Topology Essay -- Mathematics Geometry Essays

Topology Topology is the study of those properties of geometric figures that are unchanged when the shape of the figure is twisted, stretched, shrunk, or otherwise distorted without breaking. It is sometimes referred to as "rubber sheet geometry" (West 577). Topology is a basic and essential part of any post school mathematics curriculum. Johann Benedict Listing introduced this subject, while Euler is regarded as the founder of topology. Mathematicians such as August Ferdinand MÃ ¶bius, Felix Christian Klein, Camille Marie Ennemond Jordan and others have contributed to this field of mathematics. The MÃ ¶bius band, Klein bottle, and Jordan curve are all examples of objects commonly studied. These and other topics prove to be intricate and fascinating mathematical themes. Topologists are mathematicians who study qualitative questions about geometrical structures. They ask questions like does the structure have any holes in it? Is it all connected, or can it be separated into parts? Topologists are not concerned with size, straightness, distance, angle, or other such properties. An often-cited example is the London Underground map. This will not reliably tell you how far it is from Kings Cross to Picadilly, or even the compass direction from one to the other. However, it will tell you how the lines connect between them, using topological rather than geometric information (What 1). Furthermore, if one figure can be distorted into another figure without breaking, then the two figures are described as being topologically equivalent to each other. Two examples of topologically equivalent figures are a coffee cup and doughnut, and groups of the letters of the alphabet. First, an object shaped like a doughnut is a torus. A torus can... ...and. New York:Oxford University Press, 1993. "Felix Christian Klein." Available Online. http://www-groups.dcs-and.ac.uk/~history/Mathematicians/Klein.html. Accessed 12/4/99. Flegg, Graham. From Geometry to Topology. New York: Crane, Russak, and Company, Inc.,1974. "Jordan Curve Theorem and its Generalizations." Available Online. http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html. Accessed 12/6/99. "Marie Ennemond Camille (1838-1922)." Available Online. http://ukdb.web.aol.com/hutchinson/encyclopedia/91/M0046091.htm. Accessed 12/6/99. "What is Topology?" Available Online. http://www.shef.ac.uk/~pm1nps/Wurble.html. Accessed 12/4/99. West, Beverly Henderson, and others. "Topology." The Prentice-Hall Encyclopedia of Mathematics. 1982. 21: 577-585. Yaglom, I.M. Felix Klein and Sophus Lie. Boston: Birkhauser Boston, 1988.