# The Beauty of Numbers in Nature

The copyright notice says 2001, 2017, from which I conclude that the book was published before, but there is no information about what this previous publication might have been. I do not have a copy, but from the reviews of Stewart's book *What Shape Has a Snowflake?* from 2001, it looks like that was the previous version of the present one. By checking the references and recent developments of some of the topics discussed, I also assume that this is still the text from 2001, that has not been updated.

In his introduction and in the first chapter of the book Stewart starts by wondering about the shape of a snowflake (this is also a pointer to the previous title of the book). Then follow some 200 richly illustrated pages, discussing all kinds of patterns that can be found in nature, until chapter 16 (the last one, entitled *The Answer*) which includes a discussion of what we know so far about the shape of a snowflake. Since the snowflake is basically only at the start and at the finish, the new current title is in fact a better description of the bulk of the contents, and the subtitle *Mathematical patterns and principals from the natural world* is the most accurate. Nature makes ample use of patterns and humans are fascinated by them. Patterns are the toys and the feedstock of mathematicians. Finding out what the rules and the (physical) laws are that generate the patterns is what mathematicians, and scientists in general, are looking for. Since most patterns can be visualised, this is a rewarding subject to illustrate mathematical principles, while avoiding all formulas. All what is required is to detect a pattern and then ask the question "why?". This is exactly what Stewart does in this book.

There are three parts. Part 1 is an overview of what patterns we can look for, part 2 relates this to some mathematics, and finally in part 3, some more advanced issues like fractals, chaos, and cosmology, are tackled, a journey that ends back on earth with a discussion of the shape of the snowflake, since indeed all the rules that govern all these complex phenomenons can all be found in the many different appearances of a tiny snowflake.

The first problem is to find an answer to the question: what exactly is a pattern?. In the first part, Stewart presents a sample book of what a pattern could be. Patterns are everywhere. We are surrounding by patterns. Think of the stripes of the zebra, the spots of a leopard, the ripples in the sand on the beach, the hexagonal honeycomb, a five-pointed starfish, a fern, the spiral of the nautilus shell, the arrangement of the seeds of a sunflower, the trajectories of the planets,... The list is almost endless. There is obviously some mathematics that can catch the regularity of a pattern like for example Fibonacci, triangular, or square number sequences, or laws for the planetary trajectories that Kepler detected, or the number of symmetries we can detect in an object, or some self-similarity rule like in the fern. All these just describe the phenomenon, but they do not explain why the pattern occurs. They are the handles with which we get a hold on the pattern and by which we can manipulate it, trying to find an answer to the why-question.

Finding an answer to the latter requires more mathematics. These elements are introduced in part two. the pattern can be one dimensional like the gait of an animal, which also has a cyclic component. So we also need a description of periodic movement like in a pendulum. Two dimensional patterns are for example the rotational symmetry of a flower or the bilateral symmetry of the human body. Stewart gives many inspiring examples of symmetry, often corresponding to the symmetry in physical laws. However the mathematical perfection is only rarely followed by nature. The two halves of the human brain look symmetric, but they perform asymmetric tasks, the bowels of the body are not symmetric, the DNA helix is only right-handed, etc.

This raises another question: if the physical laws are symmetric, and one starts from a symmetric situation, why is symmetry lost so often? If life starts from a spherically symmetric egg, why does it evolve into something that is definitely not sphere-like? If our universe started from an homogeneous point, why is it now clustered in planets, stars, and galaxies? All these components are maybe similar, but certainly not identical. How come?

More patterns are detected in tilings. They form the basis of crystal structures (an occasion to mention the sphere packing problem and the hypothesis formulated by Kepler in his treatise on the six-pointed snowflake, and proved by Hales in 1998). We are used to the symmetry and regularity in patterns so much that when in 1970 Roger Penrose found a non-periodic tiling of the plane it came as a surprise, and when in 1982 Shechtman and coworkers detected quasi crystals they were not taken seriously: nature would never be able to do this. However his finding was confirmed and in 2011 he finally got the Nobel Prize in chemistry for his discovery. (The Nobel Prize is not mentioned in the text, which might be an illustration that the text was not updated in 2017. On the other hand, by 2000 Shechtman had received many other Prizes already which were not mentioned either. A side remark: Stewart got Shechtman's name misspelled with an 'Sch'.)

Stripes can be explained as wave phenomena, and when two wave fronts interact, they may form spots. The mathematician Turing designed a complex mathematical theory of morphogenesis, but the modern view is that the patterns must develop according to an interaction between genetic switching instructions and chemical dynamics.

In three dimensions the sphere is obviously the most symmetrical object satisfying some physical optimality condition. Soap bubbles can join in much more complex structures, some of which are definitely mathematical challenges. It is remarkable that soap bubbles (minimal surfaces) meet at either 109 or 120 degrees and no other. The double bubble conjecture describes the structure of two soap bubbles meeting as a minimal surface. The conjecture was proved assuming that the joining surfaces and their interface were all parts of spheres. The theorem was only fully proved under general assumptions in 2001. (That is not included in the text either, although it is mentioned that they were closing in on a solution. This is another indication that the text has not been updated after 2001.) Besides the sphere there are of course crystal structures with only a finite number of symmetries like the popular Buckminster ball which is the basic building block in some fullerenes.

This second part ends with a discussion of spirals, whirlpools, music, colour patterns on sea shells, and the gaits of men and animals and more generally the way that animals move.

In the third part, we meet complexity and nonlinear dynamics that can result in chaotic behaviour. Bifurcations, catastrophes and symmetry breaking can and will occur under certain conditions. In Darwinian evolution theory, several bifurcations took place. Iterative systems can create whimsical shapes that can be analysed with fractal geomrtry, but even chaotic systems can sometimes be described by very simple mathematical rules, like for example the complex structure of the Mandelbrot set. Our planetary system, turbulence, weather forecasting, and population dynamics, are all examples of chaotic systems depending on the time scale they are discussed. At a cosmic scale one has to reconcile the symmetry of its origin and its current constellation and derive estimates about its possible shape and its ultimate future. After this cosmic excursion Stewart returns to the original question since even the symmetry in the shape of a snowflake is the result of chaotic dynamics and it has a fractal structure. He discusses this in a rather philosophical way. Snowflakes are born in a cloud with complex weather conditions and what shape will result depends on complex rules, too complicate to fully analyse what exactly will be the shape of the flake. Many, but not all of them, are six-pointed, and even when six-pointed, they can have many different shapes. We are only able to classify them in general groups and find some configuration of temperature and saturation to predict what type of shapes will be produced (needles, dendrites, plates,...). A carbon atoms in a living being or in a rock, are in principle the same atoms subject to the same physical laws. What are exactly the rules that define how and why the atoms evolve and stick together in so very different forms of matter? These rules are still far too complex to be fully understood by our limited set of brain cells.

This is a marvellous book, not only because of its abundance of colour pictures, but also because of the knowledgeable text written in Stewart's pleasing style that is never pedantic, always informative and smooth. Text and illustrations are perfectly in balance. I hesitate to call it a coffee table book because of its soft cover and the normal size for a book printed in a two-column format. It is however printed on glossy paper which gives it a certain cachet lifting it above an average book.

Tvy Press is the European publisher if this book. In the US and Canada, the book is published by MIT Press under the same title with a different cover and with isbn 978-0-26-253-428-4.

**Submitted by Adhemar Bultheel |

**5 / Dec / 2017