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1736
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Topology is a field of mathematics that covers a broad
array of subjectswhose ideas are present in many different areas of mathematics
today. In particular, topology has several different branches, including
pointset topology, algebraic topology, geometric topology, and differential
topology. Topology explores such topics as knot theory and graph
theory, and the Mobius band is a famous example of a topological creation.
Famous mathematicians including Riemann, Mobius, Goldbach, Cantor, Hilbert,and
Bernoulli have all made significant contributions to topology.
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Euler's solution to the Konigsberg bridge problem in
1736 is considered the beginning of graph theory in particular and topology
in general. The German city of Konigsberg is built on both sides of a river
and included two islands which were accessible to each other by a single
bridge. One of the islands was accessible from both shores by two bridges,
andthe other island was accessible from both shores by only a single bridgefrom
each shore. Thus, there
were seven bridges in all in Konigsberg.
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Eulers solved this problem by creating
a diagram where each point (vertex) represents a land mass and
each arc (edge) represents one of the seven bridges. Using
this diagram, he was able to determine that it was impossible to cross
all seven bridges in a single trip without doubling back; using this diagram,
it is clear that the Konigsberg bridge problem is the same as asking whether
it is possible to draw the above diagram by tracing each arc exactly
once and without lifting the writing utensil.
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Euler solved this problem by reasoning that if a vertex
has an odd degree, i.e., it has an odd number of edges coming into
it, then the path must either begin or end at this vertex. Only if
a vertex has an even degree is it possible to approach the vertex
along one edge and then depart from it along another. It is clear
from Euler's diagram, also called an
Eulerian
Circuit, that each vertex in the problem has an odd degree. Therefore,traveling
across each bridge exactly once is not possible.
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Euler would later further advance the field of topology with his formula
relating the number of vertices, edges, and faces for a polyhedron. Euler
proposed the formula, v - e + f = 2, in a letter to
Goldbach in 1750.
Author: Tony Brinsko
References:
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Topology
Enters Mathematics
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Konigsberg
Bridge Problem
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Mathematization:
A Walk in the Park
Last updated December 1998