Symmetry is all around us. We see symmetry in everyday objects,
in buildings, in floor and wall tiles, in gears, and even in
automobile hub-caps. We see symmetry in many natural forms --
in the bilateral symmetry of the human form, in the rotational
and kaleidoscopic symmetry of blossoms, in the sinuous spiral
symmetry of vines and shells, and in the translation symmetry of
honeycombs and fish scales. Symmetry also reveals itself in the
decorative arts of many cultures, including, for example, in the
decorative designs of the Moors at the Alhambra, in the weavings
of the Indians of the American Southwest, and in the curious
interlocked creatures that are found in the work of the
mathematically perceptive modern graphic artist M.C. Escher.
Indeed, the symmetry evident in the decorative art of a
particular culture may serve as a cultural marker.
Symmetry appears on a grand scale in the formulation of distance
in special relativity and even in the shapes of galaxies and on
a microscopic scale in the classification of crystal structure.
Symmetry also plays a pivotal role in mathematics -- from the
verification that a general polynomial of degree five or higher
cannot be solved by formula to the classification of types of
geometry to the existence of conservation laws.
To mathematicians, symmetries are defined as transformations
that leave an object or a picture or an equation unchanged.
These transformations are called the symmetries of the object;
together they form a mathematical structure known as a group,
the symmetry group of the object.
Objects, such as the human form or a perfectly symmetric
butterfly, have bilateral symmetry because they cannot be
distinguished from their reflections across a mirror plane.
Similarly, repeating patterns, or more vividly wallpaper
patterns, are pictures that can be picked up, shifted, and put
down again so that the picture is undisturbed. Objects with
helical symmetry are those that are invariant under screw
motions about a central axis.
It is through the study of groups that different types of
pattern are distinguished. Using group theory, mathematicians
can prove that there are exactly seventeen ways to construct
repeating wallpaper patterns (or periodic planar tilings).
Indeed, group theory has been one of the most exciting branches
of mathematics during the past century, beginning with Lie's
discovery of continuous groups through to the recent
classification of finite simple groups. Investigating the vast
connections between group theory and topology, geometry, and
analysis continues to be a central theme in mathematics
research. Beginning with Galois theory and continuing to
current research, symmetry enables us to find solutions to
equations -- first to algebraic equations and now to
differential equations.
Symmetry is also central to the mathematical description of many
natural phenomena. The catalog of three-dimensional repeating
patterns is identical to the catalog of ways that atoms can
arrange themselves on crystal lattices. Chemists and
mathematicians have classified the 230 crystallographic groups
- -- the 230 forms of crystal structure -- by analyzing in
detail the possible combinations of rotations, reflections and
translations that leave a crystal lattice unchanged. Symmetry
is important in material science and elasticity where it is
incorporated into the constitutive relations that govern the
structure and behavior of solids and liquids.
Symmetry is basic to our understanding of the hydrogen atom and
molecular spectroscopy, to elementary particles and the theory
of quarks, as well as to the two crowning achievements of
twentieth century physics -- the theory of relativity and
quantum mechanics. Indeed, one might even characterize the
current search for a `unified field theory' -- a single theory
to describe all forces of nature -- as a search for the
fundamental symmetry group of the physical universe, from which
all the basic laws of physics will follow.
Symmetry has appeared in technology in surprising ways. In
computer vision the symmetries of human perception (projective
transformations) are incorporated into the design of
mathematically based image processing systems which may have
important applications to medical imaging. In applications to
control theory, rotation and translation symmetries must be
taken into account when designing feedback controllers for both
aircraft and satellites.
Just as the absence of symmetry has striking effects in art and
music, the absence or loss of symmetry is of great interest in
models of natural phenomena, often with dramatic consequences.
Symmetry breaking occurs when structures buckle, when water
boils, and (possibly even) when spots form on leopards and
stripes form on tigers. Twenty years ago, mathematicians and
physicists demonstrated a route to turbulence that involves the
development of more complicated fluid flow patterns signaled by
a succession of losses of symmetry. These kinds of exploration
have given symmetry an apparently paradoxical role -- the role
of charting the onset of complicated or chaotic behavior. The
images shown on this year's Mathematics Awareness Week poster
are formed using a combination of symmetry and chaotic dynamics.
Their detailed complex structure is due to chaotic dynamics
while their apparent regularity and familiarity is due to
symmetry.
Arrangements that show a high degree of order may fail to have
any global symmetry, yet symmetry may have a role in describing
and classifying these patterns. Tiles that can only fit
together in ways that have no translation symmetry and newly
found "quasicrystals" that display symmetry forbidden by the
conventional model of crystals are two exciting areas of current
research.
Back to Math Awareness Week 1995
