Czytaj książkę: «Forces of Nature»

The Universe in a snowflake

I love the photograph of Wilson ‘Snowflake’ Bentley (see plate section here); a tilt of the head, content, protected from the cold by curiosity, absorbed in Nature’s detail which he holds carefully in both hands, oblivious to the snow falling on his hat. No gloves. As a 15-year-old farm boy from Jericho, Vermont, Bentley spent the snow days from November to April with a battered microscope sketching snowflakes before they melted away. Frustrated by their transience, too short-lived to capture in detail, he began experimenting with a camera and, on 15 January 1885, he took the first ever photograph of a snowflake. Over the next 45 years he collected over 5000 images and dedicated his life to carefully observing and documenting the raindrops, snowfalls and mists that swept across his farm.

These delicate snapshots of a world available to everyone but rarely seen captured the public imagination. How could they not? They are magical, even today in an age familiar with photography. I challenge anyone to look at these structures, endless and most beautiful – to paraphrase Darwin – and not be curious. How do they form? What natural mechanism could mimic the work of a crazed, impatient sculptor obsessed with similarity and yet incapable of chiselling the same thing twice?

These are questions that can be asked about any naturally occurring structure, and which Darwin famously answered for living things in On the Origin of Species. In May 1898 Bentley co-wrote an article for Appletons’ Popular Science with George Henry Perkins, Professor of Natural History at the University of Vermont, in which he argued that the evidence he’d collated frame by frame revealed that no two snowflakes are ever alike. ‘Every crystal was a masterpiece of design and no one design was ever repeated’, he wrote. Their uniqueness is part of their fascination and romance, yet there is undoubtedly something similar about them; they share a ‘six-ness’. Which is more interesting? Perhaps it depends on the character of the observer.

Johannes Kepler is best known for his laws of planetary motion. He pored over the high-precision astronomical observations of the Danish astronomer Tycho Brahe, just as Snowflake Bentley pored over his photographs, and he noticed patterns in the data. These patterns led him to propose that planets move in elliptical orbits around the Sun, sweeping out equal areas in equal times and with orbital periods related to their average distances from the Sun. Kepler’s empirical laws laid the foundations upon which Isaac Newton constructed his Law of Universal Gravitation, published in 1687; arguably (I would say unarguably, but one has to keep argumentative historians happy) the first modern scientific work.

In December 1610, shortly after the publication of two of his three laws in Astronomia Nova, Kepler was walking across the Charles Bridge in Prague through the Christmas dark when a snowflake landed on his coat. The evident structure of the elegant, white near-nothing interested him, and he wrote a small book entitled On the Six-Cornered Snowflake. It is a piece of scientific writing that transcends time and provides an illuminating and entertaining insight into a great mind at play. The title page of the book is addressed ‘To the honorable Counselor at the Court of his Imperial Majesty, Lord Matthaus Wacker von Wackenfels, a Decorated Knight and Patron of Writers and Philosophers, my Lord and Benefactor’. Modern language lacks a certain flourish; I wish I had something equally magnificent with which to begin this book.

As a modern research proposal, Kepler’s Six-Cornered Snowflake would fall at the first hurdle because it begins: ‘I am well aware how fond you are of Nothing, not so much on account of its inexpensive price as for the charming and subtle jeu d’esprit of playful Passereau.¹ Thus, I can easily tell that a gift will be the more pleasing and welcome to you the closer it comes to nothing.’ Now there’s a statement of projected economic impact; the closer my research comes to nothing, the more valuable it is. Stick that on your spreadsheet… Kepler doesn’t succeed in explaining the structure of snowflakes – how could he? A full explanation requires atomic theory and a good fraction of the machinery of modern physics; we will get to that later on. What he does achieve is to make vivid the joy of science; the idea that the playful investigation of Nature has immense value, irrespective of the outcome. His book explodes with excited curiosity, fizzing with speculations on snowflakes and their similarities to other regular shapes in the natural world; five-petalled flowers, pomegranate seeds and honeycombs. He covers so much ground, bouncing thrillingly from subject to subject, that eventually, with magnificent perspicacity, he has to rein himself in: ‘But I am getting carried away foolishly, and in attempting to give a gift of almost Nothing, I almost make Nothing of it all. For from this almost Nothing, I have very nearly recreated the entire Universe, which contains everything!’

