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	Dehn twist - Wikipedia, the free encyclopedia
		In mathematics, in the sub-field of geometric topology, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
		It is a theorem of Max Dehn (and W.
		Lickorish showed that Dehn twists along 3g − 1 curves could generate the mapping class group; this was later improved by Stephen P. Humphries to 2g + 1, which he showed was the minimal number.
	en.wikipedia.org /wiki/Dehn_twist (227 words)

	Dehn, Max (Site not responding. Last check: 2007-10-20)
		Dehn was born in Hamburg and studied at Göttingen University under David Hilbert.
		Dehn found a solution to one of Hilbert's 23 unsolved problems (concerning the existence of tetrahedra with equal bases and heights, but not equal in the sense of division and completeness).
		The theorem came to be known as Dehn's lemma, but was later found not to apply in all circumstances.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/D/Dehn/1.html (151 words)

	[No title]
		The Max Dehn Papers at the Center for American History’s Archives of American Mathematics tell the story of an established Jewish mathematician from Germany leaving his homeland under pressure from the Nazis and finishing his career in the United States, moving from one mathematically low-profile position to another.
		His colleague Max Dehn, a former assistant of David Hilbert, organized the compliation of the manuscript.
		The Max Dehn Papers at the AAM include lecture notes by E. Hellinger; and correspondence, notebooks, manuscripts of publications, reprints, and lecture and course notes by Dehn.
	www.maa.org /features/050505may_archives.html (477 words)

	A Guide to the Max Dehn Papers, 1899-1979
		Collection documents the career of Max Dehn (1878-1952), relating chiefly to his research in geometry, topology, group theory, and the history of mathematics.
		Max Dehn (1878-1952) was a mathematician whose research focused on geometry, topology, group theory, and the history of mathematics.
		The Max Dehn Papers document the career of Max Dehn (1878-1952) and relate chiefly to his research in geometry, topology, group theory, and the history of mathematics.
	www.lib.utexas.edu /taro/utcah/00192/cah-00192.html (1654 words)

	Dissection Puzzles: Tiling and triangulation of a plane
		In 1900, Dehn proved that not every Prism can be dissected into a Tetrahedron (Lenhard 1962, Ball and Coxeter 1987) The third of Hilbert problems asks for the determination of two Tetrahedra which cannot be decomposed into congruent Tetrahedra directly or by adjoining congruent Tetrahedra.
		Max Dehn showed this could not be done in 1902, and W. Kagon obtained the same result independently in 1903.
		A quantity growing out of Dehn's work which can be used to analyze the possibility of performing a given solid dissection is the Dehn Invariant.
	www.katev.org /maths/stands/DISSEC1.HTM (603 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		At the beginning of the century Max Dehn solved the word and conjugacy problems for fundamental groups of orientable surfaces by making use of the underlying hyperbolic geometry.
		Dehn's ideas led to the theory of small cancellation groups and more recently to some striking connections between geometry and finite automata.
		The geometry of the known compact 3-manifolds is reflected by restrictions on the structure of their fundamental groups, and in many cases the spirit of these restrictions is captured by the fact that multiplication in the fundamental group can be carried out by finite automata.
	www.math.unl.edu /~mbrittenham2/ldt/conf/mtholy98.txt (459 words)

	Max Born -- Britannica Concise Encyclopedia - The online encyclopedia you can trust! (Site not responding. Last check: 2007-10-20)
		Born in Breslau, Germany, Max Born taught and conducted research at several German universities before he was forced to emigrate in 1933.
		The Swiss playwright and novelist Max Frisch is noted for his sparse, expressionistic explorations of the moral dilemmas of 20th-century life.
		U.S. jazz drummer and composer Max Roach was one of the most influential and widely recorded modern percussionists.
	www.britannica.com /ebc/article-9357746 (784 words)

	References for Dehn (Site not responding. Last check: 2007-10-20)
		H Guggenheimer, The Jordan curve theorem and an unpublished manuscript by Max Dehn, Archive for History of Exact Science 17 (2) (1977), 193-200.
		W Magnus, Max Dehn, The Mathematical Intelligencer 1 (1978/9), 132-133.
		W Magnus and R Moufang, Max Dehn zum Gedächtnis, Mathematische Annalen 127 (1954), 215-227.
	www-gap.dcs.st-and.ac.uk /~history/Printref/Dehn.html (72 words)

	Mathematics Colloquium --** (Site not responding. Last check: 2007-10-20)
		This simple observation motivates Max Dehn to consider the space of all homotopy classes of simple loops on a surface.
		Dehn called the space the arithmetic field of the topological surface.
		Thurston developed his theory of measured laminations by forming a topological completion of Dehns space with respect to the intersection number.
	www.math.virginia.edu /~colloq/2001-02/2-7-Luo.html (179 words)

