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Topic: Seven Bridges of K nigsberg

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	Math Forum - Ask Dr. Math
		Date: 5/23/96 at 13:21:42 From: Doctor Betsy Subject: Re: Königsberg's bridge Hi Suisui, I searched for Königsberg in a Math history archive on the World Wide Web, and found information on a page at: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Topology_in_mathematics.html This is a Math history archive, and it has some information on the Königsberg bridge.
		In 1736 Euler published a paper on the solution of the Königsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position.
		The paper not only shows that the problem of crossing the seven bridges in a single journey is impossible, but generalises the problem to show that, in today's notation, a graph has a path traversing each edge exactly once if exactly two vertices have odd degree.
	www.mathforum.org /library/drmath/view/51454.html (313 words)

	Topology Totally Explained
		This result didn't depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks.
		This problem, the Seven Bridges of Königsberg, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory.
		As with the Bridges of Königsberg, the result doesn't depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind of blob (subject to certain conditions on the smoothness of the surface), as long as it has no holes.
	topological.totallyexplained.com (1793 words)

	Math Forum: Leonard Euler and the Bridges of Konigsberg
		In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts.
		Seven bridges were built so that the people of the city could get from one part to another.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /isaac/problems/bridges1.html (336 words)

	CHAOS (OR: THE RECIPROCITY OF ANALYSIS AND SYNTHESIS IN COMPLEX AND DYNAMIC FORMS OF RELATION)
		For they clearly form the pattern of a square with a triangle pointing to it: This pattern is a remarkably accurate representation of the relation between analytic and synthetic operations.
		Bridge "7" serves both as the synthesis of the two opposing bridges, "5" and "6", and as the synthesis of the four opposing bridges "1", "2", "3", and "4".
		Of course, these logical relations were not used by Euler in proving the impossibility of a circuit in this particular case; nevertheless, the story illustrates the complexity of synthetic relations, as well as their tendency to form ordered relations, even in empirical situations where we least expect such patterns to emerge.
	www.hkbu.edu.hk /~ppp/gl/GL4.html (12320 words)

	Seven Bridges (answer) (Site not responding. Last check: )
		He asked whether it was possible to cross all seven bridges without crossing any one of them twice.
		In specific, the solution set of the Seven Bridges of Konigsberg is contained in the solution set of the less restrictive problem.
		To visualize the connection between this and the seven bridges - think of a bridge as corresponding to a pair of line segments.
	www.math.uah.edu /mathclub/puzzles/answers/bridges.html (328 words)

	Leonhard Euler Biography
		A generalization of Euler's formula for arbitrary planar graphs exists: F - E + V - C = 1 where C is the number of components in the graph.
		In 1736 Euler solved a problem known as the seven bridges of K?nigsberg publishing a paper Solutio problematis ad geometriam situs pertinentis which may be the earliest application of graph theory or topology.
		Read Euler: he is our master in everything.
	www.ebiog.com /biography/3089/leonhard-euler/bio.htm (674 words)

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