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Famous Theorems:     more detail
1. Famous geometrical theorems and problems,: With their history, by William W Rupert, 1901
2. The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries by Marilyn vos Savant, 1993-10-15
3. Geometry growing;: Early and later proofs of famous theorems by William Richard Ransom, 1961
4. Famous Problems of Elementary Geometry / From Determinant to Sensor / Introduction to Combinatory Analysis / Fermat's Last Theorem by F., W.F. Sheppard, P.A. Macmahon, & L.J. Mordell Klein, 1962
5. Famous Problems and Other Monographs: Famous Problems of Elementary Geometry/from Determinant to Tensor/Introduction to Combinatory Analysis/Three Lectures on Fermat's Last Theorem by F. Klein, 1962-06

81. Mechanical Proof Of The Pythagorean Theorem
Do we have another proof of the famous theorem? Yes we do. However, the proofis based on axioms not usually included into Euclidean Geometry.
http://www.cut-the-knot.org/pythagoras/mech.shtml
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##### Mechanical Proof of the Pythagorean Theorem
I've been asked several times whether a generalization of the Pythagorean Theorem to Euclidean spaces ( vector spaces with scalar product ) provides a proof of the "regular" Pythagorean Theorem. The answer is Yes it does but it does not belong here . As a matter of fact, proof depends on a set of selected axioms. The theory of vector spaces is derived from the set of axioms altogether different from Euclid's. For example, the Fifth Postulate is easily proven from the analytic form of the equation of a straight line. The whole of Euclidean geometry can be derived in the framework of vector spaces and analytic geometry. And as M.M.Postnikov wrote in his Lectures in Geometry , there might be a good reason to do so: ... It is usual to base an axiomatic construction of elementary geometry, essentially following Euclid in this respect, on the concepts of a point, a straight line, and a plane. Experience shows that this results in a rather complicated axiomatics containing more than a score of axioms which, what is still worse, are not used either entirely or partially anywhere else in mathematics. It turns out that a much more convenient and simpler system of axioms can be obtained if we base it on the concept of a vector . Here simplicity is attained owing to the fact that a "vector" system of axioms uses the theory of real numbers, an extensive portion of which has willy-nilly to be reproduced in "Euclidean"-type axiomatics. Besides, separate fragments of this system of axioms play an exceptionally important role in modern mathematics and must be of necessity studied sooner or later.

82. ASA - March 1996: Re: Godel's Theorem
in the foundations of mathematics, discovered by Chaitin (1975a,b) that is relatedto a famous theorem due to Kurt Godel called Godel s incompleteness theorem.
http://www.asa3.org/archive/asa/199603/0189.html
##### Re: Godel's theorem
Glenn Morton ( GRMorton@gnn.com
Fri, 15 Mar 1996 23:08:27
Ken W. Smith wrote:
In the case of SETI you have not necessarily found a handy instrument to
decode the signal. The only way you can know the signal has a message is
to find such a tool, but you may not find the right decoder and thus may
never know there is a message in there.
Consider the sequence "qi gwb fli zubf hulb sli bub" Is there a message in
there? Are there any mathematics which allow you to recognize the
existence of a message? No. That is what Yockey is saying about signals.
He says,
"The considerations in section 2.4.1 might lead us to believe that, having a definition and a measure of randomness, one could prove a given sequence to be random. In fact it is impossible to do so. It can easily beshown that a given sequence is not random. We need only find a program that compresses the

83. International Journal Of Mathematics And Mathematical Sciences
Received 17 January 2002. We investigate a class of random operator equations,generalize a famous theorem, and obtain some new results.
http://ijmms.hindawi.com/volume-30/S0161171202111252.html
 Home About this Journal Sample Copy Request Author Index ... Contents IJMMS 30:9 (2002) 511-514. DOI: 10.1155/S0161171202111252 SOME THEOREMS OF RANDOM OPERATOR EQUATIONS ZHU CHUANXI and XU ZONGBEN Received 17 January 2002 We investigate a class of random operator equations, generalize a famous theorem, and obtain some new results. 2000 Mathematics Subject Classification: 60H25, 47H10. The following files are available for this article: Pay-per-View: Hindawi Publishing Corporation Comments: webmaster@hindawi.com

