Book Review: Fibonacci’s Rabbits

“If I have seen further, it is by standing on the shoulders of giants”

Fibonacci’s Rabbits, by Adam Hart-Davis, explains in accessible language fifty significant mathematical discoveries, from ancient times through to the present day. It is divided into seven ages and demonstrates how each featured mathematician built their propositions and then proofs informed by all who had gone before. To enjoy the book fully it may be necessary to have some interest in, if not understanding of, mathematical concepts and principles.

“In mathematics, proof is everything, whereas science cannot prove anything. Scientists can disprove ideas, but they can never prove them.”

The first section outlines the long history and need for simple tallying and recording. Standardised numerals evolved from primitive marks made when counting. The reasons for and timing of the initial move from such practical applications to more complex and abstract ideas has been lost to time but ancient civilisations throughout the world are known to have used various number systems and calculations to: produce calendars, build pyramids, and study intriguing problems such as squaring the circle. Some of the questions these early mathematicians asked have still to be answered. Others led to proofs that proved useful in practical applications far in their future.

“it is easy to get trapped by habitual ways of thinking, and different approaches can lead to new insights”

Mathematics is required to be logical and rigorous. Although many of its problems appear theoretical, discoveries are often mirrored in the real world. Key patterns, sequences and shapes turn up in: plants, animals, and their natural habitats. The intrigue of the Fibonacci sequence and associated spiral enabled many new mathematical discoveries yet it is clear that Fibonacci numbers are a reflection of the way things grow – including the reproduction of rabbits.

I particularly enjoyed the chapter on zero and the mathematical problem of defining concepts. Having accepted the usefulness of ‘nothing’ whole new fields opened up.

Looking at the importance of imaginary numbers it is clear that sometimes all that is needed is a new way of simply notating a complex idea.

“In Euclid’s system, the workings of the world are not just the whims of the gods, but follow natural rules. It showed how we can find our way to the truth through logic and deductive reasoning, evidence and proof – not just intuition.”

Mathematical problems are only regarded as solved when a verified proof is discovered and written down.

“If even a single exception is found, the conjecture fails”

“for mathematicians, ‘highly likely’ is not proof”

Being unable to find an exception does not constitute a proof, however long the search has taken. There are problems that have finally been solved after hundreds of years, and others that remain outstanding.

Chaos theory and how it developed intrigued me – how one tiny event creates ripples that can radically change outcomes.

“A very small cause, which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance.”

“Really, he argues, the weather is equally as rigidly determined as the eclipse. It is just that the operation of chance with the weather is so major that we just do not have enough knowledge to predict it. Such systems seem to be chaotic, but the normal laws of the universe are still operating entirely regularly.”

Today’s intellectually challenging, high level mathematics may have no apparent practical value. And yet, as has been shown time and again over the centuries, it may offer real world insights as future applications are developed.

Mathematics is logical but logic can be mind bending.

“This statement is false”

So many of the chapters were fascinating to read, even those whose concepts I couldn’t fully comprehend.

It was fun trying to work out why certain, apparently random objects are included in the many colourful illustrations that accompany the text. This was just one entertaining puzzle in a book that provides insights into works of genius that have laid the foundations of so many discoveries and inventions.

Mathematics is an ever evolving discipline and it is exciting to know there is still so much remaining to learn.

The congenial structure and explanations provided in this book allow the reader to appreciate how important mathematics is to a wider understanding of our world and the developments we now rely on. Readers will value the tenacity of mathematicians through the ages when they realise the impact their work has had.

My copy of this book was provided gratis by the publisher, Modern Books.

Book Review: Flatland


Flatland, by Edwin A. Abbott, is one of the oddest books that I have ever read. Published in 1884 it tells the tale of a square, a resident of a two dimensional world, and his discovery of an incredible third dimension when he is visited by a sphere from Spaceland. Many of his fellow plane dwellers think him deranged, just as the resident of Lineland whom he met in a dream would not accept the existence of a second dimension because it was beyond his direct experience and comprehension.

Using mathematics the practicalities of these various worlds is explained. Each is imbued with strict hierarchies and rules alongside enhanced senses that allow order to be maintained. Much of it came across as disturbing, such as women in Flatland being the lowest of the low (almost a straight line) whereas rulers and priests were of the highest order (circles).

Children born with irregularities were generally killed at birth with any survivors forbidden to reproduce. Amongst the higher orders it was common to subject offspring to life threatening procedures in an attempt to increase the number of sides they had and thereby improve their social standing.

I struggled with the lengthy explanations in the first half of the book. They may have been clever in places and necessary for understanding but I found their presentation somewhat tedious. By the second half however the tale had become strangely readable. I had to get past my distaste for the eugenics in order to appreciate the allegorical timbre. I suspect that this story was created with a strong dose of irony, at least I hope it was.

Observations were often offered lightly, such as the suggestion that the single resident of Pointland (no dimensions) was happy because he knew no better and had no one around to disturb his simple thoughts with difficult questions. There was a recognition that the lower orders were more likely to accept the unequal society if they believed that there was a chance that they or their children could one day benefit from the way things were. Aspiration was encouraged even if many were considered expendable.

This strange little story, once it got going, turned out to be an oddly satisfying read. It posed interesting questions about how much we really know and how averse we can be to new ideas that threaten a familiar way of life. It demonstrates how those in power manage the masses for personal gain.

At a little over a hundred pages the book did not take long to read. Despite the messages it conveyed it was in no way preachy. I enjoyed the mathematical explanations and the quirkiness of the ideas. I would have some reservations about recommending this book but am glad that I persevered.