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.
IT IS ONE OF THE GREAT ACHIEVEMENTS OF MODERN SCIENCE THAT WE HAVE DISCOVERED LAWS OF NATURE, AND THAT THESE LAWS PROVIDE A COMMON EXPLANATION FOR MANY NATURALLY OCCURRING STRUCTURES. IT IS ONE OF THE GREAT MYSTERIES THAT SUCH ECONOMICAL EXPLANATIONS EXIST AND ARE AVAILABLE TO US.
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.
The complex beauty and practicality of honeycombs have long fascinated scientists. Is their hexagonal form a deliberate result of natural selection to maximise their honey storage?
‘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
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 honey bees 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
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 are a shadow of an elegant underlying mathematical law. What a beautiful answer to a simple question.
Beyond just the simple fear of a thousand honeybees’ excruciating stings, humans go to extraordinary lengths to gather the precious commodity that lies within. Perhaps nowhere is this risk more extreme than in the Himalayan cliffs of central Nepal where the world’s largest honeybee, the Himalayan Cliff honeybee, makes its home.
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 1
20 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.
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.
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.
The latest theory on the formation of honeycombs, expounded by Karihaloo, Zhang and Wang asserts that the bees’ honeycombs start life circular then join together like bubbles to create hexagonal shapes.
Knocking on the doors of chemistry
In the final lines of The Six-Cornered Snowflake, Kepler writes with lovely regret that he is ‘knocking on the doors of chemistry’; the implication being that those doors would be opened by future generations. He asserts, correctly, that the structure of the snowflakes must be due at least in part to some underlying structure or shape, but given that atomic theory didn’t move into the realm of experimentally testable science until the early nineteenth century, and the structure of atoms themselves was a twentieth-century discovery, Kepler had no way of unlocking the doors. We now know that the building blocks of snowflakes are water molecules, and water molecules are capable of extremely complex behaviour when they get together. That may be a surprising statement if we think of water as the colourless, odourless liquid in a glass. Perhaps it shouldn’t be so surprising if we think of water molecules as the objects that come together spontaneously to produce the romantic flourishes of form and exquisite diversity of snowflakes.
Single water molecules aren’t particularly complicated. They are molecules of hydrogen and oxygen, bonded together. Oxygen was first isolated in 1774 by Joseph Priestley, the son of a Yorkshire woollen cloth maker, and Henry Cavendish first identified hydrogen in 1766. The Nobel Prize in Physics in 1926 was awarded to Jean Baptiste Perrin for the confirmation of the physical reality of molecules, just about within living memory, which demonstrates how difficult it is to study the microscopic world and how quickly cutting-edge science can become common knowledge.
A water molecule consists of two hydrogen atoms bonded to a single oxygen atom: H2O (see illustration). The water molecule isn’t linear – the hydrogen atoms are displaced at an angle of 104.5 degrees. The reason for this is the presence of two extra pairs of electrons that sit on the opposite side of the oxygen atom. To see why that is, let’s have a very brief tutorial on atomic physics and quantum mechanics.
Atoms consist of a small, dense, atomic nucleus made up of protons and neutrons, with electrons attached a long way away. If the nucleus were the size of a tennis ball, the atom would be several kilometres across.
Atoms are made up of three constituents, as far as chemists are concerned (we’ll dig more deeply into this later on); they consist of a small, dense, atomic nucleus made up of protons and neutrons, with electrons orbiting a long way away. If the nucleus were the size of a tennis ball, the outer electron orbits would be several kilometres across. Hydrogen is the simplest element; its nucleus consists of a single proton. Next is helium, which contains two protons and two neutrons. Oxygen has eight protons and eight neutrons. The nucleus is surrounded by electrons, which are held in place by one of the four fundamental forces of nature: electromagnetism. Electrons are negatively charged and protons are positively charged, and the negative electric charge of the electron is precisely equal in magnitude but opposite in sign to the positive electric charge of the proton. Nobody knows why these charges are precisely equal in magnitude; it’s one of the great mysteries of fundamental physics. The atoms of each chemical element are electrically neutral, which means that the number of protons in the nucleus is equal to the number of electrons that surround it. Hydrogen atoms have a single electron, therefore, whilst oxygen atoms have eight electrons.
