integrative concepts


1. In physics, the invariant nature of physical laws governing comparable phenomena in different systems or frames of reference, or in the same phenomena within a changing system. The laws are invariant or symmetric under given transformations. Symmetry is preserved in a number of such known types. The continuous transformations that are symmetric include: the translation of a whole system in space; either active, in which the system is translated to a different, chosen frame of (spatial) reference, or passive, in which the frame itself undergoes translation. Symmetry in these cases includes the concept that all points in space are equivalent (non-distinctive), and displacements or transformations are thus in homogeneous space. Other important symmetric continuous transformations are: whole system rotation in isotropic space, all directions being equivalent; time translation or change in origin of the time coordinate (indicating that physical laws do not change with time); and Lorentz transformations, of a system to another frame of reference moving in constant relative directional velocity (indicating equivalence of all inertial frames of reference; critical to relativity theory). In symmetric Gange transformations affecting interactions of charged particles and fields there is also a special case of invariance violation in the 'approximate' symmetry of isotopic spin in strong nuclear particle interactions. The other major category (in addition to the continuous) are the discrete transformations. While the first category is characterized by parameter values that can vary continuously along the coordinate axes of the frame of reference, this is not the case in discrete transformations. The characterizing names of the principal cases of symmetric discrete transformations are space inversion and time reversal. The former is also called reflection symmetry due to mirror-imaging, as in dextrorotary and levorotary spins and rotations, and in matter, antimatter substances (however space-inversion symmetry is violated in the case of weak-interactions). The other types of symmetric discrete transformations are charge conjugation in which particles are replaced by antiparticles, and time reversal where the sign of the time coordinate is changed. Taking the three operations, CPT, together (charge conjugation, C; space inversion, P; and time reversal, T) their symmetry follows from quantum field theory. Taken in connection with invariance under the Lorentz transformations, and with a single point locality of field interaction, symmetry holds, even if CPT are taken separately and all interactions are not invariant. The foregoing and other investigations of symmetry are crucial to understanding the physical universe and the validity of many fundamental principles such as unity and the conservation laws of energy momentum, and spatial parity.
2. In logic the property relation that holds for any pair of objects regardless of order in which they occur. Such symmetric relations are the equality types: identity, equivalence and similarity. Weak symmetric relations occur in such binaries described as resembling and proximate, etc. (These are better known as tolerance relations).
3. In mathematics, commutativity and permutability, as in groups.
4. In geometry, reflection, central, axial and translational space symmetries. In these operations a figure is mapped onto itself. The set of all orthogonal transformations that may a figure onto itself is called its symmetry group.
5. In botany, leaf arrangements (phyllotaxies) characterized by twists, i.e. symmetries generated in combination, by a rotation and a translation.
6. In crystallography, the behaviour of crystals, though in rotations, reflections, parallel translations, or in any such combined operations, to replicate or repeat their organization, so that symmetry of crystalline atomic structure creates both symmetry of physical properties and symmetry in faceted form.
7. In molecular chemistry, spatial symmetry or equilibrium configuration of axes and planes of symmetry (also symmetry of nonequilibrium configurations). The symmetry of the equilibrium configuration of molecular nuclei determines the symmetry of the wave functions for various states of the molecule. Symmetry properties are important in classification and analysis of complex compounds, and in study of chemical lasers and pharmacologically active substances.
8. In biology, structural symmetry.
9. In art, harmony of composition, e.g. the Golden Mean.
10. In science, the general theory of symmetry.