Symmetries

By Mina Teicher 

Leonardo da Vinci’s Vitruvian Man, ca. 1487

It is known that mathematicians see beauty in mathematics. Many mathematicians are motivated to find the most beautiful proof, and often they refer to mathematics as a form of art. They are apt to say “What a beautiful theorem,” “Such an elegant proof.” In this article, I will not elaborate on the beauty of mathematics, but rather the mathematics of beauty, i.e., the mathematics behind beauty, and how mathematical notions can be used to express beauty—the beauty of manmade creations, as well as the beauty of nature.

I will give four examples of beautiful objects and will discuss the mathematics behind them. Can the beautiful object be created as a solution of a mathematical formula or question? Moreover, I shall explore the general question of whether visual experience and beauty can be formulated with mathematical notions.

I will start with a classical example from architecture dating back to the Renaissance, move to mosaic art, then to crystals in nature, then to an example from my line of research on braids, and conclude with the essence of visual experience.

The shape of a perfect room was defined by the architects of the Renaissance to be a rectangular-shaped room that has a certain ratio among its walls—they called it the “golden section.” A rectangular room with the golden-section ratio also has the property that the ratio between the sum of the lengths of its two walls (the longer one and the shorter one) to the length of its longer wall is also the golden section, 1 plus the square root of 5 over 2. Architects today still believe that the most harmonious rooms have a golden-section ratio. This number appears in many mathematical phenomena and constructions (e.g., the limit of the Fibonacci sequence). Leonardo da Vinci observed the golden section in well-proportioned human bodies and faces—
in Western culture and in some other civilizations the golden-section ratio of a well-proportioned human body resides between the upper part (above the navel) and the lower part (below the navel).

By Richard Taylor 

In modular arithmetic, one thinks of the whole numbers arranged around a circle, like the hours on a clock, instead of along an infinite straight line. Here we have seven “hours” on our clock—arithmetic modulo 7. To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count 3 clockwise 5 times, again ending up at 1.

Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs.

For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity. But in one of those strange pieces of serendipity which often characterize the advance of human knowledge, in the last half century modular arithmetic has found important applications in the “real world.” Today, the theory of modular arithmetic (e.g., Reed-Solomon error correcting codes) is the basis for the way DVDs store or satellites transmit large amounts of data without corrupting it. Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic. You can visualize the usual arithmetic as operating on points strung out along the “number line.”