SYMMETRY

in physics, the concept that particles such as atoms and molecules remain unchanged in properties by symmetry operations. From the earliest days of natural philosophy (Pythagoras in the 6th century BC), symmetry has furnished insight into the laws of physics and the nature of the cosmos. The two outstanding theoretical achievements of the 20th century, relativity and quantum theory, involve notions of symmetry in a fundamental way. The application of symmetry to physics leads to the important conclusion that certain physical laws, particularly conservation laws, are unaffected by symmetry operations on the geometric coordinates of the particles concerned, including time, when it is considered as a fourth dimension; i.e., the laws remain valid at all places and times in the universe. In particle physics, considerations of symmetry can be used to derive conservation laws and to determine which particle interactions can take place and which cannot (the latter are said to be forbidden). Symmetry also has applications in many other areas of physics and chemistryfor example, in relativity and quantum theory, crystallography, and spectroscopy. Crystals and molecules may indeed be described in terms of the number and type of symmetry operations that can be performed on them. The quantitative discussion of symmetry is called group theory. Valid symmetry operations are those that can be performed without changing the appearance of an object. The number and type of such operations depends on the geometry of the object to which the operations are applied. The meaning and variety of symmetry operations may be illustrated by considering a square lying on a table. For the square, valid operations are (1) rotation about its centre through 90, 180, 270, or 360 degrees, (2) reflection through mirror planes perpendicular to the table and running either through any two opposite corners of the square or through the midpoints of any two opposing sides, and (3) reflection through a mirror plane in the plane of the table. There are therefore nine symmetry operations that yield a result indistinguishable from the original square. A circle would be said to have higher symmetry because, for example, it could be rotated through an infinite number of angles (not just multiples of 90 degrees) to give an identical circle. Subatomic particles have various properties and are affected by certain forces that exhibit symmetry. An important property that gives rise to a conservation law is parity. In quantum mechanics all elementary particles and atoms may be described in terms of a wave equation. If this wave equation remains identical after simultaneous reflection of all spatial coordinates of the particle through the origin of the coordinate system, then it is said to have even parity. If such simultaneous reflection results in a wave equation that differs from the original wave equation only in sign, then the particle is said to have odd parity. The overall parity of a collection of particles, such as a molecule, is found to be unchanged with time during physical processes and reactions; this fact is expressed as the law of conservation of parity. At the subatomic level, however, parity is not conserved in reactions, owing to the weak nuclear force responsible for radioactivity. Elementary particles are also said to have internal symmetry; these symmetries are useful in classifying particles and in leading to selection rules. Such an internal symmetry is baryon number, which is a property of a class of particles called hadrons. Hadrons with a baryon number of zero are called mesons, those with a number of +1 are baryons. By symmetry there exists another class of particles with a baryon number of -1; these are called antibaryons. Baryon number is conserved during nuclear interactions. in biology, the repetition of the parts in an animal or plant in an orderly fashion. Specifically, symmetry refers to a correspondence of body parts, in size, shape, and relative position, on opposite sides of a dividing line or distributed around a central point or axis. With the exception of radial symmetry (see below), external form has little relation to internal anatomy, since animals of very different anatomical construction may have the same type of symmetry. Certain animals, particularly most sponges and the ameboid protozoans, lack symmetry, having either an irregular shape different for each individual or else one undergoing constant changes of form. The vast majority of animals, however, exhibit a definite symmetrical form. Four such patterns of symmetry occur among animals: spherical, radial, biradial, and bilateral. In spherical symmetry, illustrated only by the protozoan groups Radiolaria and Heliozoia, the body has the shape of a sphere and the parts are arranged concentrically around or radiate from the centre of the sphere. Such an animal has no ends or sides, and any plane passing through the centre will divide the animal into equivalent halves. The spherical type of symmetry is possible only in minute animals of simple internal construction, since in spheres the interior mass is large relative to the surface area and becomes too large for efficient functioning with increase in size and complexity. In radial symmetry the body has the general form of a short or long cylinder or bowl, with a central axis from which the body parts radiate or along which they are arranged in regular fashion. The main axis is heteropolari.e., with