	The Dehn Invariant (Sam Stechmann's VIGRE REU 2002 Project at Rutgers University) (Site not responding. Last check: 2007-10-20)
		In order to show that these two tetrahedra are an acceptable counterexample, Dehn needed to show that they have the same volume but are not scissors congruent.
		Theorem (Dehn) If two tetrahedra have different Dehn invariants, then they are not scissors congruent.
		So to calculate the Dehn invariant, we take the direct product of the length and the dihedral angle of each edge, and we add it all up.
	dimacs.rutgers.edu /~sam/dehninv (273 words)

	Liu Hui on the volume of a pyramid - 1. Introduction (Site not responding. Last check: 2007-10-20)
		As is well known, it is a consequence of a theorem proved by Max Dehn in 1900 that any proof of the volume of a pyramid must use infinitesimal considerations in one form or another.
		This question was the basis of the third of Hilbert's famous 23 problems for mathematicians of the twentieth century.
		Dehn solved the problem when he proved that a regular tetrahedron and a prism can in no way be divided into respectively congruent parts.
	www.staff.hum.ku.dk /dbwagner/Pyramid/Pyramid-1.html (270 words)

	DEHN Family Tree
		Meyer Abramson DEEN (1768-1833); (in Dutch pronounced as DEHN) (born 4.7.1798 in Burgsteinfurt Germany died 24.3.1833) married Sophie MELCHIOR (8.7.1809-11.3.1830), sister of Augusta MELCHIOR, boths daughters of Gerson MELCHIOR son of Moses MELCHIOR.
		DEHN studied at Göttingen under David HILBERT's supervision obtaining his doctorate in 1900 for a thesis entitled Die Legendreschen Sätze über die Winkelsumme im Dreieck.
		From 1921 until 1935 he held the chair of Pure and Applied Mathematics at the University of Frankfurt but he was forced to leave his post by the Nazi regime in 1938.
	www.loebtree.com /dehn.html (399 words)

	Hilbert's Third Problem (Sam Stechmann's VIGRE REU 2002 Project at Rutgers University) (Site not responding. Last check: 2007-10-20)
		Within a year, this problem was solved by one of Hilbert's students, Max Dehn.
		Dehn gave a counterexample using the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1) and the tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), (1,1,1) (see figures below).
		In the next section, "The Dehn Invariant", I give a brief explanation of why this is true.
	dimacs.rutgers.edu /~sam/prob3 (316 words)

	Hilbert's 3rd Problem (Site not responding. Last check: 2007-10-20)
		This would be obtained as soon as we succeed in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form to polyhedra which themselves could be split up into congruent tetrahedra.
		Dehn found that not only the volume must remain invariant under cut and paste operations, but also certain combination of side lengths and dihedral angles.
		In particular, show that the tetrahedron and the cube of equal volumes have different Dehn invariant, thus solving Hilbert's 3rd problem in the negative.
	www.csun.edu /~ac53971/courses/math623/spring02/dehn.html (408 words)

	April 7 (Site not responding. Last check: 2007-10-20)
		It was believed that there was no analogous theorem in dimension 3, and as the third of his 23 famous problems, David Hilbert (in 1900) challenged mathematicians to specify two tetrahedra of equal volume that are nonequidecomposable.
		Max Dehn, a student of Hilbert, soon settled the problem, after finding a necessary condition for two 3-dimensional polyhedra to be equidecomposable.
		Clancy is a graduate student of the Mathematics/Statistics GAU.
	www.math.unb.ca /colloq/clancy4.html (139 words)

	Christos Papakyriakopoulos
		The perfidious lemma of Dehn was every topologist's bane 'til Christos Papakyriakopoulos proved it without any strain.
		1-26, and On Dehn's lemma and the asphericity of knots, Proceedings of the National Academy of Sciences, volume 43 (1957), pp.
		Papakyriakopoulos C.D.: On Dehn's lemma and the asphericity of knots.
	www.mlahanas.de /Greeks/new/Papakyriakopoulos.htm (2724 words)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: 2007-10-20)
		Since the star-free languages form a natural, low complexity subclass of the regular languages, such a result would be in line with the low complexity of the solution of the word problem for these groups.
		(The word problem identifies the set of words which represent the identity element, and is an important logical problem arising from the work of Max Dehn) Recently, Holt (Warwick), Hermiller (Nebraska) and I started to examine Margolis and Meakin's conjecture.
		We found it to be false, indeed we found a presentation of a free group for which the set of geodesics is not star-free.
	www.math.unl.edu /pi/colloquia/abstract-20040924.txt (165 words)

	Colloquium April 17, 1998 -- Susan Hermiller (Site not responding. Last check: 2007-10-20)
		In 1912 Max Dehn first stated the word problem for groups; today this remains one of the fundamental problems in group theory.
		However, the word problem can be solved for many groups, such as free groups, surface groups, and Coxeter groups.
		In this talk I will describe rewriting systems and almost convexity, and explain how they are connected.
	www.math.utep.edu /events/abstracts/colloquia/041798/041798.html (98 words)

	USF Mathematics --- Colloquia, Fall 2000
		Hilbert raised this question again; it was the third problem of his celebrated list of 23 problems in 1900.
		The negative answer was first given by Max Dehn in the same year.
		Even today (after 170 years), the simplest proof of the problem uses Dehn's idea.
	www.math.usf.edu /Research/fall00/colloquia (700 words)