Kepler does have a clear question, however, which surely occurs to anyone who studies Snowflake Bentley’s exquisite photographs: how do structures as ordered and regular as snowflakes form from apparently form-less origins? ‘Since it always happens, when it begins to snow, that the first particles of snow adopt the shape of small, six-cornered stars, there must be a particular cause; for if it happened by chance, why would they always fall with six corners and not with five, or seven, as long as they are still scattered and distinct, and before they are driven into a confused mass?’

Kepler knew that snow forms from water vapour, which has no discernable structure. So how does the snowflake acquire structure? What is the ‘six-ness’ telling us about the building blocks of snowflakes and the forces that sculpt them? This is a modern way of looking at the world, one that any physicist would recognise. Kepler’s insight, and his delighted frustration at not possessing the knowledge to approach an answer, echoes loudly down the centuries. ‘I have knocked on the doors of chemistry,’ he writes, ‘and seeing how much remains to be said on this subject before we know the cause, I would rather hear what you think, my most ingenious man, than wear myself out with further discussion. Nothing follows. The End.’

Science is delighted frustration. It is about asking questions, to which the answers may be unavailable – now, or perhaps ever. It is about noticing regularities, asserting that these regularities must have natural explanations and searching for those explanations. The aim of this chapter, inspired by Kepler and Snowflake Bentley, is to seek explanations for the complex shapes in Nature; from beehives to icebergs; planets to free-diving grandmothers (honestly!). This will lead us to think about how such diversity and complexity can emerge from laws of Nature that are few in number and simple in form. At the end of the chapter, we will explain the structure of snowflakes.

Why do bees build hexagons?

Bees have a got a tricky problem to solve. How do you store honey, the food that will sustain your colony, through the long winter months? We know that bees build honeycombs for this purpose. Kepler was interested in the structure of honeycombs precisely because they are built, as he writes, by ‘an agent’. Since he was seeking the ‘agency’ that sculpts snowflakes, he decided to search for the reason why bees build hexagons. With the benefit of Darwin, we might propose that the answer will involve natural selection, which is a simple and powerful idea. If an inherited trait or behaviour confers an advantage in what Darwin referred to as the ‘struggle for life’, that trait will come to dominate in future generations simply because it is more likely to be passed on. The sum of an organism’s physical characteristics, behaviours and constructions is known as the phenotype, and it is on this that natural selection operates. If natural selection is the reason for the structure of honeycombs, we should be able to understand why their hexagonal shape offers an advantage to the bees that construct them.

Charles Darwin was fascinated by bees and followed precisely this path. ‘He must be a dull man who can examine the exquisite structure of a comb, so beautifully adapted to its end, without enthusiastic admiration’, he wrote in On the Origin of Species. I enjoy the directness of Victorian writing; if your mind isn’t inquisitive, you are a dullard. In the same seminal work, Darwin describes a series of experiments he conducted in order to understand the cell-making instincts of the hive bee.

‘… it seems at first quite inconceivable how they can make all the necessary angles and planes, or even perceive when they are correctly made. But the difficulty is not nearly so great as it first appears: all this beautiful work can be shown, I think, to follow from a few very simple instincts.’

To identify these simple instincts, Darwin compared the hive-making behaviours of the honeybee with a less architecturally accomplished species of bee, the Mexican Melipona domestica. The Melipona bees construct regular combs of cylindrical cells which Darwin asserted to be a simpler geometrical form, intermediate between no structure at all and the hexagons of the honeybees. ‘We may safely conclude that if we could slightly modify the instincts already possessed by the Melipona, this bee would make a structure as wonderfully perfect as that of the hive bee.’

To test the hypothesis, Darwin conducted a series of experiments in collaboration with his friend and fellow naturalist William Bernhardt Tegetmeier. They added different-coloured dyes to the beeswax, enabling them to create a visual record of the construction process, and were able to conclude that the bees first build cylindrical cells that are subsequently modified to form hexagons. Darwin was able to describe this in terms of natural selection:

‘Thus, as I believe, the most wonderful of all known instincts, that of the hive-bee, can be explained by natural selection having taken advantage of numerous, successive, slight modifications of simpler instincts; natural selection having by slow degrees, more and more perfectly, led the bees to sweep equal spheres at a given distance from each other in a double layer, and to build up and excavate the wax along the planes of intersection. The bees, of course, no more knowing that they swept their spheres at one particular distance from each other, than they know what are the several angles of the hexagonal prisms and of the basal rhombic plates. The motive power of the process of natural selection having been economy of wax; that individual swarm which wasted least honey in the secretion of wax, having succeeded best, and having transmitted by inheritance its newly acquired economical instinct to new swarms, which in their turn will have had the best chance of succeeding in the struggle for existence.’

