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Symmetry

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 (18491925) 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 (18821935) 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 (18791955) 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

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Symmetry

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/>.

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symmetry

symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. In art and design, it is often used in a somewhat loose sense, to mean a kind of balance in which the corresponding parts are not necessarily alike but only similar. A symmetrical design should produce a pleasing effect; if there is too close a correspondence, the effect may be monotonous. Ancient Greek architecture is particularly distinguished for its symmetry. In modern art, the Dutch artist M. C. Escher achieved a number of striking effects in his works exploring mathematical symmetry. A mathematical operation, or transformation, that results in the same figure as the original figure (or its mirror image) is called a symmetry operation. Such operations include reflection, rotation, double reflection, and translation. The set of all operations on a given figure that leave the figure unchanged constitutes the symmetry group for that figure. The symmetry groups of three-dimensional figures are of special interest because of their application in fields such as crystallography (see crystal). In general, a symmetry operation on a figure is defined with respect to a given point (center of symmetry), line (axis of symmetry), or plane (plane of symmetry). In biology, symmetry is studied in the correspondences between different parts of a given organism, as between the left and right halves of the human body or between the various segments of a starfish (see symmetry, biological). In physics, basic symmetries in nature underlie the various conservation laws. For example, the symmetry of space and time with respect to translation and rotation means that a given experiment should yield the same results regardless of where it is performed, what direction the equipment is pointing in, or when it is performed. These three symmetries can be shown to imply the laws of conservation of linear momentum, angular momentum, and energy, respectively.

See G. E. Martin, Transformation Geometry (1987); B. Bunch, Reality's Mirror (1989); M. C. Escher, Escher on Escher (tr. 1989).

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symmetry

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.

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Symmetry

389. Symmetry

See also 170. FORM .

asymmetry
the quality or condition of lacking symmetry. asymmetrical, asymmetric, adj.
bisymmetry
Botany. the condition of having two planes of symmetry at right angles to one another. bisymmetric, bisymmetrical, adj.
monosymmetry
1. the state exhibited by a crystal, having three unequal axes with one oblique intersection; the state of being monoclinic. See also 44. BIOLOGY .
2. Biology. the state of being zygomorphic, or bilaterally symmetric, or divisible into symmetrical halves by one plane only. See also zygomorphism. See also PHYSICS. monosymmetric, monosymmetrical, adj.
symmetromania
a mania for symmetry.
symmetrophobia
an abnormal fear or dislike of symmetry.

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symmetry

symmetry †mutual relation of parts, proportion; due or just proportion. XVI. — F. †symmetrie (mod. symétrie) or L. symmetria — Gr. summetríā, f. súmmetros commensurable, proportionable, symmetrical, f. SYM- + métron measure; see -Y3.
Hence symmetrical XVIII.

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symmetry

symmetry.
1. Exact correspondence of parts on either side of an axis, e.g. Greek temple.

2. Harmony, proportion, or uniformity between the parts of a building and its whole.

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symmetry

symmetry (sim-it-ri) n. (in anatomy) the state of opposite parts of an organ or parts at opposite sides of the body corresponding to each other.

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symmetry

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symmetry

symmetry •hara-kiri • ribaldry • chivalry • Tishri •figtree • wintry • poetry • casuistry •Babbittry • banditry • pedigree •punditry • verdigris • sophistry •porphyry • gadgetry • registry •Valkyrie •marquetry, parquetry •basketry • trinketry • daiquiri •coquetry, rocketry •circuitry • varletry • filigree •palmistry •biochemistry, chemistry, photochemistry •gimmickry, mimicry •asymmetry, symmetry •craniometry, geometry, micrometry, optometry, psychometry, pyrometry, sociometry, trigonometry •tenebrae • ministry • cabinetry •tapestry • carpentry • papistry •piripiri • puppetry •agroforestry, floristry, forestry •ancestry • corsetry • artistry •dentistry • Nyree • rivalry • pinetree

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