84. B Archimedes Of Syracuse /B
plane figures and solids. His most famous theorem gives the weightof a body immersed in a liquid, called Archimedes principal.
http://www.math.tamu.edu/~don.allen/history/archimed/archimed.html
 Next: About this document Archimedes of Syracuse Born: 287 BC in Syracuse, Sicily Died: 212 BC in Syracuse, Sicily Archimedes, the greatest mathematician of antiquity, made his greatest contributions in geometry. His methods anticipated the integral calculus 2,000 years before Newton and Leibniz. He was the son of the astronomer Phidias and was close to King Hieron and his son Gelon, for whom he served for many years. He was an accomplished engineer but loved pure mathematics. Stories from Plutarch, Livy, and others describe machines invented by Archimedes for the defense of Syracuse. These include the catapult, the compound pulley and a burning-mirror. Among Archimedes most famous works is Measurement of the Circle , in which he determined the exact value of to be between the values and . This he obtained by circumscribing and inscribing a circle with regular polygons having 96 sides. However, he required the proof of two fundamental relations about the perimeters and areas of inscribed and circumscribed regular polygons. The computation.

85. Pythagorean Theorem -- From MathWorld
Talbot, R. F. Generalizations of Pythagoras Theorem in n Dimensions. Math. Scientist12, 117121, 1987. Tietze, H. famous Problems of Mathematics Solved
http://mathworld.wolfram.com/PythagoreanTheorem.html
 INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Triangles ... Triangle Properties Pythagorean Theorem For a right triangle with legs a and b and hypotenuse c Many different proofs exist for this most fundamental of all geometric theorems. The theorem can also be generalized from a plane triangle to a trirectangular tetrahedron , in which case it is known as de Gua's theorem . The various proofs of the Pythagorean theorem all seem to require application of some version or consequence of the parallel postulate : proofs by dissection rely on the complementarity of the acute angles of the right triangle, proofs by shearing rely on explicit constructions of parallelograms, proofs by similarity require the existence of non-congruent similar triangles, and so on (S. Brodie). Based on this observation, S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean theorem.

86. Cornell Math - Math 632, Spring 2003
Burnside s famous theorem that every finite group of order p^aq^b issolvable then follows easily, as does a theorem of Frobenius.
 Spring 2003 Instructor: R. Keith Dennis Time: MWF 3:35-4:25 Room: MT 207 Prerequisite: Math 631 or equivalent. This course should be accessible to beginning graduate students and will cover material that will be essential to anyone interested in ring theory, homological algebra, representation theory, or K-theory and should be of value for students of algebraic topology and number theory as well as to others. The main content of this course is to study the "simplest'' rings (those of dimension 0), to characterize them, to relate this in various ways to other concepts in algebra, and to give useful applications. Our approach to the study of semisimple rings is homological rather than ring-theoretic as this leads to results more quickly and gives a cleaner, easier to understand approach. The philosopy of the course will be to "learn by doing'' with a corresponding set of homework assignments. Course Text: Noncommutative Algebra , Graduate Texts in Mathematics, vol. 144, Springer-Verlag, 1993.

