Forces of Nature

by Professor Brian Cox

  The most widely accepted view is that having bilateral symmetry confers an advantage over radial symmetry because it enables organisms to move more efficiently. Think of the shape of a shark. It is sculpted like a submarine for good reason; it faces the same engineering challenge of moving quickly and efficiently through water. We don’t build high-speed radially symmetric submarines; they all look like sharks. As life began to sense the world and move around in ever more complex ways, it was a body plan with a left and right and a top and bottom that could best house a central nervous system and provide the agility needed in the emerging world of predator and prey.

  There is, however, another possibility. Jellyfish also exhibit bilateral symmetry, but it is internal. From an evolutionary perspective, this is important. It may suggest that bilateral symmetry initially evolved to improve the internal functioning of organisms; it allows, for example, for an efficient separation of the gut and respiratory systems. Furthermore, the genes responsible for the development of bilaterally symmetric animals are also found in jellyfish and sea anemones. This is taken as evidence that the common ancestor may have possessed bilateral symmetry, and the external radial symmetry we see in some complex multicellular organisms was a later evolutionary development.

  Having bilateral symmetry confers an advantage over radial symmetry because it enables organisms to move more efficiently.

  This is yet another example of the wonderful pace and ever-shifting nature of science. Yesterday’s textbook explanation can become tomorrow’s historical curiosity, and this is precisely as it must be if our knowledge of the natural world is incomplete and continually growing. Progress in science implies that we understood less yesterday than we will tomorrow. That said, I think it is unarguable, and wonderful to consider, that we can trace our ancestry back through geological time to a period in Earth’s history some half a billion years ago, when we shared a common ancestor with all the multicellular animals present on Earth today, and it would have looked something like the Kimberella.

  The Universe in a snowflake

  Let’s finish where we began, bringing together everything we’ve discovered to answer Kepler’s question about the origin of the individuality and collective symmetrical beauty of snowflakes. We will follow the formation of a snowflake from its beginnings in a high cloud to its gentle arrival on the ground. Snowflakes are formed by water vapour condensing directly into ice. They are not frozen raindrops; they are crystals that grow steadily larger as they journey through the clouds.

  The structure of ice crystals at the temperatures and pressures that we find on Earth is shown in the illustration below. The 104.5-degree bond-angle of a free water molecule – a consequence of the laws of quantum theory – is the reason for a hexagonal crystalline structure, which in turn is the underlying reason for the six-fold symmetry of snowflakes. The hexagons are clearly visible in Snowflake Wilson’s photographs on here. The snowflakes are imperfect shadows of a more ‘perfect’ form – the ice lattice; itself a consequence of the structure of the water molecule, which is a physical manifestation of the underlying fundamental laws of Nature that created it – the quantum theories of the strong, weak and electromagnetic forces. When you look at a snowflake, you are seeing the primal structure of our Universe.

  And yet, despite the underlying simplicity, each snowflake is different. Why? Because of their individual formation histories. As we saw for planets, galaxies and grannies, simple laws of nature can sculpt an infinity of forms because the initial conditions and histories of formation are never precisely the same. The symmetries of the laws are obscured by history, and it is the job of the scientist to see through the distorting lens of history. A clue as to how this can be done for snowflakes can be found in what is known as the morphology diagram, shown in the illustration below. The morphology diagram shows how the structure of snowflakes varies with temperature and humidity.

  And yet, despite the underlying simplicity, each snowflake is different. As we saw for planets, galaxies and grannies, simple laws of Nature can sculpt an infinity of forms because the initial conditions and histories of formation are never precisely the same.

  The vertical axis of the diagram shows the humidity: the moisture content of the clouds within which the snowflakes form. The horizontal axis shows the temperature. Large, fluffy snowflakes with lots of intricate branches form at high humidity and temperatures between about -10 degrees Celsius and -20 degrees Celsius. At lower temperature and humidity snowflakes are small unbranched hexagons. Higher temperatures lead to needles and prism shapes. For complex, intricate snowflakes, humidity needs to be high. The diagram provides a clue as to how snowflakes can be similar and yet individual. By plotting the data in this way, we see that different patterns of crystal growth are favoured by different conditions and histories of formation.

