Nigel Lesmoir-Gordon

Fractals are both recent and timeless. They have only existed in mathematical literature for the last hundred or so years, but nature has had them from its first day. Fractal patterns exist in snowflakes, in trees, in mountain ridges, in coastlines, and even in broccoli. Although the word “fractal” was coined in 1975 by the famed mathematician Benoit Mandelbrot, work was being done in fractional and recursional geometry around the time of the invention of calculus. Nigel Lesmoir-Gordon’s The Colours of Infinity is collection of essays that explores the mathematical, physical, and imaginative boundaries of fractals and what this means for our understanding of the world today.

In short, a fractal is a figure created by infinitely modifying a line or shape according to a particular rule. In the commonly seen Sierpinski triangle, the base triangle is divided into four smaller ones. Each of those is divided in the same way, and so on, until you get an infinite array of smaller and smaller triangles. The famous Mandelbrot set is even more wondrous.

All the points in the body of the set can be contained by a simple, short equation (Z ↔ z*z + c), but you can set the visual boundaries as tight or as large as you want to. Eventually, you will always finds a copy of the original image inside itself. The rest of the set has infinite possibilities to explore, and each of the writers in this volume expound upon their experiences with fractals. If you’re a math nut, then you’ll really enjoy this one; if not, it still has a lot of pretty pictures. A very quirky read.… (meer)

This is the second review of a book in the ‘Introducing …’ series, and is related to the other title Introducing Chaos, as the topic areas overlap. As such there is a little duplication between the two books, but this one provides a deeper appreciation of fractals themselves and the part they play in the development of chaos theory and more particularly the part they play in creating the world around us.

Following the standard format of the series, the book is a combination of text and cartoon style graphics which help diffuse from the outset the fear of difficulty the subject may present.

The book begins with the startling revelation that since Plato, through Newton and into the modern science of particles and waves, we describe the world through our understanding of regular solid forms. What are called Euclidian shapes such as straight lines, cubes, spheres, triangles, squares etc. This Euclidian world of ideal shapes, though a great aid in simplification to make our modelling of the world manageable, invites us to model a world that doesn’t really exist. In the real world we rarely encounter these precise shapes. It is a world that is not naturally straight edged, and instead is fashioned with rough edges. Fractals, the fingerprints of chaos, give us a whole new way of describing, understanding and seeing this rough edged world. Once we can see in this new way we suddenly realise that fractals, and thus chaos, are literally everywhere as part of the building and operating processes of this real universe.

Fractals are within us and surround us. From the structure of our veins and arteries, the design of our lungs, the shaping of our brains and even in the nature of our behaviour. For example the behaviour of crowds of people are described by fractal patterns. These same patterns appear in the structure of rivers, the branches of trees, the nature of snowflakes and the patterns of craters on the moon.

The book helps introduce some of the key ideas of fractal understanding. For example self replication where each part of a fractal captures the essence of the whole, and thus the idea that to understand a part is to understand the totality. As another example it introduces ways in which the roughness of a fractal can be measured and places fractals intriguingly in the space between one and two dimensional objects.

It builds on these ideas by developing some of the resulting consequences. For example the coastline of Britain is a fractal and has a fractal dimension of 1.26. It goes on to illustrate that the measured length of this coastline depends entirely on the length of unit of measurement used. The smaller the unit of measurement, the greater the length, with the consequence that the length of the coastline can’t be stated with any certainty and tends towards infinity.

Imagine for example driving around the coastline, compared to walking. When walking you will follow little indentations invisible to the driver. Now imagine the coastline walked by an ant, or the coastline at the atomic level.

The fractal thus becomes a way of seeing infinity.
This idea of uncertainty is a powerful one, and one that is essential for a real understanding of change and in turn calls for us to change our thinking..

For example the book provides an alarming example of uncertainty in the solar system. Whilst Newton was able to describe the nature of gravitation between two bodies, it’s simply impossible to calculate the attraction between three or more bodies, a limitation not defined by our cleverness, but the nature of mathematics. In truth nature itself can’t predict what happens when three or more bodies interact. This is real chaos - and an interesting subject for thought in a solar system of rather more than three bodies.

This book helps reveal new perspectives on how we can see and understand these real processes. The latter part of the book then explores how some of this understanding is being applied in areas as diverse as medicine, engineering, data compression and earthquake prediction.

As with Introducing Chaos Theory it concludes with intriguing references to the understanding of fractals that appears to be inherent and locked into ancient cultures and beliefs. For example whilst modern buildings rarely stray away from Euclidian cuboid forms, gothic cathedrals and churches are for the most part fractal in design whilst traditional African societies are modelled on fractal forms.

This is an intriguing subject which I am sure has great relevance for the understanding of organisational change. This introductory book will allow you to sample the concepts within a day and who knows where the thoughts it stimulates might lead.… (meer)