Mathematics Awareness Week 1995

Theme Essay: Mathematics and Symmetry

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This essay is being disseminated to the mathematics community by the Joint Policy Board for Mathematics for use in conjunction with Mathematics Awareness Week 1995. Text was provided by:
  • Paul Davis, Worcester Polytechnic Institute
  • Martin Golubitsky, University of Houston
Review and ideas were provided by:
  • Jeffrey Adams, University of Maryland
  • Martin Isaacs, University of Wisconsin
  • Peter Olver, University of Minnesota
  • Marjorie Senechal, Smith College
  • Doris Schattschneider, Moravian College

Mathematics and Symmetry
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.

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