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Topic: H S M Coxeter

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	Coxeter
		He continued to study for a doctorate at Cambridge under H F Baker, and this was awarded in 1931.
		Coxeter polytopes are the fundamental domains of discrete reflection groups, now called Coxeter groups, and they give rise to tesselations.
		H S M Coxeter was elected to the Royal Society of London in 1950.
	members.tripod.com /sfabel/mathematik/database/Coxeter.html (763 words)

	science.ca Profile : Harold Scott Macdonald (H. S. M.) Coxeter (Site not responding. Last check: 2007-08-04)
		Coxeter was known as H. Coxeter, though friends and relatives called him Donald.
		One of Coxeter’s major contributions to geometry was in the area of dimensional analogy, the process of stretching geometrical shapes into higher dimensions.
		Coxeter loved his work and once said of his career, “I am extremely fortunate for being paid for what I would have done anyway.” His advice to young people thinking about a career in mathematics: “If you are keen on mathematics, you have to love it, dream about it all the time.”
	www.science.ca /scientists/scientistprofile.php?pID=5 (1281 words)

	Mathematics meets Art for Prof. Coxeter
		Coxeter has been trying to find out how accurately Escher, who knew almost no mathematics, was able to repeat the same arc angles of intersection.
		Coxeter, whose voice is still brushed with the sounds of his native England, encounters Escher on something akin to home turf: His colleagues describe Prof.
		Coxeter was one of those prodigies who seem to tumble out of the womb number-crazed.
	www.math.toronto.edu /coxeter/art-math.html (1065 words)

	Coxeter, Harold Scott MacDonald (1907-2003)
		In 1926, at the age of 19, Coxeter discovered a new regular polyhedron, having six hexagonal faces at each vertex.
		Coxeter was a close friend of the artist M. Escher, whom he met in 1954, and also of Buckminster Fuller, who used Coxeter's ideas in his architecture.
		Indeed Coxeter's work was motivated by a strong artistic temperament and a sense of what is beautiful.
	www.daviddarling.info /encyclopedia/C/Coxeter.html (348 words)

	PIMS Changing the Culture 2000: Public Lecture
		The present article analyzes the structure, using the elements of trigonometry and the arithmetic of the biquadratic field containing the square roots of 2 and 3; subjects of which he steadfastly claimed to be entirely ignorant.
		He was first given the name MacDonald Scott Coxeter, but a godparent suggested that his father's name should be added, so Harold was added at the front.
		Coxeter also has a love or art and music and his ideas have influenced the architecture of Buckminster Fuller and the art of M.C. Escher.
	www.pims.math.ca /education/2000/CtC/coxeter (640 words)

	H.S.M. Coxeter (Site not responding. Last check: 2007-08-04)
		H.S.M. Coxeter was born and educated in England, but his professional connections with North America began early.
		Shortly after finishing his doctoral studies at Cambridge University, and while he was a research fellow there, he spent two years as a research visitor at Princeton University.
		Coxeter is also available here at the Great Canadian Scientists Web Site.
	www.math.toronto.edu /~coxeter (422 words)

	morley thm (Site not responding. Last check: 2007-08-04)
		As a consequence of this, it was finally proven that a general angle could not be trisected with straight edge and compass, thus ending the search for one of the classic problems of antiquity.
		H S M Coxeter has suggested that from this time on people felt uneasy about the mention of trisecting an angle.
		This, he thinks, probably contributed to the reason that Morley's Theorem was not discovered until near the dawning of the 20th century.
	www.pballew.net /morley.html (306 words)

	H.S.M. Coxeter (Site not responding. Last check: 2007-08-04)
		H.S.M. Coxeter researched and wrote at length on polyhedra and polytopes (higher-dimensional generalizations of polyhedra).
		Coxeter (left) and me (right) examining a few of the constructions at the 1984 conference "Shaping Space" at Smith College, Northhampton, Massachusetts.
		Professor Coxeter's bio is online at the University of Toronto, and there are related articles about him in the web pages of great Canadian scientists and The Globe and Mail.
	www.georgehart.com /coxeter.html (182 words)

	Transfer Functions Between Group Categories
		Given a subgroup H of the group G, construct the permutation representation of G given by the action of G on the (right) coset space cos(G, H).
		The permutation representation is obtained by using the Todd-Coxeter procedure to construct the coset table for H in G. Note that G may be an infinite group: it is only necessary that the index of H in G be finite.
		Given a subgroup H of the group G, construct the image of G given by its action on the (right) coset space of H in G, returning it as a permutation group.
	www.umich.edu /~gpcc/scs/magma/text246.htm (866 words)

	Xah: Introduction to Real Projective Plane
		Theorem: The correspondence between the points of a range and their harmonic conjugates with respect to two fixed points M and N is an opposite correspondence with invariant points M and N. Xah's Note: Given H(MN,XX'), there exist a quadrangle P,Q,R,S such and such by the definition of Harmonic sets.
		When l is horizontal, it is said that R and S have move to the point at infinity.
		Now when R and S move to the right, observe that S moves increasingly faster then R. The distance between R and S is approaching infinity.
	xahlee.org /projective_geometry/projective_geometry.html (5013 words)

