Symmetry | Research & Encyclopedia Articles

Symmetry

Symmetry is an intrinsic characteristic of an object that means the object remains unchanged after a symmetry operation is performed on it. Group theory is the area of mathematics that is concerned with the systematic study and formalization of symmetry. A symmetry operation is a mathematical transformation that, when performed on a symmetric object, produces an object that is identical to the original object. Symmetry operations are defined relative to a given point, center of symmetry, line, axis of symmetry, or plane, plane of symmetry, and when performed on symmetric objects preserve distances, angles, sizes, and shapes. In mathematics there are several kinds of symmetries but plane symmetry, those whose operations take place in a plane on two-dimensional figures, and spatial symmetry, those whose operations take place in three-dimensional space on solid shapes, are the most common.

Although symmetry is an important conceptual tool of modern science the history of this concept can be dated back to the times of early astronomy when scientists believed in the Ptolemaic system, the theory that the earth was the center of the universe. Early astronomers believed that because the circle is the most perfect of geometrical shapes due to its high symmetry, that astronomical objects must move in circles. They found that in this model the motions of the planets were much to complex to be considered simple circles. In 1513 Nicolaus Copernicus wrote the beginnings of the Copernican theory, a theory that put the Sun, not the Earth, at rest in the center of the universe. Using this theory it was possible to construct a model of planetary motions in terms of simple circles around the Sun. This was the first application of symmetry to a scientific problem. Group theory, a method employed in the analysis of abstract symmetrical physical systems, was formally developed by Evariste Galois near the end of his life in 1832. Although he is generally considered to be the father of group theory Carl Friedrich Gauss developed the theory about 1800 but never published it. In the early 1900s German mathematician Herman Weyl formulated the most commonly used definition of symmetry: "An object is said to be symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation as before. Any such operation is called a symmetry of the object." Later, scientists went on to formulate the symmetry of the laws of nature. The first symmetry is known as time translation symmetry that assumes that physical laws do not change with time. The second is that the laws of physics must be the same everywhere, on this side of the solar system as well as the other, and this is called translational symmetry. Lastly, the laws of physics to not change regardless of the direction that one faces. This is called rotational symmetry. Although these three symmetries have been accepted as valid symmetries of nature, Albert Einstein employed a symmetry that said the laws of physics should be the same for all observers regardless of their state of motion. He used this symmetry as the basis for developing his theory of relativity. This is the ultimate example of a symmetry principle leading to a radical revision of fundamental concepts.

There are four basic symmetry operations: rotation, inversion, reflection, and translation. Rotation means to turn an object by an angle around a defined point or line for a solid. Rotation about an n-fold symmetry axis is usually indicated as C_n. An inversion is a symmetry operation that creates a new set of inverted points P' such that OP OP' = OQ² = k² with respect to the inversion circle:

In three-dimensional space an inversion is carried out with respect to an inversion sphere. The center of symmetry under inversion is represented by i. Reflection is the symmetry operation by which an object is reflected in a mirror so that the signs of its coordinates are reversed to form an image. Reflection can be horizontal or vertical and is usually represented by n_h or n_v depending upon whether the reflection is horizontal or vertical. Translation is the symmetry operation by which the points of an object undergo a constant offset without rotation or distortion. Translation is signified by E. An improper rotation is a combination of a rotation followed by an inversion and is sometimes called a rotoinversion. An improper rotation about an n-fold symmetry axis is signified as S_n. If one object remains symmetric after more symmetry operations than another object then we say it is more symmetrical. For example, a circle would be more symmetric than a square.

These symmetry operations together produce 32 crystal classes corresponding to 32 point groups in group theory. Not only can the study of symmetry be applied to arrangements of objects but also to coefficients of equations. Quintic equations can be proved to be unsolvable by applying group theory to polynomial equations. There are two main symmetry principles that are of the utmost importance in mathematics and physics, the Noether's symmetry theorem and the symmetry principle. Noether's symmetry theorem states that each symmetry of a system leads to a physically conserved quantity. This important theorem in physics shows that symmetry under translation corresponds to conservation of momentum, symmetry under rotation leads to conservation of angular momentum, and that symmetry in time corresponds to conservation of energy. The other important principle, the symmetry principle, sometimes called the Schwarz reflection principle, indicates that symmetric points are preserved under a Möbuis transformation.