Symmetry, Pure and (not so) Simple

By ANTHONY D. BLAOM

Right and Left by Winslow Homer (1836 -1910)

My chief interests lie in Differential Geometry and Dynamical Systems. A common theme underlying these interests has been symmetry and the breaking of symmetry. Sundry comments below focus on this theme. For specific information on my research, see my Homepage.

"Dissymmetry makes the phenomenon"

Beauty has been defined as "symmetry slightly broken" (Hermann Weyl?). The image of a human face immediately comes to mind. In science, the relaxation of an imposed or imagined symmetry can lead to breakthroughs. For example, Kepler's failure to predict planetary orbits using Archimedes' "perfect" solids (see figure below) led him ultimately to consider elliptic orbits with a Sun asymmetrically positioned at one focus.

Stereochemistry, the study of molecules, such as sucrose, possessing either a "left" or "right" orientation (think of corkscrews for left or right handers) originated in attempts to explain aberrations in X-ray diffraction properties initially taken to be centrosymmetric. One sugar substitute tricks the body by having a molecular structure identical to sucrose, except for a reversed orientation. To quote Pierre Curie, "dissymmetry makes the phenomenon."

For further examples in science along theses lines see I. Hargatti & M. Hargatti's book "In Our Own Image: Personal Symmetry in Discovery" reviewed here.

Symmetry in mathematics

A considerable amount of contemporary mathematics concerns itself with symmetry, one way or another. One reason for this is that "solving" a mathematical problem, e.g., finding the area of a randomly oriented triangle, often amounts to solving a simplified but equivalent problem, as in finding the area of a triangle with one side horizontal (area = half base times height!). But this requires us to recognize when two problems are equivalent, for example, to determine when two triangles are congruent. The symmetries of a mathematical object - whether a triangle, a differential equation, a polynomial, an algorithm, or whatever - are, by definition, the self-equivalences of the object. An equilateral triangle, for example, has precisely six such equivalences. These symmetries tell you a lot about the object and how to solve related mathematical problems.

Groups: The mathematical language of symmetry

When the "essential essence" of a symmetry is distilled, one obtains an abstract mathematical construct called a group. A Platonist might describe groups as the "forms" or "cookie cutters" from which all symmetries are cut. Associated with any symmetry is its symmetry group; two objects with the same kind of symmetry have the same symmetry group.

By definition, a group is any set, together with a sensible rule for multiplying elements together. (Here "sensible" has a technical meaning we won't go into.) For example, the symmetry group for the three-pronged figure shown left is the set {1, 2, 3} with the multiplication table

1·1 = 1

2·1 = 2

3·1 = 3

1·2 = 2

2·2 = 3

3·2 = 1

1·3 = 3

2·3 = 1

3·3 = 2 .

We emphasize that groups are mathematical entities in their own right (they represent cookie cutters, not the cookies themselves). Nevertheless, for each specific object with given symmetry group, one has an interpretation of the group elements as self-equivalences. In the present example, 1 means "do nothing," 2 means "rotate anticlockwise 120 degrees" and 3 means "rotate clockwise 120 degrees." The product 2·3 means "first rotate anticlockwise 120 degrees, then rotate clockwise 120 degrees"; whence 2·3 = 1 (do nothing). You get the idea.

The group above has only three elements but groups come in a multitude of shapes and sizes. Many important groups in geometry have an infinite number of elements.

Confused? Don't worry too much. If you have the gist of the group concept proceed!

Symmetry in abundance

Group theory is of particular significance to the study of objects having plenty of symmetry. The sphere is a good example; its self-equivalences include arbitrary rotations in three dimensions about the centre. Notice that for any two points on the sphere, there is a self-equivalence (i.e., rotation) taking one of the points to the other. We say that the sphere is homogeneous, a technical word meaning "having lots of symmetry"! A special role is played by the set of self-equivalences fixing a single point (in this case, the rotations about a fixed axis); it is called the isotropy subgroup of the symmetry.

An elementary result of group theory states:

A homogeneous object is completely determined by just two things: (1) Its symmetry group, and (2) The isotropy subgroup.

This remarkable fact reduces questions regarding an object with sufficient symmetry (whether it be a geometric figure, the roots of a polynomial, or whatever) to questions in group theory. In other words, if we are experts in group theory, we can forget everything we know about the original problem! (Well, ... sort of: Ultimately an understanding of the original problem will be needed to interpret our conclusions.)

Geometry beyond Euclid

Planar Euclidean geometry, i.e., the geometry studied in high school, is the study of properties of geometric figures in the plane invariant under rigid motions. Considering rigid motions as a group of symmetries for the plane makes it into a homogeneous object. But in light of the fact cited above, we can turn this construction on its head: Ask your friendly neighborhood Group Theorist for a list of all known groups and their subgroups, and we can generate an exhaustive list of homogeneous objects. Those corresponding to "smooth spaces" in some sense (things like planes, spheres, tori, other surfaces, hyperplanes, hypersurfaces, etc.) will be candidates for "alternative geometries."

Generally speaking, notions from Euclidean geometry, such as "straight line", "polygon" and "rigid motion", can be generalized to the alternative geometries constructed above. For example, in spherical geometry (already identified above as homogeneous) the straight lines are the great circles; this is the geometry of ocean navigators. Another example is cylindrical geometry. Here the underlying space is an infinite cylinder, the straight lines are azimuthal circles or helixes, and the rigid motions are combinations of translations along the cylinder, and rotations about the cylinder's axis. Actually cylindrical geometry is, for certain reasons, close to conventional planar geometry, and is consequently viewed as uninteresting. A more exotic but extremely important example is hyperbolic geometry. Here the underlying space is a hyperboloid (see figure). If the hyperboloid is appropriately "flattened out" to form a disk, then the straight lines of the geometry consist of circular arcs orthogonal to the disk's boundary; see the left figure below for a single straight line and the right figure for many such lines. On the hyperboloid itself these lines are the shortest curves joining any two of their points.

(Picture Credit: Curtis D. Bennett)

Differential geometry: The mathematics of inhomogeneity

The zoological diversity of groups and their subgroups is reflected in a manifold of geometric "flavors." In addition to Euclidean, spherical, cylindrical and hyperbolic, we have elliptic, projective, conformal, Weyl, and Hermitian, to name a few. The underlying space of each geometry is a homogeneous space, i.e., a space with lots of symmetry.

Curie's observation "dissymmetry makes the phenomenon" now engenders the following question: What happens in geometry - whether it be Euclidean, hyperbolic or whatever - when the underlying symmetries are destroyed? In a certain sense the source of asymmetry or inhomogeneity we have in mind is curvature. We will not describe much detail here, but we will give the resulting spaces a name: they are Cartan geometries, named after the brilliant mathematician Elie Cartan.

Although symmetry is lost, Cartan geometries have a lot of structure. For example, if there is a notion of straight line in the original geometry, then such a notion will persist in the Cartan geometry which perturbs it. In the Euclidean, spherical and hyperbolic cases (among others) there continues to be a notion of "angle" and "distance between two points." Spaces of this kind, called Riemannian manifolds, are of central importance to theoretical physicists. One interesting question is: Given an arbitrary Riemannian manifold, how do I recognize if it is homogeneous or not? If so, how do I recover the full group of symmetries? If the manifold is homogeneous and two-dimensional (say), will it be a plane, sphere, hyperboloid, or something else? How will I tell? Questions like these are close to the frontiers of research in differential geometry.