	Coxeter Groups I
		The classification of the finite Coxeter groups in three dimensions is intimately related to the classification of the regular solids.
		This is not the case if the Coxeter group is the Weyl group of a Kac-Moody Lie algebra, since in that case the roots themselves are part of the structure of the Lie algebra.
		Prove that the regular Euclidean polyhedra are classified by isomorphism classes of (a) a Coxeter diagram associated to a finite Coxeter group together with (b) a single node of the diagram on its boundary.
	www.math.ubc.ca /~cass/coxeter/crm1.html (3666 words)

	Totally Tessellated: Escher Biography, page 3/3
		Although Escher did not have a strong background in mathematics, his careful explorations of tilings of the plane were extensive and representative of mathematical research.
		During his lifetime M. Escher designed over a hundred tessellated patterns, many of which were transformed into the famous works of art that we recognize.
		During this gathering, x-ray crystallographers used some of M. Escher's works of art to explain the concepts of symmetry and transformations, very important in the field of x-ray crystallography.
	library.thinkquest.org /16661/escher/biography.3.html (575 words)

	Pythagorean Triples
		This calculator generates primitive triangles from m and n values (and the multiples of each if required) so will be slower for larger values of the hypotenuse.
		The only condition we can have is that m – n = 1 or m = n + 1 - there are no other triangles with hypotenuse one more than a leg except those generated by consecutive m n values in the m-n formula.
		A tiny block of clay, about the size of a postcard (5 inches x 3.5 inches or 12cm x 9cm) with 15 rows of 4 columns of "numbers" is dated to about 1800 BC and so is probably the world's oldest surviving mathematical artefact.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Pythag/pythag.html (6979 words)

	References for Coxeter (Site not responding. Last check: 2007-08-04)
		H S M Coxeter: published works, The geometric vein (New York-Berlin, 1981), 5-13.
		G F D Duff, H. Coxeter Celebrates 90th Birthday, Notices of the American Mathematical Society 44 (3) (1997), 340-341.
		I Hargittai, Lifelong symmetry: a conversation with H S M Coxeter, The Mathematical Intelligencer 18 (4) (1996), 35-41.
	www-groups.dcs.st-and.ac.uk /history/References/Coxeter.html (103 words)

	The Golden Geometry of Solids or Phi in 3 dimensions
		H S M Coxeter, Introduction to Geometry, 1961, John Wiley, is a classic!
		The classic and encyclopaedic book on tilings is Grunbaum and Shepard's Tilings and Patterns W H Freeman and Co, 1989.
		Golomb is the inventor of polyominoes and this is the revised and expanded second edition of the original of 1965 that sparked off the polyomino puzzle craze.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/phi3DGeom.html (4544 words)

	Pythagoras, Phoenician/Greek Mathematician
		R S Brumbaugh, The philosophers of Greece (Albany, N.Y., 1981).
		H Wussing, Pythagoras, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).
		H S M Coxeter, Polytopes, kaleidoscopes, Pythagoras and the future, C.
	phoenicia.org /pythagoras.html (3184 words)

	Quasicrystal Research
		Consider a system S in physics with a (Hamiltonian, Lagrangian) dynamics invariant under a group G of symmetry operations.
		As shown by H S M Coxeter and V Kac, the point group of the hypercubic space-time lattice is the Coxeter group with the diagram
		Fig: Dynkin diagram for the hyperbolic Coxeter group which generates all integral Lorentz transformations.In [15] we transform this group to the Dirac spinor SL(2,C) representation.
	homepages.uni-tuebingen.de /peter.kramer (2044 words)

	Alter blog » Import: à lire
		H S M Coxeter, Non-Euclidean Geometry, source: Baez.
		M Gockeler/T Schucker, Differential Geometry, Gauge Theories and Gravity, source: Baez.
		M S Longair/Malcolm S Longair, Theoretical Concepts in Physics, source: Baez.
	www.5etdemi.com /alter/2005/12/import-a-lire (672 words)

	Harold Scott MacDonald Coxeter - Wikipedia, the free encyclopedia
		Harold Scott MacDonald "Donald" Coxeter CC (February 9, 1907 – March 31, 2003) is regarded as one of the great geometers of the 20th century.
		He was born in London but spent most of his life in Canada.
		In 1997 he received Sylvester Medal from the Royal Society and was made a Companion of the Order of Canada.
	en.wikipedia.org /wiki/Coxeter (395 words)

