A brief history of maths
Brian Cogan reviews The Mathematical Explorer, an unusual piece of educational software
I first encountered The Mathematical Explorer under what were ideal circumstances for me - recuperating in bed after minor surgery. I loaded the program on to my laptop with a complete absence of guilt. I spent much of the next couple of days visiting many of the famous problems in the history of mathematics, learning about the people who solved them (and those who failed) and exploring the application of ideas such as Turtle fractalisation and public-key cryptography.
Professor Stan Wagon, a Mathematica expert and celebrated educator and author, wrote The Mathematical Explorer. He has included topics varying in difficulty from the Riemann Hypothesis to games from recreational mathematics. The front-end is the familiar Mathematica Help browser. A scaled down Kernel is included so I didn't need any additional software to execute the numerous examples or plot graphs etc. Each of the 16 main sections ends with examples and solutions and suggestions for finding further information, with references to books, articles and links to web sites. There are also links to the web-based version of Stephen Wolfram's The Mathematica Book. Famous solved problems as well as some unsolved problems are described as well as much discussion of topics such as chaos theory, fractals, prime numbers, methods for calculating pi etc. There are many cultural and historical details included.
Several features in particular appealed to me. First was the humour with which fairly intimidating mathematical results are presented. One example of this is 'The Shocking BPP' formula, first published in 1995, and used to compute digits of pi without computing prior digits. This surprising and recent result is explored in some detail and the author gives his own version of the original formula. He also includes Mathematica code for generating these formulas. However, we are also given the mnemonic: 'How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics' where the number of letters in each word gives successive digits of pi.
Another humorous and whimsical example is where we meet the Chicken McNugget problem. Chicken McNuggets are sold in boxes of 6, 9, and 20 so it is possible to buy, say, 29, 12 or 15 McNuggets - these are called McNugget numbers. The problem is to discover the largest number of Chicken McNuggets that cannot be bought i.e. what is the largest non-McNugget number? These light-hearted examples present opportunities for some tricky analysis and for the development of problem solving skills.
Another attractive feature is that, unlike a static, classical textbook, each of the 16 main sections is interactive. The reader can experiment with different parameters, plot graphs, solve new equations, and add commands - a great learning experience. Also, the reader is never patronised and many recent results are described as well as some unsolved problems.
Brief biographies and many pictures of more than 40 mathematicians from Archimedes to Andrew Wiles (who proved Fermat's Last Theorem) are included. The many cultural and historical references make for a very rich and satisfying experience. There is also a wealth of fascinating information included in an additional demos section. For example a polyhedron explorer allows you take solids such as a cube, an octahedron, or a dodecahedron and find stellated, indented, truncated and shrunken versions. Other demos include a celestial mechanics sampler, examples of cellular automata and an analysis of a chaotic circuit. Although designed to demonstrate the full version of Mathematica I thoroughly enjoyed these bonus applications.
I particularly enjoyed the sections on prime numbers, calculating pi, chaos, and the Four Colour Theorem. In this last section, Kempe's false proof of the Four Colour Theorem is described in detail. Surprisingly, the author then uses this as the basis for an algorithm for colouring maps that works most of the time.
Other sections on calculus, square wheels, check digits, secret codes, Escher patterns, plane curves, recreational mathematics, Fermat's Last Theorem, the Riemann hypothesis, and unusual number systems were also very interesting. I learned a great deal about unsolved problems - such as Catalan's conjecture. This says the only consecutive powers are 8 and 9, i.e. 23=8 and 32=9, and there are no other non-trivial, integer examples! There is something here for everybody with an interest in maths and its history. Teachers of maths subjects, anxious to introduce novelty, culture and history into courses, will get inspiration here. Engineers and scientists who are not mathematicians but enjoy mulling over famous mathematical problems and trying their hand at some of the unsolved ones may possibly find unforeseen applications for the new ideas presented. And students wishing to broaden and deepen their appreciation of the subject beyond the limits imposed by curricula, lectures and examinations will be entertained, challenged and informed.