At the northern tip of Alexander Island, in the Antarctic Peninsula, every one of these icebergs begins its life far from the oceans as the lightest of snowfalls on the icecaps of Greenland. For an iceberg born in the twenty-first century that may mean that the snow that made it fell from the sky whilst the Egyptians were still building the pyramids. These ancient fragile flakes of snow are then slowly transformed as layer after layer of snow is lain down in the same area.
Now we need a little sprinkle of quantum theory. You can picture the electric charge of the atomic nucleus as creating a kind of box within which the electrons are trapped. Electrons, along with all of the fundamental building blocks of the Universe, obey the laws of quantum theory, which describe how they move. It turns out that the basic rules of quantum theory are counterintuitive and fly in the face of common sense. But that is okay because there is no reason at all to expect the laws that govern the Universe to be in accord with ‘common sense’. The most fundamental rule governing the behaviour of subatomic particles is that they don’t like to stand still. Unfettered, they are very likely to wander off, and the more we try to pin them down, the more they are inclined to wander. The presence of the nucleus tames the anarchic electrons somewhat, by confining them to the ‘nuclear box’.
The structure of a water molecule, showing oxygen’s eight electrons, two of which are shared with the hydrogen atoms.
Another rule governing the behaviour of electrons is that they don’t much like each other’s company. This is known as the Pauli exclusion principle, also a consequence of the laws of quantum theory. Electrons will arrange themselves around the nucleus such that they stay away from each other, as best they can. There is a caveat, though, which is important for understanding the structure of atoms. Electrons of opposite spin are allowed to get close together (or ‘pair up’). Of course they cannot get too close because they have the same electric charge and ‘like-charges repel’. Spin is a property of subatomic particles that is easy to name but hard to picture. You could think of electrons as little spinning tops, if you like, but that’s a bad analogy on many levels, so you probably shouldn’t. Having said that, spin is a measure of how much an electron is spinning – it is just that the notion of a spinning point is not something we can easily imagine. For particles such as electrons, which are known as ‘spin ½’ particles or fermions, spin can have only two values; these are known as spin-up and spin-down. Spin is a direct, if rather subtle, consequence of the merger between Einstein’s Theory of Special Relati
vity and quantum theory, achieved by physicist Paul Dirac in his equation describing the electron in 1928. The details don’t matter here; what matters is that the negatively charged electrons get trapped by the positive electric charge of the protons in the atomic nucleus and that electrons tend to keep away from each other, although opposite-spin electrons can get closer together than same-spin electrons can. This is enough information for us to get a basic understanding of a water molecule. Oxygen has eight electrons. Two of the electrons sit close to the nucleus and do not play much of a role in binding the two hydrogen atoms to the oxygen. The remaining six are shared out as in the diagram above.3
The hexagonal crystalline structure of ice 1h. Water molecules are attached together by hydrogen bonds, with oxygen atoms from one water molecule lining up with hydrogen atoms from another.
One of the basic concepts in chemistry, which again goes all the way back to the fundamental laws of quantum theory, is that electrons can be shared between atoms. This results in the formation of a chemical bond. Two hydrogen atoms will share their single electrons with an oxygen atom if they can, pairing up to fill the two remaining outer slots around the oxygen nucleus; the result is a water molecule, which is shown in the top illustration. The reason for the 104.5-degree ‘kink’ is the presence of the other two pairs of electrons in the outer level of the oxygen atom. They take up residence on the opposite side of the oxygen atom to the hydrogen atoms, giving the water molecule its distinctive shape, and its many unusual properties.
Forces of Nature Page 2
