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Topic: Karl Menger

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	Karl Menger Summary
		Menger was born on January 13, 1902 in Vienna, Austria, the son of eminent economist Carl Menger and novelist Hermione Andermann.
		Menger's prestige as a mathematician received a major boost in 1927 when he was chosen as a member of the elite Vienna Circle, a group of some three dozen scientists from wide-ranging disciplines.
		With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions to the notions of angle and of curvature in terms of directly measurable physical quantities, namely ratios of distance values.
	www.bookrags.com /Karl_Menger (962 words)

	Karl Menger
		Abacci > Abaccipedia > Ka > Karl Menger
		Karl Menger (January 13 1902 - October 5 1985) was a mathematician of great scope and depth.
		With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions to the notions of angle and of curvature in terms of directly measurable physical quantities, namely ratios of distance values.
	www.abacci.com /wikipedia/topic.aspx?cur_title=Karl_Menger (203 words)

	Karl Menger
		Karl Menger (Jan. 13 1902 - Oct. 5 1985) was a mathematician of great scope and depth.
		His most famous popular contribution was the Menger Sponge (mistakenly known as Sierpinski's Sponge), a three dimensional version of Sierpinski's Carpet.
		You may be looking for the article on his father, the economist Carl Menger.
	www.ebroadcast.com.au /lookup/encyclopedia/ka/Karl_Menger.html (75 words)

	Menger biography
		Menger became interested in the topic and was encouraged by Hahn to work on the topic.
		Menger's work led him to a definition of dimension independently of Urysohn, however Urysohn had died in a drowning accident before he could publish his work and Menger was not aware of it.
		Menger was not sorry to leave Amsterdam since he had become involved in a priority dispute with Brouwer and they were not on the best of terms.
	www-history.mcs.st-and.ac.uk /history/Biographies/Menger.html (503 words)

	Illinois Institute of Technology - Applied Mathematics Department: Karl Menger
		Karl Menger was born on January 13, 1902 in Vienna.
		Karl Jr., born in 1936, Rosemary and Fred, twins born in 1937, and Eve, born in 1942.
		One of Menger's theorems on isometric imbeddings is
	www.math.iit.edu /Menger/menger.html (4665 words)

	The Vienna Colloquium
		The "Vienna Colloquium" run by the mathematician Karl Menger (son of the economist Carl Menger) in the 1930s, brought together many different minds from mathematics but also the physical sciences, philosophy, statistics and economics, that eventually set in motion modern general equilibrium theory from its early roots in the Lausanne School.
		Among the relevant participants in the Colloquium besides Karl Menger, was the banker and amateur economist Karl Schlesinger as well as the young Oskar Morgenstern, the polymath John von Neumann and the statistician Abraham Wald.
		Menger, 1994 Reminiscences of the Vienna Circle and the Mathematical Colloquium, Dordrecht: Kluwer.
	cepa.newschool.edu /het/schools/vienna.htm (791 words)

	Illinois Institute of Technology - Department of Applied Mathematics - Remembering Menger 2007
		On Monday, April 9, the Department of Applied Mathematics at IIT will host a half-day event to honor Professor Karl Menger, an outstanding mathematician of the twentieth century, and a former faculty member at IIT.
		Professor Karl Sigmund of the University of Vienna will deliver the Inaugural Karl Menger Memorial Lecture on "Menger, Games, and Morals" at 4:30 pm.
		Made possible with the generous support of the Menger family, IIT Department of Applied Mathematics, IIT Department of Institutional Advancement, and IIT alumni.
	www.math.iit.edu /events/menger2007.html (394 words)

	Karl Menger on Hans Hahn
		Menger's introduction exhibits the historical context and tells about Hahn's interest in parapsychology.
		Finally, one day in the early 1920s, the newspapers claimed that two professors of physics at the University of Vienna, Stephan Meyer and Karl Przibram, had exposed the entire spiritualistic swindle.
		What had happened, the newspapers elaborated, was that Professor Meyer had invited many people to his house for a séance.
	www-gap.dcs.st-and.ac.uk /~history/Extras/Menger_on_Hahn.html (2715 words)

