Symmetry has long fascinated the great minds in all disciplines from art to theoretical physics. We are drawn to symmetrical objects such as snowflakes, planets and even the human body. But how can we define symmetry itself? The mathematician Professor Hermann Weyl gave an excellent definition of this elusive concept: an object is symmetrical if it looks the same after it is transformed in some way. For instance, if a snowflake is rotated by 60 degrees, it is indistinguishable from the original so it has a particular kind of symmetry.
Among everything found in nature, living things exhibit the greatest variety and most fascinating range of symmetries. Symmetry in form and behaviour is a key factor in the survival of all organisms, from the leaping of a lion to the capturing of light by photosynthetic plants. A lion’s need for symmetrical movement and balance is clear, but the advantages of symmetry for plants and nature’s tiniest creatures are far more subtle.
Honeybees are experts in the use of symmetry for efficiency. Worker bees construct hexagonal cells to house their supply of honey. But there is more to the hexagonal grid than mere beauty. Only recently, in 1999, the mathematician Thomas Hales proved that this arrangement minimises the amount of wax required. Ever since Pappus of Alexandria discussed it around 300 AD, this ‘honeycomb conjecture’ had been an open problem in mathematics.
Even the simplest of organisms display extraordinary self-similarity. For instance, the chambered nautilus builds its shell using ever-larger compartments, each of the same shape, creating a curve known as the Fibonacci spiral. Apart from providing protection, the shell controls the buoyancy of the animal. Older compartments are sealed and contain gas, allowing the animal to float freely and move easily in water, improving its chances of finding food and escaping from predators.
Such Fibonacci spirals occur in plants as well, which use the spiral to arrange leaves and petals to maximise exposure to light and to attract insects. With each successive leaf projecting outwards at roughly 137.5 degrees to the previous leaf, this arrangement gives minimal overlap between neighbouring leaves. Indeed, the number of petals, leaves or seeds in each spiral is often one of a special series of numbers known as Fibonacci numbers. Evolution, it seems, has converged upon this theoretical maximum to maximise growth, survival and reproduction.
Although symmetry underpins some of the greatest triumphs of evolution, asymmetry also has an important role in living systems. Many biological molecules, such as lactose, can exist in two different forms that are mirror-images of each other, known as enantiomers. Although they have the same structural formula, they are not identical. In order for an organism to produce both enantiomers, two different enzymes would be required. Therefore, in order to conserve energy, many living systems only produce one and pass on this capability of only processing one enantiomer to their offspring. After millennia of evolution, all organisms on Earth produce only the enzyme required to digest the version of the compound that is found in their food. Enantiomers can have quite different properties: the artificial sweetener aspartame has one enantiomer that tastes sweet and another that tastes bitter. This indicates that the taste receptors have a complementary asymmetry, in order to bind to the two aspartame enantiomers. In answer to Lewis Carroll’s Alice, who wonders whether looking-glass milk would taste good: it would probably not taste like milk and would be impossible to digest.
Non-living objects display an equally striking range of symmetries. It is a wonder that we are surrounded by such a wide variety of symmetrical objects, but this is not a coincidence. Not only are the objects we encounter symmetrical, but so are the physical laws that describe their behaviour. For example, the snowflake is symmetrical simply because the physical laws which governed its creation are symmetrical.
But how can a physical law be symmetrical? A law is symmetrical in space if it is unchanged by moving from one point to another. If everything in the universe could be translated (shifted in one direction), preserving the relative positions of planets, stars and galaxies, the motions and interactions between them would proceed uninterrupted—the physical laws that describe the motion are unchanged by a translation.
One of the startling observations about the laws of physics is that they are not only symmetrical in space, but also in time. This idea that an experiment will give the same result even if performed at two different points in space or in time has enormous implications for physics. In particular, this invariance of the laws of physics is the principle on which Einstein’s theory of relativity is based. Since there is no experiment which can distinguish between one point in space-time from another, Einstein concludes that there is no such thing as absolute time or space and different observers may disagree on the duration of time or even the order in which events took place.
Almost every physical law known today is symmetric in time, and postponing an experiment has no effect on its result. Indeed, these laws are unchanged even if the direction of time is reversed. However, this is contrary to our experience since time appears to flow in one direction, with the past inaccessible from the present. The universe began at a single, fixed point in time. The only known physical law that is not time-symmetric is the second law of thermodynamics, which states that entropy (a measure of disorder) cannot decrease with time. Our sense of time seems incompatible with our current description of the universe but it is a fundamental part of reality: we all know a tower of bricks can fall over but will never reconstruct itself, even though this is permissible by the laws of motion. Time appears to flow in the direction of increasing entropy. The nature of this arrow of time is one of the great mysteries of modern physics and must be resolved if science is to achieve its ultimate goal of discovering a ‘Theory of Everything’.
It is therefore no surprise that symmetry is a standard tool in theoretical physics; in 1918 Emmy Noether published a theorem that explained the deep connection between symmetry and physical laws. Noether’s theorem allowed physicists to take a symmetry property of the universe, such as spatial symmetry, and find a corresponding conservation law, such as the conservation of momentum. Similarly, rotational symmetry leads to the conservation of angular momentum and time translational symmetry leads to the conservation of energy. Such principles allow physicists to solve a wide range of problems. Noether’s theorem guarantees that any new physical laws we discover, as long as they are symmetric in space and time, will conserve momentum and energy.
Symmetry is the most enduring aspect of all scientific theories. While refinements are made and outdated theories refuted, the underlying symmetries remain. Indeed, Einstein’s theory of relativity is a consequence of a few basic principles of symmetry. It seems symmetry is an inherent part of our Universe and it continues to guide the discovery of new physical laws.
Jack Williams is a 1st year undergraduate in the Department of Mathematics