	Indagationes Mathematicae. New Series, 1999; 10 (4) (Site not responding. Last check: 2007-08-04)
		(Communicated by Prof H S M Coxeter at the meeting of April 26,1999) / Chouikha, A Raouf / Coxeter, H S M
		(Communicated by Prof M S Keane at the meeting of February 22, 1998) / Karimov, Umed H / Repovs, Dugan / Keane, M S
		(Communicated by Prof M S Keane at the meeting of June 21, 1999) / King, Alastair / Schofield, Aidan / Keane, M S
	www.ucm.es /BUCM/compludoc/W/10001/00193577_1.htm (358 words)

	atlas: atlas::weyl::Transducer Class Reference
		In the full group, this has necessarily cardinality 2m, with m = m(s,t) the coefficient in the Coxeter matrix, and xs.t goes down iff xs is the unique elt.
		This is the number of positive roots for the Levi subgroup L_r, minus the number of positive roots for L_{r-1}.
		When x' is not equal to s, this is an equality of minimal coset representatives.
	www-math.mit.edu /~dav/html/classatlas_1_1weyl_1_1_transducer.html (692 words)

	The Design of 2-Colour Wallpaper Patterns Using Methods Based on Chaotic Dynamics and Symmetry
		Often we drop the superscript s if the sequence s is implicit from the context or n = 1.
		M J Field and M Golubitsky, Symmetry in Chaos, (Oxford University Press, New York and London, 1992).
		M J Field, I Melbourne and M Nicol.
	arpam.free.fr /mfield.html (5289 words)

	Interactive Coset Enumeration
		This function creates a coset enumeration process for enumerating the cosets of the subgroup H of the finitely presented group G. Note that no actual coset enumeration is started for the created coset enumeration process.
		This means that a coset enumeration process P for the cosets of H in G is transformed into a coset enumeration process for the cosets of < H, w > in G, where < H, w > denotes the subgroup of G generated by H and w.
		Given a finite index subgroup H of a group G, the action of G on the set of right cosets of H in G by right multiplication defines a permutation representation rho : G -> S of G onto a suitable subgroup S of the symmetric group on [G:H] letters.
	www.umich.edu /~gpcc/scs/magma/text471.htm (5151 words)

	Departmental Logo for Mathematics at York (Site not responding. Last check: 2007-08-04)
		"A theorem novel for Euclidean geometry was discovered in 1899 by Frank Morley, professor of mathematics at Johns Hopkins University, and proofs were subsequently published by many men [For a proof and references to published proofs see H S M Coxeter, An Introduction to Geometry, John Wiley and Sons, 1961 (S 3 COX), pp.23-25].
		The theorem states that if the angle trisectors are drawn at each vertex of a triangle, adjacent trisectors meet at the vertices of an equilateral triangle (see figure).
		M Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press 1972 (S 0.9 KLI).
	www.york.ac.uk /depts/maths/images/logo.htm (193 words)

	Intuition: A Link Between Psi and Spirituality
		I think it is of significance that Charlson, in retrospect, has come to view his intuitions as having a connection with higher spiritual sources.
		This is a viewpoint shared by other inventors, such as Chester Carlson (inventor of the xerox process) and Arthur M. Young (inventor of the Bell Helicopter) -- who have also made significant financial contributions to psi research and related fields.
		H.S. Coxeter, "Cases of Hyperdimensional Awareness," in Charles Muses and Arthur M. Young (Eds.), Consciousness and Reality.
	www.intuition.org /revision.htm (2365 words)

	H. S. Vandiver Papers (Site not responding. Last check: 2007-08-04)
		The H. Vandiver Papers at the Archives of American Mathematics are open for research after the completion of an extensive preservation and access project.
		From the H.S. Vandiver Papers at the Archives of American Mathematics, Center for American History, The University of Texas at Austin.
		By far the richest section of the H. Vandiver Papers is his correspondence, containing over 2500 letters written from 1910 to 1965.
	www.maa.org /features/071406vandiver.html (401 words)

	References > M
		S Esposito, E Majorana Jr, A van der Merwe, and E Recami, eds Notes on Theoretical Physics Kluwer 2004.
		w H Wagner "Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models" PRL 17 (1966) 1133-1136 [>pt].
		w K S Thorne "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity" AJP 56 (1988) 395-412 [>gr, wh].
	www.phy.olemiss.edu /~luca/Refs/m.html (10499 words)

	Amazon.ca: Introduction to Geometry, 2nd Edition: Books: H. S. M. Coxeter (Site not responding. Last check: 2007-08-04)
		Coxeter (Author) "About 300 B.C., Euclid of Alexandria wrote a treatise in thirteen books called the Elements..." (more)
		It is now a little dated but only in the topics that it does not cover.
		Like all of Coxeter works each topic is clear and to the point.
	www.amazon.ca /Introduction-Geometry-2nd-H-Coxeter/dp/0471504580 (675 words)

	Cut The Knot!
		First published in 1899, the theorem was brought to broad attention in 1969 through the popular Mathematical Snapshots by H.
		where gcd(m,n) is the greatest common divisor of m and n.
		He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem.
	www.maa.org /editorial/knot/Pick.html (790 words)

	Doris Schattschneider's Bibliography of Tesselations
		Haak, S. "Transformation Geometry and the Artwork of M. Escher." Mathematics Teacher 69 (1976): 647-652.
		Hargittai, I. and M. Hargittai, Symmetry, A Unifying Concept.
		Stevens, Peter S., Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions.
	www.geom.uiuc.edu /software/tilings/TilingBibliography.html (707 words)

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