	The Institute For Figuring // Online Exhibit: Mathematical Paper Folding
		Menger attended a course of lectures by Professor Hans Hahn at the University of Vienna entitled What’s New Concerning the Concept of the Curve; under Hahn’s encouragement he embarked on an exploration of the concept of dimension that him to an expanded definition of this seemingly obvious term.
		Though it manifestly occupies a volumetric space, the Menger sponge is essentially a linear object – it possesses a topological dimension of 1.
		Menger proved that it is indeed the universal curve - that is, any possible one-dimensional curve is mathematically identical to some part of its infinitely complex internal morphology.
	theiff.org /oexhibits/menger01.html (828 words)

	Von Smith zu Menger
		Karl Marx (1818-1883) mußte nicht weit suchen, als er intellektuelle Munition für sein Revolutionsprogramm suchte.
		Menger hatte beim Schreiben der Grundzüge gehofft, daß er damit die theoriefeindlichen Ökonomen der Historischen Schule in Deutschland überzeugen könnte, ihre unwissenschaftliche Haltung aufzugeben.
		Menger hatte damit nicht nur das Wasser-Diamanten-Paradoxon aufgelöst, sondern insgesamt die Smithsche Lehre auf ein sicheres theoretisches Fundament gestellt.
	www.mehr-freiheit.de /idee/menger.html (2860 words)

	Carl Menger / Biography
		The "value" of a good, Menger claimed, would be the equal to the least urgent use to which it was applied.
		While Menger did have an adjustment process for the establishment of price, he did not have an "auctioneer" outright but rather had his agents inching their way along, making errors and correcting them, over a long period of time to discover the "equilibrium" prices at which exchange could be effectuated to everyone's complete satisfaction.
		Menger, 1884) and the debate was carried on into the 20th Century by their disciples.
	www.cooperativeindividualism.org /mengerbio.html (805 words)

	Methodology of the Austrian Economists, Carl Menger
		In this manner Menger proposed to discover "exact" laws governing economic phenomena: not laws of mathematical precision, but laws which follow necessarily from the essential nature of the factors involved, and thus are invariably true regardless of time and place.
		Menger's concern with the essence or nature of economic phenomena meant attention to the reason for their existence, their origins.
		Menger's son Karl, a mathematician, has pointed out that mathematical economists are limited to functional relations, while the Austrians demand causal explanations.
	www.mises.org /mofase/ch2.asp (1949 words)

	MS.153
		The collection of photographs was given to the University by Karl Menger, the son of Carl Menger.
		Carl Menger was a founder of the Austrian school of economics.
		Carl Menger's son Karl (1902-1905) was a distinguished mathematician, and his grandson, Fred M. Menger, is a professor of chemistry at Emory University.
	www.library.jhu.edu /collections/specialcollections/manuscripts/msregisters/ms153.html (451 words)

	math lessons - Karl Menger
		Karl Menger (Born in Vienna, Austria, January 13 1902 -- Died in Highland Park, Illinois, USA, October 5 1985) was a mathematician of great scope and depth.
		His most famous popular contribution was the Menger Sponge (mistakenly known as Sierpinski's Sponge), a three dimensional version of Sierpinski's Carpet.
		With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions to the notions of angle and of curvature in terms of directly measurable physical quantities, namely ratios of distance values.
	www.mathdaily.com /lessons/Karl_Menger (188 words)

	Illinois Institute of Technology - Applied Mathematics Department - Opportunities
		The Inaugural Karl Menger Memorial Lecture was given by Prof.
		Karl Sigmund, University of Vienna, on the afternoon of Monday, April 9, 2007.
		Mike McCourt '07 won the IIT Karl Menger Award for exceptional scholarship by a student.
	www.math.iit.edu /more-news.html (1694 words)

	About \| Mengermania
		Mengermania tracks the progress of building a level four Menger sponge.
		A Menger sponge is a three-dimensional fractal named for its creator, Karl Menger.
		Each time the process of dividing and removing cubes is completed, a new "level" of sponge is created.
	mengermania.c82.net /about.php (263 words)

	Karl Menger at AllExperts
		He was the son of the famous economist Carl Menger.
		He was a student of Hans Hahn and received his PhD from the University of Vienna in 1924.
		His most famous popular contribution was the Menger sponge (mistakenly known as Sierpinski's sponge), a three-dimensional version of Sierpinski's carpet.
	en.allexperts.com /e/k/ka/karl_menger.htm (364 words)

