Symmetry

Resources

Symmetry is a property of some images, objects, and mathematical equations whereby reflections, rotations, or substitutions cause no change in properties or appearance. For example, the letter M is symmetrical across a line drawn down its center, a ball is symmetrical under all possible rotations, and the equation y = x² (a parabola) is symmetrical under the substitution of -x for x This equation’s mathematical symmetry is equivalent to its graph’s physical symmetry. The ability of mathematical symmetries to reflect the physical symmetries of the real world is of great importance in a variety of scientific fields such as biomechanics, geophysics, and particle physics.

Many real objects and forces at all size scales— subatomic particles, atoms, crystals, organisms, stars, and galaxies—exhibit symmetry, of which there are many kinds. Line or bilateral symmetry, the simplest and most familiar, is the symmetry of by any figure or object that can be divided along a central line and then restored (geometrically) to wholeness by reflecting its remaining half in a mirror.

Symmetries are not only defined in terms of reflection across a line. A sphere, for example, can be rotated through any angle without changing its appearance, and in mathematics is said to possess O(3) symmetry. The quantum field equations whose solutions describe the electron, which is, like a sphere, the same viewed from any direction, also have O(3) symmetry.

In particle physics, the mathematics of symmetry is an essential tool for producing an organized account of the confusing plethora of particles and forces observed in Nature and for making predictions based on that account. An extension of the parabola example shows how it is possible for mathematical symmetry to lead to the prediction of new phenomena. Consider a system of two equations, y = x² and y = 4. There are two values of x that allow both equations to be true at once, x = 2 and x = -2 The two (x, y) pairs (2, 4) and (-2, 4) are termed the solutions to this system of two equations, because both equations are simultaneously true if and only if x and y have these values. (The two solutions correspond to the points where a horizontal line, y = 4, would intersect the two rising arms of the parabola.) If this system two equations constituted an extremely simple theory of matter, and if one of its two solutions corresponded to a known particle, say with “spin” = x = 2 and “mass” = y = 4, then one might predict, based on the symmetry of the two solutions, that a particle with “spin” = -2 and “mass” = 4 should also exist. An analogous (though more complex) process has actually led physicists to predict, seek, and find certain

KEY TERMS

Noether’s theorem —Mathematical theorem stating that every conservation law in Nature must have a symmetrical mathematical description.

fundamental particles, including the Ω^- baryon and the η⁰ muon).

Symmetry, however, not only is a useful tool in mathematical physics, but has a profound connection to the laws of nature. In 1915, German mathematician Emmy Noether (1882–1835) proved that every conservation law corresponds to a mathematical symmetry. A conservation law is a statement that says that the total amount of some quantity remains unchanged (i.e., is conserved) in any physical process.

Momentum, for example, is conserved when objects exert force on each other; electric charge is also conserved. The laws (mathematical equations) that describe momentum and charge must, therefore, display certain symmetries.

Noether’s theorem works both ways: in the 1960s, a conserved quantum-mechanical quantity (unitary spin) was newly defined based on symmetries observed in the equations describing a class of fundamental particles termed hadrons, and has since become an accepted aspect of particle physics. As physicists struggle today to determine whether the potentially all-embracing theory of “strings” can truly account for all known physical phenomena, from quarks to gravity and the big bang, string theory’s designers actively manipulate its symmetries in seeking to explore its implications.

See also Cosmology; Relativity, general; Relativity, special.

Resources

BOOKS

Barnett, R. Michael, Henry Mühry, and Helen R. Quinn. The Charm of Strange Quarks. New York: American Institute of Physics Press, 2002.

OTHER

Cambridge University. “Cambridge Cosmology.” 1996. <http://www.damtp.cam.ac.uk/user/gr/public/cos_home.html> (accessed November 27, 2006).

University of Winnipeg. “Symmetry.” September 29, 1999. <http://theory.uwinnipeg.ca/mod_tech/node9.html> (accessed November 27, 2006).

Larry Gilman

Symmetry

In the most general sense, symmetry can be defined as a property that an entity has whereby it preserves some of its aspects under certain actual or possible transformations. A sphere is symmetrical because a rotation about its axis preserves its shape. A crystal structure is symmetrical with respect to certain translations in space. The existence of symmetries in natural phenomena and in human artifacts is pervasive. However, nature also displays important violations of symmetry: Some organic molecules come only or predominantly in left-handed varieties; the bilateral symmetry of most organisms is at best only approximate.

The general concept of symmetry applies not only to objects and their collections, but also to properties of objects, to processes they may undergo, as well as to more abstract entities such as mathematical structures, scientific laws, and symbolic and conceptual systems, including mythology and religion. Symmetry symbols pervade ancient cosmologies. Thus the concept of axis mundi (the world axis) is a famous mytho-poetic archetype expressing the idea of centrality in the arrangement of the Cosmos. Whether axis mundi is represented as a sacred mountain, tree, or ladder, it invariably signifies a possibility for humans to connect with heaven. The central image of Christianity, the cross, belongs in the same broad category, as far as its symbolic connotations are concerned. The concept of triadicity so essential to many religions is closely linked to symmetry considerations.