  The morphology diagram shows how the structure of snowflakes varies with temperature and humidity.

  To better comprehend this, we need to understand how the snowflake crystals grow. The process by which the geometry of the water molecules is transferred to the snowflake is known as faceting. A small ice crystal in a cloud grows because other water molecules bump into it and stick to it through hydrogen bonds. Faceting occurs because rough, uneven bits of the crystal have more available sites for water molecules to bind to; smooth bits, on the other hand, have fewer. This means that rough regions of the initial crystal will grow faster than smooth regions, and become smoother as the jagged spots are filled in. Faceting produces flat, hexagonal prism shapes like those labelled ‘solid plates’ in the morphology diagram. The plates are flat because water molecules are more likely to bind to the rectangular thin edges, which are known as prism facets, than to the hexagonal top and bottom surfaces, which are known as basal facets. In low-humidity conditions this is the dominant method of growth, which is why snowflakes can remain broadly hexagonal with few intricate branches. If you look back to Snowflake Bentley’s photographs on here, the snowflake in the top right-hand corner, labelled 780, is of this type.

  The interplay between the laws of Nature, which are simple and deeply symmetric, and history, which is long and messy, produces the complex world we inhabit.

  Complexity arises from another form of crystal growth called the branching instability. If a bump forms on the crystal surface, the tip of the bump is slightly more likely to accumulate water molecules because it sticks out further into humid air. This causes the bump to grow rapidly, which is why it is referred to as an instability. Branching competes with faceting for a limited supply of water molecules, and it is this competition that leads to the complexity of snowflakes. The corners of the hexagonal prisms are subject to the branching instability – they tend to grow faster than the flat sides, causing them to become concaved. This is resisted by faceting, because there are more sites available for bonding in the centre of the concaved surface between the edges. More water molecules are available at the points, but water molecules are more likely to stick to the centre of the resulting curves. The two processes compete with each other for water molecules. If the air is humid and there is a plentiful supply of water molecules, branching dominates because the rate of growth of the instabilities outstrips the rate at which water molecules diffuse down to the faceted surface below; the corners of the hexagonal prisms therefore tend to grow rapidly, producing intricate, star-like branches. If the air is less humid, the growth rate of the branches falls below the diffusion rate and faceting dominates; the crystal evolves towards a smoother, simpler shape.

  As the snowflake grows, it will pass through many different regions in a cloud and experience different conditions, passing through more humid and less humid air, and through regions of differing temperature. Each of these different regions will favour a different type of growth; sometimes faceting will dominate, and other times branching will win out. Look again at Snowflake Bentley’s images. You can read the history of each snowflake from the inside out; they all begin with little hexagons; when
they are small, faceting always wins. If they enter humid air, branching drives an explosion of intricacy. They may drift into a less humid region and the smooth, faceting growth reasserts itself. This is the reason why every snowflake is different. Each one follows a unique path through the clouds, and every detail of this path is written into its structure. The snowflakes retain an element of the underlying symmetry of the crystal because conditions do not vary in clouds over distances of a few centimetres, which is the size of a snowflake. Each corner therefore experiences precisely the same conditions, which lead to the same structural growth. If one side of a snowflake experiences a different history to the other – perhaps it is involved in a collision – then the symmetry of the snowflake is lost. There are, of course, many snowflakes that reach the ground in a battered, asymmetric state, but we don’t take pictures of those!