Darwin concluded that bees build hexagonal honeycombs because they are the most economical way of dividing up their honey storage area. Hexagons use less wax, and the bees that use less wax are more efficient and more likely to survive and pass on their inherited behaviour to the next generation. This makes sense, because building a wax hive is extremely honey-intensive; for every gram of wax a bee produces it has to consume up to eight grams of honey. There is clearly an impetus to build efficiently, since using as little wax as possible maximises the honey available for food – an advantage that will have shaped the behaviour of honeybees over generations.

Is this correct? It’s certainly plausible. If bees used cylinders to build their honeycomb there would be gaps between each cell and the whole structure would be less efficient. Similarly, pentagons and octagons also produce gaps and so cannot be optimal. It is possible to imagine that each cell could be constructed in a bespoke shape by each bee to fit perfectly with its neighbour. In this ‘custom-made’ scenario each cell would be a different shape, but the gaps in the honeycomb could still be minimised. A problem with this strategy might be that one bee has to finish before the next bee can create a cell to fit. That’s an inefficient use of time. A repeatable single shape that leaves no gaps would seem to be preferred. The square, the triangle and the hexagon are the only regular geometrical figures that can fit together in a plane without leaving gaps.²

But why do bees use hexagons? Sometime around 36 BC, the Roman scholar Marcus Terentius Varro wrote down the earliest-known description of the honeycomb conjecture. This states that the most efficient way to divide a surface into regions of equal area (cells) with the least total perimeter (wax) is to use a regular hexagonal grid or honeycomb. No proof was offered, and the assertion remained conjecture for the next 2000 years until, in 1999, a mathematician at the University of Michigan named Thomas Hales found a proof: a hexagonal pattern is the most efficient engineering design. Natural selection, selecting for efficiency and creating structures that are a shadow of an elegant underlying mathematical law. What a beautiful answer to a simple question.

‘BEES, THEN, KNOW JUST THIS FACT WHICH IS USEFUL TO THEM – THAT THE HEXAGON IS GREATER THAN THE SQUARE AND THE TRIANGLE AND WILL HOLD MORE HONEY FOR THE SAME EXPENDITURE OF MATERIAL IN CONSTRUCTING EACH.’

— PAPAS OF ALEXANDRIA, AD 340

Well … possibly, but there may be more to it. In 2013, three engineers – Karihaloo, Zhang and Wang – published an article entitled ‘Honeybee combs: how the circular cells transform into rounded hexagons’. The claim is that honeybees, just like the Melipona bees that Darwin dismissed as inferior architects, make cells that are initially circular in cross section. The hexagons appear because the bees’ body heat softens the wax until it reaches 45 degrees Celsius, a temperature at which wax begins to flow like a viscous fluid. The circular cells of molten wax then act in a similar way to soap bubbles, joining together at an angle of 120 degrees wherever they meet. If all the bubbles or wax cells are identical in size and spacing, the circular cells spontaneously reform into a sheet of hexagons. Karihaloo and his team demonstrated this by using smoke to interrupt honeybees in the process of making a hive, revealing that the most recently built cells were circular, whilst the older ones had developed into hexagons. This transition from cylindrical to hexagonal structure appears to be what Darwin observed, but the explanation for the transition is different.

Natural selection is still the basic explanation for the hexagons, but the bees don’t have to go to the trouble of building the most efficient packing shape because physics will do that for them, given a nice sheet of circular cells of similar size and spacing and some body heat. To me, this is even more elegant and efficient; the bees allow physics to finish their work! As the authors of the study write: ‘We cannot … ignore, nor can we not marvel at the role played by the bees in this process by heating, kneading and thinning the wax exactly where needed.’ Is this the solution to a problem that has fired the imagination of so many for so long? The origin of the hexagons continues to be debated, and Karihaloo et al. will probably not be the last word in the literature.

This is as it should be, and illustrative of something that is often missed in the presentation of science. Scientific results are always preliminary. No good scientist will believe that they have offered the last word on a given subject. A result is published if the authors and a group of their peers consider it to be a valuable contribution to the field. Crucially, this does not mean that it’s correct; it means that it’s not obviously wrong. Rather than closing down a question, publication is intended to be a red flag to bullish colleagues. As one reads in Kepler’s partial, yet evident, delight in not discovering a satisfactory explanation for the structure of a snowflake, there is joy in hearing what you think, my most ingenious colleague.