics in Remote Antiquity
The Grammar of Size, Order, and Shape
Euclid as a Springboard
Number Lore in Antiquity
The Rise and Decline of the Alexandrian Culture
The Dawn of Nothing
Mathematics for the Mariner
The Geometry of Motion
Logarithms and the Search for Series
The Calculus of Newton and Leibnitz
The Algebra of the Chessboard
The Algebra of Choice and Chance

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Historical Topics for the Mathematics Classroom

MathClassroomDecember’s Book of the Month is Historical Topics for the Mathematics Classroom by J.K. Baumgart, D.E. Deal, B.R. Vogeli, A.E. Hallerberg, eds., published by the National Council of Teachers of Mathematics.

This book has a summary of the history of each major branch of mathematics (Number, Computation, Algebra, Geometry, Trigonometry and Calculus) followed by brief essays on specific topics of significance within each branch. The last two chapters of the book give a brief, integrated view of the direction of mathematics development over the last century.
Originally published in 1969, and updated in 1989, the book continues to offer high school and university teachers an excellent tool for their own development.
(Adapted from a review by Tim Keenan).

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Imaginary

OberwolfachDecember’s Site of the Month: IMAGINARY

Imaginary is a site containing “vignette-like” short articles on modern mathematics for a general audience. It emanates from the Mathematics Research Centre at Oberwolfach, where groups of mathematicians meet to work on particular areas or projects. The articles are written by experts in the field and then edited for a general readership.

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Geometry and the Imagination

GeoAndImagNovember’s Book of the Month is Geometry and the Imagination by Hilbert and Cohn-Vossen. First published in 1952.

The AMS website says: “This remarkable book has endured as a true masterpiece of mathematical exposition. … The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians.
[It] is full of interesting facts, many of which you wish you had known before. … The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in \mathbb{R}^3. …
One of the most remarkable chapters is “Projective Configurations”. … Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. …
A particularly intriguing section … is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? … The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained! …”

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Accromath

AccromathNovember’s site of the month: Accromath

This is the site of a French-Canadian journal aimed at a mathematics teacher audience.

Now in its 10th volume, the journal has articles in French that can be read online or download as pdf.

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The Laboratory of Mathematical Instruments

MacchineMatOctober’s site of the month: The Laboratory of Mathematical Instruments

October’s book of the month: Bartolini Bussi, Maria G., Maschietto, Michela (2006). Macchine Matematiche. Springer

Japanese Translation: Kyokusen no jiten : seishitsu rekishi sakuzuhoÌ, Tankoban Hardcover

A mathematics teacher coming to Modena, Italy, might enjoy a visit to the Laboratory of Mathematical Instruments at the University of Modena and Reggio Emilia (UNIMORE).
The Laboratory is run by Michela Maschietto and Maria G. Bartolini Bussi, in collaboration with the no-profit Association with the same name.

The Laboratory has 2 different sites:
– one in the Building of the Department of Physics, Informatics and Mathematics (via Campi 213/B9),
– one in a central building of UNIMORE (via Camatta 15).
In the former you can see about one hundred of curve drawing devices and instruments for geometrical transformation. In the latter you can see a beautiful exhibition of working models of instruments for perspective drawing — many reconstructed from the Renaissance period.

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For further information write to:

Michela Maschietto: michela.maschietto@unimore.it
Mariolina Bartolini Bussi: bartolini@unimore.it

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Trading in a Town

Originating authors are Alberto A. Pintoa (University of Porto, cLIAAD-INESC Porto LA) and Telmo Parreira (University of Minho, cLIAAD-INESC Porto LA).

Have you ever been frustrated by the lack of choices when you go to buy something? Why do producers of the things we buy tend to make their products as similar to each other as possible? If we model how shoppers in a city will choose in which store they will make their purchases, the resulting mathematics leads us to an appreciation of Hotelling’s Law that making products similar to your competitor is actually a rational decision. We are also led into the mathematics of game theory and Nash’s Equilibrium.
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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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