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Symmetry 
(Under construction)
Definition
Orderly,
mutually corresponding arrangement of various parts of a body, producing
a proportionate, balanced form.
Symmetry is a fundamental organizing principle in nature and in culture. The analysis of symmetry allows for understanding the organization of a pattern, and provides a means for determining both invariance and change.Related concepts: Duality, Daology, Complementarity, Parity, Ying Yang Theory,
A pattern, whether in nature or art, relies upon three characteristics: a unit, repetition, and a system of organization.
In art, symmetry can be considered as a geometrical duality.
In theoretical physics, symmetry is sort of complementarity.
The principle of symmetry is of great importance in the fields of biology, mineralogy, physics and mathematics.
Duality in Symmetry:
Asymmetry:
Symmetry is recongnized as the symmetry because
it often relies upon asymmetry.
Asymmetry is both the absence of symmetry, and a fundamental
basis for symmetry. Symmetry analysis may result in the
identification of a fundamental region that is the smallest element required
to explain the repetition that forms a pattern. The fundamental region
is asymmetrical.
- Symmetry Breaking:
In nature, symmetry is imperfect, although mathematicians may treat it as an ideal. In art, too, it seems that the approximation of symmetry, rather than its precision, teases the mind as it pleases the eye.
In mineralogy, laws of symmetry apply to the angular structure of crystals. All classes of crystal are divided into six systems that are based on the length of their axes and other details of symmetry. See Crystal; Metallography.
In biology, the regular distribution of various parts of an animal's body on two opposite sides of a linear axis, or a median plane, is known as bilateral symmetry. The proportional arrangement of similar parts of a body around a central axis, as in the case of jellyfish or starfish, is known as radial symmetry. The bodies of protozoans, such as those of the order Radiolaria, which have a round form about a central point or nucleus, are said to have a spherical symmetry.
Physics: In physics, a system is said to exhibit symmetry if it remains unchanged in the course of operations such as mirror reversal, reversal in the direction of time, and space-time translation. Many physical systems obey such symmetries, to which the conservation laws of physics are also related. This relationship has come to be of particular importance in particle physics, where certain symmetries called internal symmetries are observed. Such symmetries exist in the mathematical "space" of that realm and underlie the conservation of such quantities as charge, parity, baryon and lepton number, and total strangeness, even as certain particles are substituted for one another. In current theoretical physics, however, such symmetries are now known to be only approximate. Except for baryon and lepton number, that is, they are violated in their physical manifestations. When internal symmetries do not operate the same way but instead can be different at each point in space-time, they are called gauge symmetries. Theorists currently hope to reduce all such symmetries to gauge symmetries in their effort to develop a grand unification theory that can incorporate all of the fundamental interactions of matter
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 chemistry—for 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.
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In geometry, symmetry is a feature of certain plane and solid shapes. So-called symmetry operations are those mathematical transformations that produce a figure identical to the original or a mirror image of the original figure. Symmetry operations are defined with respect to a given point (center of symmetry), line (axis of symmetry), and plane (plane of symmetry).
The Four Basic Symmetries
POSSIBILITIES FOR THE COMPOSITION of a design are limitless, and may rely upon choices. But possibilities for the repetition of that design, whether symmetrical or asymmetrical, are limited by the laws of pattern formation and are subject to the constraints of symmetry.IN ALL PATTERNS there are four basic symmetry operations that may be performed upon a fundamental region, design or motif. Mathematicians call these rigid motions because they suggest movements without distortion of size or shape around a point, along or across a line, or to cover a plane.
HERE, THE LETTER F (and the blank space around it) is taken as our fundamental region to demonstrate the four basic symmetry operations or rigid motions:
translation
rigid motion with repetition
along a line
reflection
rigid motion with repetition
across a line (axis)
glide reflection
rigid motion with reflected
repetition along a line
rotation
rigid motion with repetition
around a point
by David Y. Gao
