Topology :: Math Mathmatics

Topology Topology is a modern branch of geometry. It has been called qualitative geometry because instead of thinking about the traditional characteristics of an object (like angles, length, etc.), topologists study features that can’t be altered by stretching, twisting or shrinking the object. After any alteration all points in the object that were connected must still be connected and all points separated by a hole must remain separated. Topology also attempts to explain objects that cannot exist in three dimensions using mathematical equations, since it is nearly impossible to imagine such objects within our frame of reference. The dimension of an object can be thought of in two ways: intrinsic and extrinsic. The perception of a â€Å"creature† occupying, say a line, is one-dimensional, since he can only move in one dimension. However, we draw a line on a plane, so extrinsically it is two-dimensional (1). So how do objects occupying the same dimension diff er topologically? A doughnut shaped object, called a torus, and a sphere are topologically different. Both of these objects are extrinsically two-dimensional, since we only deal with the surfaces of the object. There is no â€Å"inside.† The reason for the topological difference is the hole in the middle of the torus. No permitted alterations (stretching, twisting, shrinking) can be made to the sphere that will transform into a torus. Topology emerged out of Euler’s work on graph theory in the early 1700’s. Leonhard Euler was born on April 15, 1707 in Switzerland. His father was a minister and wanted his son to follow in his footsteps. He sent his son to the University of Basel in 1720, when Leonhard was only 14. It was here that his interest and natural capabilities in mathematics really began to show. After completing his studies and showing very promising mathematical talent, Euler moved to St. Petersburg, Russia to teach mathematics, at the age of only 19. He remained in Russia for several years (4). And it was here that he made contributions to mathematics that would later be seen as the first steps towards topology. Graph theory studies how points are connected without giving any regard to the distance between them or the actual shape of the line connecting them.