Symmetry (mathematics)

Summary: An ancient mathematical concept, there are various forms of symmetry.

“Symmetry,” which comes from the Greek word roots meaning “same” and “measure,” describes a picture, shape, or other object that looks the same when viewed from another perspective or that can be transformed in some way without changing its important properties. The word “symmetry” can refer to this property, to the transformation itself, or more holistically to an aesthetically pleasing sense of balance. Eighteenth-century mathematician Adrien-Marie Legendre revolutionized the concept of symmetry when he connected it to transformations. There are a wide variety of uses of the word “symmetry” in different domains, including art, architecture, and science, and many of these have existed from antiquity. The concept of symmetry is inherent to modern science and architecture, and its evolution reflects in many ways the dynamic nature of these fields.

Visual Symmetry

In the context of geometric figures drawn in the plane, there are three fundamental types of symmetry:

• 1. A figure has “reflection” symmetry if it coincides with its own mirror image across some line. The capital letters M and W have a single reflection symmetry, while the letter H has two symmetries, horizontal and vertical.
• 2. A figure has “rotational symmetry” if it can be rotated around a fixed point, leaving the figure unchanged. For example, the capital letters N, Z, and S are unchanged when rotated 180 degrees. The pattern of black squares in traditional crossword puzzles also has this half-turn symmetry.
• 3. A figure has “translational symmetry” if it can be slid or moved without changing. A typical example is a repeating pattern on wallpaper.

Construed in the broadest terms, symmetry plays a role in almost all art and is related to balance and harmony. One of the many ways in which the narrower geometric notion of symmetry applies to art is tessellations. A tessellation is a covering of the plane by copies of a limited set of tiles. Such figures are often highly symmetric. Tilings by squares, hexagons, and triangles are common enough, both in art and on kitchen floors, and more fanciful tessellations involving animal and plant shapes are also possible. Tessellations, dynamic symmetry, and mathematical sophistication are especially evident, for example, in the art of M.C. Escher (1898–1972).

Abstract Symmetries

Symmetry is not just a geometric concept. Any structure or object can have symmetry. Abstractly, a symmetry is any transformation of an object resulting in an object that is “the same” in the sense of having all the same properties that are important in context. Often, the object is a geometric figure, and the relevant properties are length, angle, and area, but it need not be so.

Consider the game rock-paper-scissors. Renaming the scissors gesture to “paper,” renaming paper to “rock,” and renaming rock to “scissors” would leave the rules of the game unaltered. This is an abstract symmetry of the game. Then, there are enough symmetries to identify any move with any other, so all three options are intrinsically “equally good.” In this example, there is symmetry but no geometry whatsoever.

Symmetry and Groups

In higher mathematics, notions of symmetry are expressed in the language of group theory. A “group” is a set (G) of objects that can be composed together (in other words, if x and y are elements in a group, x×y is also an object in the group), subject to three conditions: associativity, identity, and inverse criteria. The salient feature of this definition is that the set of all the symmetries of any object satisfies these conditions. The associativity property is automatic from function composition; but what about the other two? These are restatements of the convention that the transformation that does nothing is a symmetry and the idea that symmetries are “undo-able.” Symmetries leave an object “structurally the same as it was,” so there will always be another symmetry to undo any given symmetry.

The symmetries of any object that preserve any desired features form a group, called the “symmetry group” of the object. Often, one can understand a complicated object much better by studying the size and structure of its symmetry group.

Klein and the Erlangen Program

Felix Klein (1849–1925) greatly strengthened the connection between geometry and group theory. His insight was that, if one really wants to understand a geometric structure, then one should study the group of symmetries that preserve the structure. This philosophy has proved very fruitful and is now known as the Erlangen program.

For example, in ordinary Euclidean plane geometry, the focus is on lengths and angles. The group of symmetries that preserve lengths and angles consists of translations, rotations, reflections, and combinations of these. Given any two points, each with an arrow pointing away from it in a given direction, one can always translate and rotate the plane so that the image of the first point lies on the second point, and the arrows are pointed in the same direction. This is the sophisticated way to understand the notion that every point and direction in the plane are functionally the same as every other point and direction.

The Erlangen program has played a fundamental role in the development of nineteenth- and twentieth-century geometric thinking, clarifying the relationships and distinctions between geometry and topology; projective and affine geometry; and Euclidean, hyperbolic, and spherical geometry.

Symmetry and the Universe

Those who study the shape of space are greatly concerned with symmetry. Consider the question of whether the universe is “homogeneous.” That is, do the laws of physics treat every place the same as every other place? Is every direction physically like every other direction? What answers to those questions are believed to be correct determines what shapes, structures, and geometries are viable candidates to model the universe.

Time symmetry is another issue of importance in physics research; one wants to know to what extent the physical laws of the universe treat the past and future symmetrically. On a small enough scale, particle interactions have time symmetry. If one watched a “movie” of particle interactions on a small enough time-scale, it would be impossible to tell whether the movie was playing forward or backward. On the other hand, the large-scale events observed in everyday life do not possess such past-future symmetry; for example, eggshells break but do not spontaneously assemble, people age but do not become more youthful. This discrepancy between small-scale symmetry and large-scale asymmetry is rather mysterious, and one can hope that reconciling the two will lead to greater understanding of physics.

Symmetry and Architecture

Symmetry has long been connected with architecture. In Greek and Latin, symmetry was used to indicate a common measure or a notion of something well-proportioned, rather than as a reflection. However, reflection symmetries can be found in many buildings from different cultures, where the left side is a mirror image of the right side. Architects have also used symmetry in external views, layout, stability, or building details, such as stairs or windows. Some authors claim that the first recorded instance of the use of symmetry as a mirror reflection was in 1665, when Gian Lorenzo Bernini was asked to design an altar for the church of Val-de-Grace, while others assert that it was first found in Claude Perrault’s 1673 treatise on columns. Perrault is best known as the architect of the east wing of the Louvre.

Concepts such as the symmetry groups of the plane also originate in architecture. Beginning with mathematician Edith Muller’s 1944 analysis, experts continue to debate how many of the 17 groups can be found in the mosaics of the Alhambra at Granada, a fourteenth-century Moorish palace. Some assert that all 17 can be found there and in many other examples in Islamic architecture and art. A formal mathematical proof that there are no additional symmetry groups was proven independently by Evgraf Fedorov in 1891 and George Pólya in 1924. Partly because of a prohibition against using anthropomorphic forms, symmetry appears in many instances of Islamic-influenced architecture, such as the Taj Mahal.

The connections between symmetry and architecture continue into the twenty-first century. In numerous texts in the twentieth and twenty-first centuries, mathematicians such as Hermann Weyl illustrate concepts using architectural references. Architects and engineers also frequently use symmetry, though architects working in the modernist aesthetic reject symmetry in their designs.

Bibliography

Cohen, Preston Scott. Contested Symmetries and Other Predicaments in Architecture. Princeton, NJ: Princeton Architectural Press, 2001.

Gardner, Martin. The Ambidextrous Universe: Symmetry and Asymmetry From Mirror Reflections to Superstrings. 3rd ed. New York: W. H. Freeman, 1990.

Hon, Giora, and Bernard Goldstein. From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept. New York: Springer, 2008.

Weyl, Hermann. Symmetry. Princeton, NJ: Princeton University Press, 1952.