Understanding Structure in Mathematical Objects

There are two types of mathematicians- algebraists and geometricians. The former excel at the manipulation of abstract symbols representing numerical quantities. Their reasoning centers on formal logic and neither relies upon nor allows for the messy realities of empiricism. Geometricians, in contrast, are concerned with spatial relationships. Their work is best represented visually and hence invites the use of intuition, although formal proofs must be provided. The goal of undergraduate mathematics, as sourly postulated by Benoit Mandelbrot, is to take the geometricians and hound them until they become algebraists.

It is true that mathematics can appear dominated by endless streams of meaningless numbers and esoteric symbols. The structure and harmony of the mathematical world are often not obvious to visual thinkers. However, mathematics is in essence the study of relationships, and when viewed as such its importance becomes apparent. There are branches of mathematics that illustrate structure and relationship well, but they are rarely emphasized until graduate or upper-level undergraduate coursework.

Graph theory is the study of nodes and networks. A graph is simply a set of dots, called vertices or nodes, and a set of edges connecting them. The edges may be “directed” or “weighted”, meaning that they have a direction or number associated with them. The applications are endless. The vertices of a graph could be airports, and the edges flight paths. The edges might have an associated direction (if a flight is one way), and would certainly have an associated number (the distance). Computer science relies heavily on graph theory to represent the structure of a network; the vertices are computers connected by edges if they are networked. Graphs are used to model the spread of disease. In this case, the vertices are people, and two vertices will be connected by an edge if those two people came in contact with the possibility of spreading the disease. Such a model can help track the progression of a rare infection and hopefully stop it before it spreads widely.

Not long ago, the field of fractal geometry found its way into popular culture. Beautiful “pictures” of objects generated by complex iterations were sold as posters and integrated into other works of art. These objects, called fractals, displayed a definite pattern while at the same time exhibiting a natural, “organic” sort of beauty that the rigid objects of classical geometry could never show. While fractal artwork has long since lost its novelty, its usefulness remains. Fractals can be used to model natural phenomena like coastlines and clouds as well as social phenomena such as stock markets. Computer graphics algorithms often use fractal geometry to make virtual objects appear more real.

Fractals were brought into the public eye by the prolific work of Benoit Mandelbrot, a geometrician whose visual intuition apparently survived the rigors of undergraduate mathematics. Though he learned the mechanics and formality of symbol manipulation, he never lost his ability to wonder at the structure exhibited by nature. His observations were sometimes expressed with a trace of poetic sentiment.

“A cloud is made of billows upon billows upon billows that look like clouds,” Mandelbrot observed. “As you come closer to a cloud you don’t get something smooth, but irregularities at a smaller scale.” Inspired by the interplay of chaos and structure that is nature, Mandelbrot forged a new area of mathematics. His discoveries are an excellent example of how formal abstract thinking and visual intuition work together to form a greater understanding of the structure of mathematical objects and the natural world they model.