Symmetrical symbols, drawings, and buildings have fascinated men and women for millennia. In prehistoric times, craftsmen created symmetrical pieces of jewelry, inspired possibly by the human body and the animal body. One of the oldest symmetrical objets d’art crafted by human hand is a bangle, decorated with an extremely complicated drawing, that was found in the Ukraine and dates from the 11th millennium BCE. Ancient architecture also boasts examples of symmetry, such as the pyramids of Giza (3000 BCE) and the arrangement of rocks at Stonehenge (2000 BCE).

But symmetry does not belong exclusively to the domain of art, nor does it belong only to architecture. Scientists also claim ownership. And once scientists set to work, it is usually mathematics that provides the language and furnishes the tools to explore nature’s phenomena.

In elementary geometry there are three broad sorts of symmetry. First, there are figures, such as the letters M or W that are “reflection symmetric”: The two halves are mirror images of each another. The line separating the halves—the vertical line through the middle of each letter—is called the axis of symmetry. Second, there are shapes, such as the letters S or Z, that have rotational symmetry. They coincide with themselves after having been rotated 180 degrees about some point. The point about which they are rotated is called the center of rotational symmetry. Finally, an infinite sequence of shapes or signs, such as KKKKKKKKK or QQQQQQQQQ, is said to be translation symmetric, since the pattern coincides with itself when slid (translated) to the right or to the left. More complicated types of symmetry also exist, and different symmetries can be combined with each other.

Wallpaper patterns, for example, can have reflection, rotational, and translation symmetry all at the same time.

During the summer of 2003 a conference entitled “Symmetry Festival” took place in Budapest at which scientists and artists gathered for a week of interdisciplinary discussions. They scrutinized examples of symmetry, among them batik weaving, the Talamana system of proportions in Indian sculptures, and, a staple of this kind of art, the pictures of M. C. Escher. This was also an occasion to dispel, once and for all, some of the myths that have become so dear to us all. It turns out, for example, that the pentagram, the five-pointed figure widely believed to have been used by the Pythagoreans as a secret sign, did not in fact have this significance at all. The only source attributing the pentagram to the Pythagoreans dates from the 2nd century AD, 700 years after the death of Pythagoras. More credible sources identify this sign to be the seal of King Solomon. It then mutated into the six-pointed Star of David, which today decorates the national flag of Israel. The myths surrounding the golden section, or “the divine proportions,” apparently also belong to the realm of fantasy and fable. It came to be regarded as an ideal proportion only during the 19th century, at which point the Romanticists projected it back to the medieval times that they so admired.

Is it true, then, that symmetry is an ideal state? Most of the conference participants were of the opinion that complete symmetry is rather boring. Only when a painting, or a piece of music or even a ballet, breaks its symmetries does art truly become interesting. A saying in Zen-Buddhism claims that real beauty appears only when symmetry is intentionally broken. The same holds for the sciences. Many phenomena arise at the borderline between symmetry and asymmetry. Pierre Curie, the noted French physicist and Nobel laureate, once said: “C’est la dyssymétrie qui crée le phénomène” (“It is dissymmetry that creates the phenomenon”). In the mid-19th century Louis Pasteur discovered that many chemical substances had “chirality.” This term refers to the fact that right-handed and left-handed molecules may exist—mirror im-

ages of each other—which could not, however, be exchanged with one another, much in the same way that a right hand does not fit into a left glove.

A very sad reminder of the fact that left and right are not always interchangeable occurred during the 1960s. It was discovered that two versions of the pharmacological ingredient thalidomide, used in a medication called Contergan, existed—a right-handed version and a left-handed version. One form of the component is an efficient drug against nausea; the other causes the most dreadful birth defects.

The most significant symmetry break, according to one of the conference participants, occurred around 10 billion to 20 billion years ago. Matter and antimatter had been in equilibrium until, all of a sudden, something—nobody knows exactly what—happened that caused a disturbance in the symmetry. The result, so he claimed, was the Big Bang.