The abstract notion of symmetry also lies at the very foundation of natural science. The fundamental significance of symmetries for physics came to the fore early in the twentieth century. Prior developments in mathematics contributed to this. Thus, in his Erlangen Program (1872), the German mathematician Felix Klein (1849–1925) proposed interpreting geometry as the study of spatial properties that are invariant under certain groups of transformations (translations, rigid rotations, reflections, scaling, etc.). Emmy Noether (1882–1935) applied Klein's approach to theoretical physics to establish in 1915 a famous theorem relating physical conservation laws (of energy, momentum, and angular momentum) to symmetries of space and time (homogeneity and isotropy). By that time, Albert Einstein's (1879–1955) Theory of Relativity had engendered the notion of relativistic invariance, the kind of symmetry all genuine physical laws were expected to possess with respect to a group of coordinate transformations known as the Lorentz-Poincaré group. With this came the realization that symmetry (invariance) is a clue to reality: Only those physical properties that "survive" unchanged under appropriate transformations are real; those that do not are merely perspectival manifestations of the underlying reality.

With the development of particle physics the concept of symmetry was extended to internal degrees of freedom (quantum numbers), such as C (charge conjugation, the replacement of a particle by its antiparticle) and isospin (initially the quantum number distinguishing the proton from the neutron). Along with P (parity, roughly a mirror reflection of particle processes) and T (time-reversal operation), these were long believed to be exact symmetries, until the discovery in 1956 of C - and P -symmetry violations in certain weak interactions, and the discovery in 1964 of the violation of the combined CP -symmetry. However, theoretical considerations preclude violation of the more complex CPT -symmetry.

The emergence of quantum electrodynamics (QED), the first successful quantum relativistic theory describing the interaction of electrically charged spin-1/2 particles with the electromagnetic field, made the notion of gauge symmetry central to particle physics. The exact form of interaction turns out to be a consequence of imposing a local gauge invariance on a free-particle Lagrangian with respect to a particular group (U(1) in the case of QED) of transformations of its quantum state. Extending this principle to other interactions led to the unification of electromagnetic and weak forces in the Weinberg-Salam-Glashow theory on the basis of the symmetry group SU(2) × U(1) and to quantum chromodynamics (a theory of strong quark interactions based on the group SU(3)), and eventually paved the way for the ongoing search for a theory unifying all physical forces.

See also Laws of Nature

Bibliography

mainzer, klaus. symmetries of nature: a handbook for philosophy of nature and science. berlin and new york: walter de gruyter, 1996.

rosen, joe. symmetry in science. an introduction to the general theory. new york: springer-verlag, 1995.

yuri v. balashov

Symmetry

Symmetry is a property of some images, objects, and mathematical equations whereby reflections , rotations, or substitutions cause no change in properties or appearance. For example, the letter M is symmetrical across a line drawn down its center, a ball is symmetrical under all possible rotations, and the equation y = x2 (a parabola ) is symmetrical under the substitution of -x for x. This equation's mathematical symmetry is equivalent to its graph's physical symmetry. The ability of mathematical symmetries to reflect the physical symmetries of the real world is of great importance in physics , especially particle physics.

Many real objects and forces at all size scales—subatomic particles, atoms , crystals, organisms, stars, and galaxies—exhibit symmetry, of which there are many kinds. Line or bilateral symmetry, the simplest and most familiar, is the symmetry of by any figure or object that can be divided along a central line and then restored (geometrically) to wholeness by reflecting its remaining half in a mirror.

Symmetries are not only defined in terms of reflection across a line. A sphere , for example, can be rotated through any angle without changing its appearance, and in mathematics is said to possess O(3) symmetry. The quantum field equations whose solutions describe the electron , which is, like a sphere, the same viewed from any direction, also have O(3) symmetry.

In particle physics, the mathematics of symmetry is an essential tool for producing an organized account of the confusing plethora of particles and forces observed in Nature and for making predictions based on that account. An extension of the parabola example shows how it is possible for mathematical symmetry to lead to the prediction of new phenomena. Consider a system of two equations, y = x2 and y = 4. There are two values of x that allow both equations to be true at once, x = 2 and x = -2. The two (x, y) pairs (2, 4) and (-2, 4) are termed the solutions to this system of two equations, because both both equations are simultaneously true if and only if x and y have these values. (The two solutions correspond to the points where a horizontal line, y = 4, would intersect the two rising arms of the parabola.) If this system two equations constitued an extremely simple theory of matter , and if one of its two solutions corresponded to a known particle, say with "spin" = x = 2 and "mass" = y = 4, then one might predict, based on the symmetry of the two solutions, that a particle with "spin" = -2 and "mass" = 4 should also exist. An analogous (though more complex) process has actually led physicists to predict, seek, and find certain fundamental particles, including the Ω– baryon and the Η0 muon).