	Amazon.com: Principles of Economics (Institute for Humane Studies Series in Economic Theory): Books: Karl Menger (Site not responding. Last check: 2007-10-26)
		Menger's pre-requisites for a thing to become a good are the foundation for the bulk of modern marketing (at least, modern marketing that is of any value to consumers), and his explanation of exchange is vastly superior to the robotic theories offered by Jevons, Marshall, and the other predecessors of modern mainstream economic thought.
		Menger claims that property is not an arbitrary arrangement but "the only possible solution of the problem that is, in the nature of things, imposed upon us by the disparity between requirements for, and available quantities of, all economic goods".
		Menger's emphasis on learning and gradual progress and learning is important because it allows for a middle ground where the world can be imperfect, but not hopelessly so.
	www.amazon.com /Principles-Economics-Institute-Studies-Economic/dp/0910884277 (3032 words)

	fan5583's Xanga Site (Site not responding. Last check: 2007-10-26)
		Karl Menger (1840-1921) was born on February 28th 1840 in Galicia, which is now part of Poland.
		Karl Menger has been called the father of Austrian economics because of his emphasis on individual economic freedom.
		The book caused uproar among other German economists, who were sadistic to Karl Menger and the "Austrian School" as they called it because it was different and set apart from the ways of economics in other parts of the country.
	www.xanga.com /fan5583 (2146 words)

	machine project » The Business Card Menger Sponge
		Menger’s Sponge - named for its inventor Karl Menger and sometimes wrongly called Sierpinski’s Sponge – was the first three dimensional fractal that mathematicians became aware of.
		Recipe for a Menger Sponge: Take a cube, divide it into 27 (3 x 3 x 3) smaller cubes of the same size; now remove the cube in the center of each face plus the cube at the center of the whole.
		The Business Card Menger Sponge is being presented in its first appearance to the general public by Machine Project and the Institute For Figuring to coincide with OSME 4, the fourth international conference on Origami in Science, Mathematics and Education, being held at Caltech.
	machineproject.com /2006/08/21/the-business-card-menger-sponge-2 (439 words)

	Math prodigy wins $1,000 award
		The Karl Menger Memorial Prizes are funded by income from the Karl Menger Fund, which was established by the family of the late Karl Menger.
		Karl Menger (1902-1985), a leading mathematician of the 20th century, was born in Vienna and received his PhD in mathematics from the University of Vienna.
		The list of winners of Menger Awards, going back to 1990, may be found at http://www.ams.org/prizes/menger-award.html.
	www.eurekalert.org /pub_releases/2006-05/ams-mpw051806.php (747 words)

	Amazon.com: Reminiscences of the Vienna Circle and the Mathematical Colloquium (Vienna Circle Collection): Books: Karl ... (Site not responding. Last check: 2007-10-26)
		Karl Menger (1902--1985), a pure mathematician of distinction, also took an active interest in both philosophy and economics.
		The memoir devotes particular attention to Wittgenstein (with whose family Menger was acquainted) and to Gödel, whom he was instrumental in bringing to America.
		It testifies both to Menger's power to inspire and to the critical eye he always turned on even the philosophers he most approved of.
	www.amazon.com /Reminiscences-Vienna-Mathematical-Colloquium-Collection/dp/079232711X (994 words)

	Sierpinski Gasket
		In 1926 when Karl Menger first looked at his creation, the Menger sponge, could he have ever imagined that in the future people would instruct machines to draw the sponge to depths he could only dream of.
		The Menger sponge is the 3D equivalent of the Sierpinski carpet which is in turn the 2D equivalent of the 1D Cantor set.
		The Menger sponge is often used as a challenge to the computer graphics community.
	astronomy.swin.edu.au /~pbourke/fractals/gasket (2172 words)

	Menger's Sponge (Site not responding. Last check: 2007-10-26)
		Menger's sponge (sometimes wrongly called Sierpinski's Sponge) is a fractal solid that can be described as follows.
		Take a cube, divide it into 27 = 3 x 3 x 3 smaller cubes of the same size and remove the cube in the center and the six cubes that share faces with it.
		Menger's Sponge is named for its inventor, Austrian mathematician
	world.std.com /~j9/sponge/menger-sponge.html (122 words)

	Menger, Karl :: People : Gourt
		Karl Menger - Biography, bibliography, fields of research and references.
		Karl Menger - A memorial note by Seymour Kass.
		Menger - A short biography of Karl Menger (1902-1985)
	science.gourt.com /Math/History/People/Menger,-Karl.html (118 words)

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