  As a physicist, I have to observe that snowflakes are four-dimensional objects; their structure can only be understood with reference to their history, and their history is encoded visibly into their structure. You can read a snowflake like a history book. Precisely the same observation can be made about living things. It is impossible to understand the structure of a manatee unless you understand its evolutionary history. Why does a manatee have finger bones embedded in its flippers? Because they evolved from the legs of a small land-dwelling ancestor. Living things are a snapshot, a temporal shadow of a much grander 4-dimensional story; they encode the entire history of life on Earth, stretching back four billion years, into their structure. No wonder they are complex and difficult to understand. Every twist and turn of history is faithfully recorded.

  The interplay between the laws of Nature, which are simple and deeply symmetric, and history, which is long and messy, produces the complex world we inhabit. The triumph of modern science is that we can separate the two, and this has led to discoveries of overwhelming importance. The seeds of this approach are clearly visible in the writings of Kepler, all those years ago. ‘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 … ?’ he asks. And there it is: Nature is beautiful, deep down, and we want to glimpse that underlying beauty. Let’s not guess. Let’s not make something up. Let’s think, observe, experiment, pay attention, look for similarities and differences across the natural world and try to understand them. Most of all, let’s be comfortable, delighted, exhilarated when faced with the unknown and devote our time to exploring the infinite territory beyond. There are treasures beyond imagination in the simplest things, if we care to look closely.

  1 It is generally accepted that ‘Passereau’ is used here by Kepler as a pun, connecting a playful sparrow with French poet Jean Passerat, who wrote a New Year’s poem on the subject of Nothing. Obscure, but fun!

  2 There are many ways of tiling a plane using tiles of more than one shape; Penrose tiling is a particularly interesting example, in which the plane can be tiled by a set of ‘aperiodic’ tiles that form a pattern that never repeats.

  3 The details of where the electrons reside in any particular molecule are generally very hard to compute. The details follow from solving the Schrödinger equation in the spherically symmetric potential generated by the atomic nucleus. If you’d like more details, and are interested in delving more deeply into quantum theory, there is much more in my book with my colleague Jeff Forshaw entitled The Quantum Universe.

  4 There is currently some discrepancy between different measurements of the proton radius, which may signal something interesting that we don’t understand. See, for example, http://arxiv.org/pdf/1502.05314.pdf, which is a technical paper, but I recommend a glance because it demonstrates rather beautifully the precision of modern particle physics.

  5 George Orwell, Politics and the English Language.

  6 1024 means 1 followed by 24 zeros. So we could write the number 100 as 102, 1000 as 103 and so on. Notice how big a number 1024 is. 102 is 10 x 10. 103 is 10 x 10 x 10. So 1024 is 10 x 10 x 10 …. 24 times.

  7 These are not symmetries of three-dimensional space, like the rotational symmetry of a cube. They are more abstract symmetries.

