Topology
Mathematics is a field so vast and diverse that it is impossible to be an expert in all areas. It is also a field that is constantly evolving and branching outward. The field of topology is one of the newest intensively studied branches of mathematics. “A simple way to describe topology is as rubber sheet geometry” [2]. “Topology is an offshoot of geometry that originated during the 19th century and that studies those properties an object retains under deformation - specifically, bending, stretching and squeezing, but not breaking or tearing” [1]. Under these conditions, one could say that a square is topologically equivalent to a circle because a square can be bent and stretched into a circle [3]. However, a square is not topologically equivalent to a torus because a torus cannot be formed unless a hole is bored through the medium, or two pieces are joined together. Topologists obviously have expanded upon these simple concepts over time to create theorems further removed from our ordinary experiences. Some of these shapes and objects exist in four dimensional space or higher dimensions and cannot exist in our world. Theoretically these shapes would be as commonplace as a tree or rock in a higher dimensional universe. However, in our universe topologists turn to mathematics to understand these shapes [6].
The first mathematical problem, which led to the origins of topology, was the Konigsberg bridges problem. The people of Konigsberg wondered if they could walk around the city in a way that they would also cross every bridge exactly once. The city map looked something like this [2]:
Euler determined that it was indeed impossible to accomplish this feat. He rationalized this problem...
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...nal space.
Works Cited
[1] http://www.britannica.com/bcom/eb/article/2/0,5716,115452+1,00.html Encyclopedia
Britannica: Topology. Accessed December 6, 1999.
[2] http://www.forum.swarthmore.edu/~isaac/problems/bridges1.html The Beginnings of Topology. Accessed December 6, 1999.
[3] http://www.geom.umn.edu/docs/doyle/mpls/handouts/node13.html Topology.
Accessed December 6, 1999.
[4] http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/topology_in_mathematics.html
Topology Enters Mathematics. Accessed December 6, 1999.
[5] http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Klein.html Felix
Christian Klein. Accessed December 7, 1999.
[6] http://www.pepperdine.edu/seaver/natsci/faculty/kiga/topology.htm What is
Topology. Accessed December 7, 1999.
[7] Yaglom, I. M. Felix Klein and Sophus Lie. Birkhauser, Boston. 1988.
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