Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing
This is not the first and will not be the last book with a collection of illustrations of mathematical applications at a low level to show the mathematician as well as the non-mathematical reader the fun that mathematics can be, but also how much we depend on mathematics in our daily life. By showing the recreational applications and the fun games and puzzles, the attention of the reader is attracted. Later it is shown that the same principles are used to solve real life problems. It hopefully will remove the perception about mathematics that lives in the mind of many youngsters, and unfortunately in the minds of educated adults, as being a subject that is difficult, boring, and very remote from reality. In their minds mathematics consists of a dull sequence of abstract theorems and proofs with the only purpose of proving other even more abstract theorems. The common idea underlying the selection of subjects that Chartier has chosen to include is that they are all linked to "computing". This makes it immediately less abstract. Some applications are recreational, just for fun, but they are gracefully linked to others that are really of major practical importance. The mathematics are low level and kept to a minimum. These are the mathematical "bits" in the book. Chartier has collected a dozen of these applications with computational aspects into short chapters, the "bytes" that make up this book. The quibble is that the problems can bite you and leave an itching curiosity in the back of your mind or they are presented as a set of bites like you taste different cakes and pies, to let you decide which one will be further explored in more detail.
One of the first examples discussed is illustrating the strategy very well. One is warned for the danger of rounding errors on a digital computer. The playful aspect is the seemingly counter example to Fermat's Last Theorem that appears in an episode of the Simpsons. Checking it on a pocket calculator shows that the relation holds, but that is only because of rounding errors. Chartier explains how to easily see that the relation cannot be true without evaluating the powers and challenges the reader to check similar relations for validity. But then he also illustrates that neglecting the rounding errors can lead to disastrous results in real life. This strategy is used throughout: there is the catchy playful aspect of the example, followed by a teasing problem to challenge the reader, and finally linking it to examples of real life importance. Here are some other examples: The performance of computers has increased enormously but the speed at which powers of 2 grow is unexpected or counter intuitive like the classic example of doubling the number of grains on the squares of a chess board, but the same effect plays in spreading news via twitter and conversely, dividing by 2 decays really fast, for example placing a number at the right position in a long sequence of numbers as in sorting algorithms. Another example is a fractal and the use of fractals in creating realistic backgrounds in animation movies. The design of fonts and the parabolic trajectory described by the catapulted birds in the angry birds game is an incentive to introduce equations for lines and curves. Doodles evocate Euler's formula for graphs (or polyhedra) and this leads to traveling salesman problems and the analysis of labyrinths. Certain forms of image processing subdivide the colour of the pixels into a reduced number (four) classes of different colours, which gives the image a stylized artistic touch (obviously relating it to the four colour problem). So, using M&Ms of four different colours as pixels, one may produce a picture of that type. Other image processing techniques like distortion or approximating a portrait as a linear combination of a given set of portraits are considered. These require the introduction of matrices, vectors, linear transformations, avoiding to define them as just mathematical objects. Google's pagerank is introduced first by methods of predicting the outcome of a basketball tournament, but in the end its principle and the corresponding eigenvalue problem is pretty well explained.
Thus Chartier has brought some examples that are presented as fun examples of mathematics but at a much more advanced level, they do matter in our computerized world and it is well illustrated that mathematics plays a role. Linear transformations as matrix products, eigenvectors, graph theory, prime numbers, etc., these are all important elements in practical applications. The reader is constantly challenged to think about it or answer certain questions and to solve some problems (some solutions are provided at the end). Most of all, it's such a lovely little booklet that does not give you the time to get bored with. The average chapter length, including the many illustrations, is only 10 pages. Just enough to catch your interest and get bitten by the mathematics.