  8 The smallest size of a round lump of rock and the height of the tallest mountains on Earth

  Imagine a big cube of rock sitting on the surface of a second much bigger ball of rock, like a planet maybe (we are thinking of a cube for the sake of being specific but any shape will do). If the cube is too big then its weight will cause the rock underneath to fail and the cube will sink. Obviously it takes a lot of weight before rock starts to deform and give way. For granite, the maximum pressure before failure is around 130 million Newtons /m2 (written 130 MPa), which is a little more than 1000 times atmospheric pressure. We will assume that our big ball of rock has compressive strength of around 100 MPa, and we label it using the symbol P. Now we need to know how heavy the cube is, given that its height is h. Its weight is equal to its mass multiplied by GM/R2 (from Newton’s law), where M is the mass of the big ball and R is its radius. If the density of the cube is d = 3000kg/m3 (typical of rock) then its mass is d x h3. The mass of the ball will likewise be M = 4/3 x 3.14 x R3 x d (3.14 is the mathematical number pi, and we have used the formula for the volume of a sphere). For our purposes 3.14/3 is close enough to 1 as to make no difference (the goal here is to make a rough estimate, not a highly accurate computation). Together, these results mean that the weight of the cube is d x h3 x G x 4R x d. Now, this weight bears down on the ground below, which will give way if the weight is bigger than the compressive strength of the rock supporting it, which is P x h2. In other words, the ground will give way under the cube if h3 x G x 4R x d2 is bigger than P x h2. This implies that h must be smaller than P/G/4/R/d2. If this maximum value for h is less than 10 per cent of the radius of the ball, the surface of the ball will not be too much deformed from spherical by the cube. (i.e. the cube will be a small bump on the surface of a bigger ball). Putting h/R = 0.1 tells us that the planet’s radius R must be bigger than the square root of P/G/4/d2/10 per cent. Putting in the numbers gives a radius equal to just over 600 kilometres. This number should not be taken too literally, because we used typical numbers for the density and the compressive strength and these will vary across the variety of planets, asteroids and comets. But that should not detract from what we have achieved. Our calculation is telling us that lumps of rock larger than about 600 kilometres in radius will tend to look pretty smooth because big structures on their surface will tend to sink down and be absorbed. While we are at it, we can quickly go ahead and estimate the size of the biggest mountains on Earth and Mars. We have already worked this out above. The maximum size of a cubic mountain on Earth would be P/G/4/R/d2. On Earth, the combination GM/R2 (where M is the mass of the Earth and R is its radius) is called the acceleration due to gravity, g, and it is close to 10 m/s2. This means that our cubic mountain would sink if it were taller than P/d/g, which is around 3.3 kilometres. If the mountain is cone shaped instead of cubic then this number increases by a factor of 3 to around 10 kilometres, which is very close to the height of the largest mountains on Earth. On Mars, the surface gravity is around 40 per cent that of the Earth, which means that its tallest mountains should be more like 10 kilometres/40 per cent = 25 kilometres high, which is the height of Olympus Mons.

  Somewhere in spacetime

  ‘Suddenly, from behind the rim of the Moon, in long, slow-motion moments of immense majesty, there emerges a sparkling blue and white jewel, a light, delicate sky-blue sphere laced with slowly swirling veils of white, rising gradually like a small pearl in a thick sea of black mystery. It takes more than a moment to fully realize this is Earth . . . home.’

  – Edgar Mitchell, Apollo 14, in orbit around the Moon, February 1971.

  ‘Alas I have little more than vintage wine and memories.’

  – Uncle Monty, Withnail and I, Camden,
1987.

  Do you remember a perfect summer’s day? Not precisely where or when, but a moment of languid warmth, pepper scents, itchy grass, airborne seeds and brushing insects. Great artists are able to summon the tangled past into vivid present experience because memories are indelible. Claude Monet captured notes of a thousand childhood summers in his Coquelicots (Poppies). Depicting a rural landscape near the village of Argenteuil, it remains one of Monet’s most loved and most recognised paintings.

  Monet painted a series of similar pictures during the summer and autumn of 1873. The precise date and time of the scene are unknown, but the poppy fields around Argenteuil are in full bloom in late May and early June, and the bright sky and lack of shadow suggest that the scene was set at around noon. Let’s take artistic licence of our own, though, and label the moment when a little boy ambled through the poppies with his mum, and Monet placed a carefully considered dab of red paint on his canvas. Let’s say it was noon, 26 May 1873, in a field close to the village of Argenteuil, France. Almost 150 years later, that day has slipped from living memory and exists only in Monet’s painting. The place is still there, but the moment is gone. Whimsical common sense, you may say, but is it correct?

  In 1905, Albert Einstein published his Theory of Special Relativity, which contains the famous equation E=mc2. I take great comfort in the fact that there is such a thing as a famous equation; it allows me to imagine that I glimpse a flicker of intellectual depth illuminating the all-enveloping darkness of popular culture. Special Relativity deals with moments, or more precisely events. An event is something that happens at a particular location in space and at a single instant in time. Monet’s dab of paint on the canvas is an event; it has a location and there is a time that it happened. Einstein’s theory tells us how we should measure the distance between events and how to think about their connection to each other. It is a theory of space and time.