	USC College : Robert Penner
		Professor Penner's early research interests were in the fields of low-dimensional topology and dynamical systems.
		His doctoral thesis solved an old problem of Max Dehn and led to a monograph describing the basic theory of train tracks in surfaces.
		Turning more towards geometry, Professor Penner discovered and developed the decorated Teichmueller theory of punctured surfaces in a series of papers.
	www.usc.edu /assets/college/faculty/old_profiles/253.html (170 words)

	Hamish Short (Site not responding. Last check: 2007-10-20)
		Early in the 20th century, Max Dehn formulated the three classical problems in group theory: the word problem, the conjugacy problem and the isomorphism problem.
		This course is dedicated to the exposition of the geometric methods that have been developed to understand the first and most important of the three problems.
		This session will be dedicated to introduce some of the basic tools of the theory and their related results.
	www.crm.es /wordproblem/hamish_short.htm (155 words)

	Abstracts by Walter Neumann (Site not responding. Last check: 2007-10-20)
		Euclid's attempt to formalize the concept of volume eventually led to Hilbert's 3rd problem, which asked if a regular tetrahedron can be cut into finitely many polyhedral pieces that can be re-assembled to a cube.
		Although the problem was solved by Max Dehn the year Hilbert posed it, problems arising from it remain an active area of research, with ramifications in fields ranging from geometry to number theory.
		We shall start with a survey of this history and follow it to recent connections with other invariants of three-manifolds, including the Jones polynomial invariant of knots.
	www.math.uiuc.edu /Colloquia/03SP/neumann_apr15-03.html (261 words)

	A Springer-Verlag Poster by F. J. Craveiro de Carvalho (Site not responding. Last check: 2007-10-20)
		Among others it includes Max Dehn, F. Klein, E. Zermelo and, last but not least, David Hilbert.
		Max Dehn's life has always interested me. He solved one of Hilbert's famous problems, the third, and things like Dehn twists, Dehn surgery, Dehn's lemma, for instance, crop up frequently in Topology and Group Theory related areas.
		Unfortunately he was one of the few mathematicians unable to obtain a job in accordance with his mathematical talent after being forced to flee from Germany in 1939.
	at.yorku.ca /t/o/p/d/62.htm (292 words)

	Hilbert's 23 Unsolved Problems
		Are there two tetrahedra which cannot be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra.
		Max Dehn showed this to be impossible in 1902, and W.F. Kagon came up with the same result independently in 1903.
		If the ordering and incidence axioms are retained, the congruence axioms weakened, and the equivalent of the Parallel Postulate omitted, finds whose axioms are closest to those of Euclidean Geometry.
	www.andrews.edu /~calkins/math/biograph/199899/tophilpr.htm (498 words)

	AMERICAN MATHEMATICAL MONTHLY -March 2000
		A famous old result of Max Dehn from 1902 says that a rectangle can be tiled by squares if and only if the ratio of its sides is a rational number.
		At first glance, it seems as if the answer should be a direct corollary of Dehn's work, or at least an easy corollary of one of the many subsequent proofs of Dehn's theorem.
		This is the easy part because we just modify a very clever proof of Dehn's result given by Swiss geometer, Hadwiger.
	www.maa.org /pubs/monthly_mar00_toc.html (609 words)

	Jakob Nielsen papers
		This inventory was made by Sigurd Elkjær and Kurt Ramskov, July 1998, and translated into English by Ramskov in October 1998.
		Jakob Nielsen (1890-1959) was born on the peninsula Als (then ruled by the Germans) and he studied in Kiel, where he has good contact with Max Dehn.
		Dehn, Max: 20 reprints, a copy of the manuscript for the lecture Über Kurvensysteme auf zweiseitigen Flächen presented in Breslau 11 February 1922, notes to Dehn's lectures, and 24 letters from Dehn to Nielsen 1916-31.
	www.math.ku.dk /arkivet/jnielsen/jnpapers.htm (3173 words)

	Born, Max -- Britannica Student Encyclopedia (Site not responding. Last check: 2007-10-20)
		South African-born American microbiologist who won the 1951 Nobel Prize for Physiology or Medicine for his development of a vaccine against yellow fever.
		The Austrian theatrical director Max Reinhardt was one of the first in his profession to achieve recognition as a creative artist.
		He worked in Berlin, Germany; Salzburg, Austria; New York City; and Hollywood, Calif.; and he helped to found the Salzburg Festival in 1920.
	www.britannica.com /ebi/article-9310310 (647 words)

	THE BROOKLYN RAIL - ART (Site not responding. Last check: 2007-10-20)
		Although always a painter, I studied math with Max Dehn, dance with Merce Cunningham, music with John Cage and Lou Harrison, photography with Edward Steichen and Hazel Frieda Larsen, and on and on.
		An important area was my study with the great mathematician Max Dehn.
		I told Max I was having some difficulties with assignments.
	www.thebrooklynrail.org /arts/jan05/rockburne.html (3816 words)

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