Symmetry

58 Algorithm A mathematical rule for computation involving a succession of procedural steps Array In maths, a regular matrix Attractor In dynamical systems, a set to which a system tends to evolve Basic Nets In planedivision this is the framework that creates the unitcell which provides the mode of repetition Bilateral Commonly, having two equal but reversed sides technically, reflected about a mirrorline in 2dimensions or about a mirrorplane in 3dimensions Bifurcation The process of division into two branches Chaos theory Mathematical theories dealing with apparent randomness deriving from precise, deterministic causes, and hidden consistencies in complex, nonlinear dynamical systems Chiral Of a shape that is not superimposable on its mirrorimage Congruence In geometric symmetry this means that the separate elements involved in the symmetry correspond in every detail, and that the distance between any two points on any part of these elements is regular Conservation Law A law which states that the total value of a given quantity is not changed in any reaction Continuous Term applied to symmetry groups with an infinite number of symmetry operations, i.e., those of a circle Conic sections Curves of the second degree since they intersect with a straight line at only two points. Curves Can be thought of as a point that moves along a continuous path, or locus they are symmetrical when the direction of this locus is selfconsistent. Dihedral Finite, centred arrangement around mirrorlines Dilation Symmetrical transformation achieved by enlargement or reduction by means of lines radiating from a centre Discrete Term applied to symmetry groups that involve discrete steps, and have no infinitesimal operations, as in an equilateral triangle Dorsiventral Reflection about a single mirrorplane in 3dimensions Feigenbaum mapping Selfsimilar mapping that is unchanged by renormalisation, i.e. has constant scaling ratio Feigenbaum number A mathematical constant, 4.6692016 symbolised by d the ratio between successive periodic doublings in Feigenbaum mapping Fractal A geometrical figure that is recursive and scaling, i.e., that repeats itself on an ever reduced scale Gnomon A geometrical shape which, when added to or subtracted from another, results in a figure similar to the original Golden Section, Golden Ratio That division of a line that leaves the ratio of the smaller segment to the larger equal to that of the larger segment to the whole line Group Theory The mathematical language of symmetry see note below Invariance A constancy in maths, an expression or quantity that is unchanged by a particular procedure in physics, an equality of laws in space or time, virtually synonymous with symmetry Isometry Any movement or transformation that maps a figure onto a congruent figure. It is qualified by being either direct or opposite Isomorphic Having the same abstract structure, even when described in different terms Movement The change of a congruent object from one symmetrical position to another may be direct or opposite Periodicity The regular spacing of elements in symmetries Phase transition A critical transition of a system from one state to another, usually associated with a change in symmetry, i.e., melting, boiling, magnetism Phi The golden number, 512 0.6180339887 symbolised by f it can be squared by adding 1, and its reciprocal found by subtracting 1 Point symmetries Symmetries around a point or line Reflexion An indirect or opposite isometric movement about a mirrorline in 2 dimensions or around a mirrorplane in 3dimensions Rotation An isometric movement around a point the element of symmetry can be rotated 2, 3, 4 or more positions Spirals and helices Are symmetrical by virtue of the regularity with which they wind around a centre or axial line respectively Strange attractors A chaotic attractor, or one that has noninteger dimensions see Attractor Symmetry groups The group of all isometries, under which it is invariant with composition as the operation Tortuous curves Regular curves in 3d measured by their changes of direction across three continuous points Transformation A rule for a movement in a symmetry Translation Transformations that slide objects along without rotating them Wave equation A differential equation which describes the passage of harmonic waves through a medium. The form of the equation depends on the nature of the medium and on the processes by which the wave is transmitted GlossAry One of the most remarkable aspects of symmetry groups is the extent to which they are represented in nature indeed at the fundamental level nature could actually be said to be defined by symmetry. Group theory, the basic mathematical description of symmetry, classifies the various types according to the operations involved, i.e., rotations, reflections, repetitions, and the various combinations of these. The principle division in this general scheme is between discrete and continuous symmetries. Remembering that a symmetry is defined by the movement required to restore an object to its original position, a discrete symmetry relies on a series of discrete steps to achieve this, as in the regularities of, say, an equilateral triangle. The pointgroups and lattice groups that have been variously encountered in this book belong in this discrete category. Continuous symmetries, by contrast, are constant over infinitesimal movements of angle and distance circles and spheres, in 2d and 3d respectively, are of this kind. Continuous symmetry groups are mathematically described by a particularly elegant branch of algebra known as Lie Group Theory. In the late 19th century the French mathematician Elie Cartan 18691951 used Lie groups to classify every possible variant in this class of symmetry. This exhaustive work remained as a somewhat obscure branch of mathematics until the early 1960s, when it was recognised by Murray GellMan of Caltech as the perfect instrument to deal with the plethora of subatomic particles that were being discovered at that time. It soon became apparent that the Lie symmetries SU3 fitted perfectly with the emerging field theories of quantum physics, to the extent that the existence and properties of some particles were predicted prior to their actual discovery. The four basic forces gravitational, electromagnetic, and the weak and strong nuclear are similarly described, using the gauge symmetry groups U1 X SU2 X SU3. This means that present cosmological views, centred around the socalled Standard Model of fundamental particles and antiparticles, are conceived entirely in terms of symmetry groups. The challenge for contemporary cosmologists is to unify the basic forces with this new periodic table, in one grand symmetrical scheme. A Note on Group Theory