Symmetry, however, not only is a useful tool in mathematical physics, but has a profound connection to the laws of Nature. In 1915, German mathematician Emmy Noether (1882–1835) proved that every conservation law corresponds to a mathematical symmetry. A conservation law is a statement that says that the total amount of some quantity remains unchanged (i.e., is conserved) in any physical process.

Momentum , for example, is conserved when objects exert force on each other; electric charge is also conserved. The laws (mathematical equations) that describe momentum and charge must, therefore, display certain symmetries.

Noether's theorem works both ways: in the 1960s, a conserved quantum-mechanical quantity (unitary spin) was newly defined based on symmetries observed in the equations describing a class of fundamental particles termed hadrons, and has since become an accepted aspect of particle physics. As physicists struggle today to determine whether the potentially all-embracing theory of "strings" can truly account for all known physical phenomena, from quarks to gravity and the Big Bang, string theory's designers actively manipulate its symmetries in seeking to explore its implications.

See also Cosmology; Relativity, general; Relativity, special.

Resources

books

Barnett, R. Michael, Henry Mühry, and Helen R. Quinn. TheCharm of Strange Quarks. New York: Springer-Verlag, 2000.

Elliot, J.P., and P.G. Dawber. Symmetry in Physics. New York: Oxford University Press, 1979.

Silverman, Mark. Probing the Atom Princeton, NJ: Princeton University Press, 2000.

other

Cambridge University. "Cambridge Cosmology." [cited February 14, 2003]. <http://www.damtp.cam.ac.uk/user/gr/public/cos_home.html>.

Larry Gilman

KEY TERMS

Noether's theorem

—Mathematical theorem stating that every conservation law in Nature must have a symmetrical mathematical description.

Symmetry

Symmetry is a visual characteristic in the shape and design of an object. Take a starfish for an example of an object with symmetry. If the arms of the starfish are all exactly the same length and width, then the starfish could be folded in half and the two sides would exactly match each other. The line on which the starfish was folded is called a line of symmetry. Any object that can be folded in half and each side matches the other is said to have line symmetry.

When an object has line symmetry, one half of the object is a reflection of the other half. Just imagine that the line of symmetry is a mirror. The actual object and its reflection in the "mirror" would exactly match each other. A human face, if vertically bisected into two halves (left and right), would reveal its bilateral symmetry: that is, the left half ideally matches the right half.

It is possible for an object to have more than one line of symmetry. Imagine a square. A vertical line of symmetry could be drawn down the middle of the square and the left and right sides would be symmetrical. Also, a horizontal line could be drawn across the middle of the square and the top and bottom would be symmetrical. As shown in the figure below, a star or a starfish has five lines of symmetry.

Alternately, the ability of an object to match its original figure with less than a full turn about its center is called rotational or point symmetry. For example, if a starfish embedded in the sand on a beach is picked up, rotated less than 360°, and then set back down exactly into its original imprint in the sand, the starfish would be said to have rotational symmetry.

Many natural and manmade items have reflection and/or rotation symmetry. Some examples are pottery, weaving, and quilting designs; architectural features such as windows, doors, roofs, hinges, tile floors, brick walls, railings, fences, bridge supports, or arches; many kinds of flowers and leaves; honeycomb; the cross section of an apple or a grapefruit; snowflakes; an open umbrella; letters of the alphabet; kaleidoscope designs; a pinwheel, windmill, or ferris wheel; some national flags (but not the U.S. flag); a ladder; a baseball diamond; or a stop sign.

Mathematicians use symmetries to reduce the complexity of a mathematical problem. In other words, one can apply what is known about just one of multiple symmetric pieces to other symmetric pieces and thereby learn more about the whole object.

see also Transformations.

Sonia Woodbury

Bibliography

Schattschneider, D. Visions of symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher. New York: W. H. Freeman and Company, 1990.

Internet Resources

Search for Math: Symmetry. The Math Forum at Swarthmore College. <http://www.forum.swarthmore.edu/>.

sym·me·try / ˈsimitrē/ • n. (pl. -tries) the quality of being made up of exactly similar parts facing each other or around an axis: this series has a line of symmetry through its center a crystal structure with hexagonal symmetry. ∎  correct or pleasing proportion of the parts of a thing: an overall symmetry making the poem pleasant to the ear. ∎  similarity or exact correspondence between different things: a lack of symmetry between men and women | history sometimes exhibits weird symmetries between events. ∎  Physics & Math. a law or operation where a physical property or process has an equivalence in two or more directions.DERIVATIVES: sym·me·trize / -ˌtrīz/ v.