unlike ends, one of which bears the mouth and is termed the oral, or anterior, end, and the other of which, called the aboral, or posterior, end, forms the rear end of the animal and may bear the anus. The main axis is hence termed the oral-aboral, or anteroposterior, axis. Except in animals having an odd number of parts arranged in circular fashion (as in the five-armed starfishes), any plane passing through this axis will divide the animal into symmetrical halves. Animals having three, five, seven, etc., parts in a circle have symmetry that may be referred to, respectively, as three-rayed, five-rayed, seven-rayed, etc.; only certain planes through the axis will divide such animals into symmetrical halves. Radial symmetry is found in the coelenterates and echinoderms. In biradial symmetry, in addition to the anteroposterior axis, there are also two other axes or planes of symmetry at right angles to it and to each other: the sagittal, or median vertical-longitudinal, and transverse, or cross, axes. Such an animal therefore not only has two ends but also has two pairs of symmetrical sides. There are but two planes of symmetry in a biradial animal, one passing through the anteroposterior and sagittal axes and the other through the anteroposterior and transverse axes. Biradial symmetry occurs in the comb jellies. In bilateral symmetry there are the same three axes as in biradial symmetry but only one pair of symmetrical sides, the lateral sides, since the other two sides, called the dorsal (back) and ventral (belly) surfaces, are unlike. Thus, only one plane of symmetry will divide a bilateral animal into symmetrical halves, the median longitudinal, or sagittal, plane. Bilateral symmetry is characteristic of the vast majority of animals, including insects, fishes, amphibians, reptiles, birds, mammals, and most crustaceans. The concept of symmetry is also applied in botany. A flower is considered symmetrical when each whorl consists of an equal number of parts or when the parts of any one whorl are multiples of that preceding it. Thus, a symmetrical flower may have five sepals, five petals, five stamens, and five carpels, or the number of any of these parts may be a multiple of five . The number of parts in the pistillate whorl is frequently not in conformity with that in the other whorls, but in such cases the flower is still called symmetrical, provided that the other whorls are normal. A flower in which the parts are in twos is dimerous; in threes, fours , or fives, trimerous, tetramerous, or pentamerous, respectively. Trimerous symmetry is the rule in the monocotyledons, pentamerous the most common in the dicotyledons, although dimerous and tetramerous flowers also occur in the latter group. Bilateral symmetry of the orchid (Vanda) When the different members of each whorl are alike, the flower is regular and is referred to as actinomorphic, or radially symmetrical, as in the petunia, buttercup, and wild rose. Differences in size or shape of the parts of a whorl make the flower irregular (as in the canna and Asiatic dayflower ). When a flower can be divided by a single plane into two equal parts, it is zygomorphic, or bilaterally symmetrical, as in the snapdragon, orchid (see photograph), and sweet pea. in crystallography, fundamental property of the orderly arrangements of atoms found in crystalline solids. Each arrangement of atoms has a certain number of elements of symmetry; i.e., changes in the orientation of the arrangement of atoms seem to leave the atoms unmoved. One such element of symmetry is rotation; other elements are translation, reflection, and inversion. The elements of symmetry present in a particular crystalline solid determine its shape and affect its physical properties. Translations involve displacement of the crystal in a direction that replaces each atom by one of its identical neighbours, so that the atoms seem unmoved. Rotations turn the crystal around an axis of symmetry passing through the crystal; the only rotations compatible with translational symmetry move the crystal through an angle of 360 divided by n, with n equal to 1, 2, 3, 4, or 6. Reflections exchange the parts of the crystal on the two sides of a plane of symmetry (mirror plane) within the solid. Inversions move every atom to another position in the crystal; the old and new positions of the atom lie upon a line, at the middle of which is the centre of inversion. So-called improper rotations are rotations followed by reflections (known as rotoreflections) or rotations followed by inversions (called rotoinversions). A crystal can be classified according to its elements of symmetry; for example, it may belong to one of 230 space groups, 32 point groups, 14 Bravais lattices, and 7 crystal systems. A crystal can be represented diagrammatically by an orderly stacking of unit cells; the shape of the unit cell determines which of the seven crystal systems the crystal belongs to. Unit cells of the same shape may have points (each representing an atom or a group of atoms) at their centres or on their faces, in addition to those at their corners. These additional lattice points divide the 7 crystal systems into 14 Bravais lattices; the Bravais lattices are subdivided into 32 crystal classes, or point groups. Each point group corresponds to one of the possible combinations of rotations, reflections, inversions, and improper rotations; with the inclusion of translational elements, 230 space groups are produced.