is the distance between the centre of mass of the Earth and your centre of mass, but it’s a very good approximation to simply insert the radius of the Earth into Newton’s equation. This is because you are only around a couple of metres tall, and the average radius of the Earth is 6,371,000 metres, so moving your centre of mass around by a few tens of centimetres isn’t going to change the calculation much.
Plugging in the numbers, Newton’s equation tells us that the force on you at the Earth’s surface – your weight – is approximately 736 Newtons (a force of 1 Newton produces an acceleration of 1 m/sˆ2 on a 1kg mass).
We now need to introduce another of Newton’s laws – his third law of motion, also published in the Principia: To every action, there is an equal and opposite reaction. This says that the Earth exerts a force on you and you exert an equal and opposite force on the Earth. We can now understand what happens when the human towers get higher and higher. If one person stands on another’s shoulders, there is a downward force on the lower person of around 730 Newtons. If another person of the same mass climbs up, the force on the person at the base doubles to 1460 Newtons. If another two people climb up to form a tower five people high, the force on the base person is 2920 Newtons, and so on. Clearly, at some point, the person at the base isn’t strong enough to hold the tower up, and the whole thing will collapse. This is where the skill of the castellers comes in. By having a base, made up of many individuals, the forces can be distributed across the human structure, and this allows the towers to get higher before catastrophe strikes. There is clearly a trade-off; a larger base can support a larger layer above, which in turn can support a larger layer above, and so on. But a larger layer weighs more, and exerts a larger force on the layer below. The ingenious geometrical solutions to this gravitational conundrum emerge through a combination of trial and error, instinct and skill, and this is what makes the Tarragona Castells competition so compelling. For our purposes, it is the principle that matters. As the tower gets higher, the forces on the base increase, and ultimately a limit will be reached.
Perhaps you can see where this is leading. High human towers are more difficult to sustain because the force on the base becomes increasingly large as the mass of the tower increases. This suggests that the size of structures that rise above the surface of a planet is limited by the structural strength of the rock out of which the planet is made, and the mass of the planet, which sets the gravitational pull and therefore the weight of the structure. On Earth, the tallest mountain as measured from its base on the sea floor is Mauna Kea, on the island of Hawaii. This dormant volcano is 10 kilometres high, over a kilometre higher than Mount Everest. Mauna Kea is sinking because its weight is so great that the rock beneath cannot support it. Mars, by contrast, is a less massive planet. At a mere 6.39 x 1023 kg, it is around 10 per cent of Earth’s mass and has a radius about half that of Earth. A quick calculation using the equation here will tell you that an object on the surface of Mars weighs around 40 per cent of its weight at the Earth’s surface. Since Mars has a similar composition to Earth, its surface rock has a similar strength, and this implies that more massive mountains can exist on Mars because they weigh less – and this is indeed the case. The Martian mountain Olympus Mons is the highest mountain in the Solar System; at over 24 kilometres in altitude, it is close to the height of three Everests stacked on top of each other. Such a monstrous structure is impossible on Earth because of the immense weight – a result of the Earth’s greater mass and therefore stronger gravitational pull at the surface.
We see that there must be a limit to the height to which a structure can rise above the surface of a planet. The more massive the planet, the stronger the gravitational pull at its surface, and the lower the height of structures that the surface can support. As the planets get more and more massive, their surfaces will get smoother and smoother because of the stronger gravity. Less-massive planets can be more uneven. We are approaching an answer to our question; we have a mechanism for smoothing out the surface of a planet, but why should this mean that planets get smoothed into a sphere?
Imagine a mountain on the surface of a planet. Let’s say it is at the North Pole. Now, in your mind’s eye, imagine rotating the planet through, say, 90 degrees, so the mountain sits on the Equator. Has anything changed? All the arguments about the maximum height of the mountain still apply, because the gravitational force at the surface depends only upon the radius and mass of the planet and the mass of the mountain. There is no reference to any angle in Newton’s equation (here).
In more sophisticated language, we can say that Newton’s law of gravitation possesses a rotational symmetry. By that, we mean that it gives the same results for the gravitational force between any two objects regardless of their orientation. This is an example of what physicists and mathematicians mean when they speak of the symmetries of an equation or law of Nature, and it means that the calculation for the maximum height of a mountain at any place on the Earth’s surface must give the same answer irrespective of the position of the mountain because Newton’s law of gravitation is symmetric under rotations. The symmetry of the law of gravity is reflected in the symmetry of the objects it forms. Gravity smooths mountains democratically, symmetrically, with the result that lumps of matter with a gravitational pull strong enough to overcome the rigidity of the substance out of which they are made end up being spherical. This is the reason why the Earth is spherically symmetric.
There is a deep idea lurking here that lies at the heart of modern theoretical physics. Thinking of things in terms of symmetry is extremely powerful, and perhaps fundamental. Consider the possibility that the laws of Nature possess certain symmetries, which are the fundamental properties of the Universe. This would be reflected in the physical objects they create. For example, imagine a Universe in which only laws of Nature that are symmetric under rotations through 90 degrees are allowed. In such a Universe, objects that remain the same under rotations through 90 degrees are created; cubes exist but spheres are forbidden. This isn’t quite as crazy as it sounds. As far as we can tell, our Universe does possess a set of extremely restrictive symmetries, and the subatomic particles that exist and the forces that act between them are determined by these underlying symmetries.6 In fact, all of the laws of Nature we regard as fundamental today can be understood by thinking in this way. There is certainly a strong case to be made that Nature’s symmetries can be regarded as truly fundamental. The Nobel Prize-winning physicist Steven Weinberg wrote, ‘I would like to suggest something here that I am not really certain about but which is at least a possibility: that specifying the symmetry group of Nature may be all we need to say about the physical world, beyond the principles of quantum mechanics.’ Nobel laureate Philip Anderson wrote, ‘It is only slightly overstating the case to say that physics is the study of symmetry.’ Nobel laureate David Gross wrote, ‘Indeed, it is hard to imagine that much progress could have been made in deducing the laws of Nature without the existence of certain symmetries … Today we realise that symmetry principles are even more powerful – they dictate the form of the laws of Nature.’ The complexity we perceive when casually glancing at the Universe masks the underlying symmetries, and it is one of the goals of modern theoretical physics to strip away the complexity and reveal the underlying simplicity and symmetry of the laws of Nature.
Returning to the task in hand, this reasoning leads to a prediction about the size and shape of planets and moons that can be checked: they should be spherical if they are large enough, and therefore massive enough, for their gravitational pull to overcome the structural strength of the rock out of which they are made. The strength of rock is ultimately related to the strength of the force of Nature that holds the constituents of rock together – molecules of silicon dioxide, iron and so on. This is the electromagnetic force; what other force could it be? There are only four forces, and the two nuclear forces are confined within the atomic nuclei themselves. Big things like planets are shaped by the interplay between gravity, trying to squash them into spheres; and electromagnetism, trying to resist the squashing. We can perform a calculation to estimate the minimum size that a lump of matter must be to form into a near-spherical shape by equating the weight of a mass of rock near its surface to the structural strength of the rocks below.7 Our answer is approximately 600 kilometres.
We can check this by direct