Symmetry is a very powerful concept in physics. According to Noether’s theorem, for every symmetry there is a corresponding conservation law. For instance, the fact that the laws of physics are invariant with respect to time implies that energy is conserved, the fact that they are invariant with respect to spatial translation implies the conservation of momentum, and the fact that they are invariant with respect to rotation implies the conservation of angular momentum.
Group theory is the mathematical theory of symmetry. The symmetry group of an object is a collection of transformations which leave the object invariant. For instance, an object which possesses bilateral or mirror symmetry remains unchanged if it is reflected about its plane of symmetry. Mirror symmetry is an example of a discrete symmetry. On the other hand, translational symmetry is an example of a continuous symmetry, since the amount of translation is arbitrary. For instance, an infinite pipe possesses translational symmetry along its length, since it looks the same if it is moved in the direction of its length. It is this symmetry of the universe which gives rise to conservation of momentum.
Symmetries in physics do not need to have a geometric interpretation. The most important examples of non-geometric symmetries are known as gauge symmetries, and the group theories pertaining to these symmetries are known as gauge theories. Gauge theories have been used to explain conservation of electric charge as well as the strong and weak nuclear forces. The most important gauge theory is the standard model, which includes electromagnetism as well as the strong and weak forces, in fact all known forces except gravity. The gauge theory of the standard model is
SU(3) x SU(2) x U(1).
Another important concept in physics is symmetry breaking. A physical theory may possess a high degree of symmetry, but some or all of this symmetry may be spontaneously broken. An example of this is a pencil balanced on its tip. The physics of the pencil is symmetric, meaning that there is no preferred direction for the pencil to tip over. However, the pencil is unstable, so it will tip over in some direction or other. When this happens, the symmetry is said to be broken. Another example is the formation of a crystal as a substance freezes. When the substance is in the liquid state, there is no preferred direction, so everything looks the same in all directions. However, once it freezes, three principal axes are picked out for the formation of the crystalline structure, breaking the symmetry.
One of the great recent discoveries of modern physics was the unification of electromagnetism and the weak force by Glashow, Weinberg, and Salam in 1967. They discovered that at very high energies, the electromagnetic and weak forces are indistinguishable, but at lower energies, the symmetry of these two forces breaks, whence the weak force becomes much weaker than the electromagnetic force as well as having a very short range. Many physicists believe there is an even bigger symmetry which is responsible for unifying the strong force with these two forces and perhaps and even bigger symmetry yet, known as supersymmetry, further unifying gravity. The very early universe is believed to have possessed supersymmetry during the Planck Era, the first 10^-43 seconds after the Big Bang.