87. Discrete Comprehensive 1989
What famous theorem is this? . Define the Ramsey numbers r(n,m) and prove thatthey exist. (b). Let X be a set with n elements, and ka positive integer.
http://bigcheese.math.sc.edu/~sumner/comps/discrete89/DiscreteComp.html
##### Combinatorics Comprehensive Exam August 18, 1989
Directions: You may take up to 4 hours on this exam.
• (a). State and prove (any proof) Hall's Theorem on the existence of an SDR.
(b). Show that every Latin Square.
(d). Show that if n is a prime power, then there exists a set of n - 1 mutually orthogonal Latin squares.
• For fixed n, let F and that for all A,B F. )
Determine f(k) with proof. What famous theorem is this?
• . Define the Ramsey numbers r(n,m) and prove that they exist.
(b). Let X be a set with n elements, and k a positive integer. Suppose that we arbitrarily k-color all of the non-empty subsets of X. Show that if n is large enough, then there must exist two disjoint, non-empty subsets A, B of X such that all of A, B, and have the same color.
• (a). State the Kruskal-Katona Theorem.
(b). Construct a family F of 4-subsets, (i.e. A F (c). Prove that the k-cascade representation of an integer m, is unique for all m.
• n n Then the Binomial Inversion theorem allows you to express in terms of the . Give this expression and prove it is correct. State any other inversion theorems that you use in the process.
• For the array below, how many different ways are there to color the squares using the colors red, blue and green? Two colorings are considered distinct if neither can be obtained from the other by a rotation of the square or any flip about either diagonal or the center row or column (Hint: there are eight such symmetries).
• 88. Math: Geometry
A discussion of Morley s famous theorem and the research of which it was a tinypart. 5 proofs are given including J. Conway s, D. Newman s, and A
http://www.puredirectory.com/Science/Math/Geometry/
##### Math: Geometry
A Gallery of Interactive On-Line Geometry
At the Geometry Center.
ArXiv Front: MG Metric Geometry
Metric geometry section of the mathematics e-print arXiv.
ArXiv Front: SG Symplectic Geometry
Symplectic geometry section of the mathematics e-print arXiv.
Bette Veteto's Homepage
History of Mathematics, very strong on geometry.
Bob's Pages
A collection of interactive geometry applets. Topics include polyhedra, Poncelet's porism, Soddy's hexlet, Mandelbrot set, Steiner porism, Pappus's chain, Repulsion polyhedra and stereo pictures.
Center of Points in 2- or Higher-dimensional Space
It provides important formulations for finding a center of points in 2 or higher dimensional space. It can be used in drawing delauney triagulation and voronoi diagram in two or higher dimensions....

89. Flt.html
For example, I find myself telling my students how I think Euclid couldhave discovered his famous theorem on perfect numbers. Also
http://www.mapleapps.com/categories/mathematics/numbertheory/html/flt.html
 flt.mws Section 9. Bibliography and web references. # ICTMT5talk.mws Fermat's little theorem, ICTMT5, Klagenfurt University, Austria. August 2001. Fermat's little theorem For Mark Daly - former colleague and friend - who helped me (and still does) when I was really stuck John Cosgrave, Mathematics Department, St. Patrick's College, Drumcondra, Dublin 9, IRELAND. email: John.Cosgrave@spd.ie web site: http://www.spd.dcu.ie/johnbcos A thing of beauty is a joy for ever. Its loveliness increases; it will never Pass into nothingness. [John Keats (1795-1821), Endymion (1818)] Fermat's 'little' theorem is one of the jewels of Number Theory, and to mark the anniversary of Fermat's birth ( August 2001), I offer this talk. My talk is not intended as an introduction to Number Theory, nor indeed even as an introduction to Maple, although in both cases it could serve as such. Indeed in the time available (some 30 minutes) it will be possible for me to cover only a very small selection of the topics listed below. Anyone who is interested may access the this Maple .mws file at my web site, and also a html text conversion (which may be read by anyone who has a web browser, and does not require having Maple); they are in the Public and Other Lectures section of the Maple section of my site. Because of the widely reported work of Andrew Wiles, many non-mathematicians have heard of Fermat's

90. MTH-3E12 : Analytic Number Theory
We shall aim to prove a famous theorem of Kuratowski which gives exact conditionsfor the planarity of a graph. Of further interest will be graph colourings.
http://www.mth.uea.ac.uk/maths/syllabuses/0405/4E2304.html
 MTH-4E23 : Graph Theory 1. Introduction: This fourth year course is a thorough introduction on modern Graph Theory. This subject plays an important role in many branches of mathematics and the sciences generally. Graph Theory therefore has applications almost everywhere. There are not many prerequisites and Algebra I (MTH-2A23) is sufficient; students with other backgrounds who would like to attend this course should contact the lecturer. 2. Timetable Hours, Credits, Assessments: The course is a 20 UCU unit of 33 lectures and additional hours for group discussions on demand. Assessment is by course work (20%) and examination (80%). 3. Overview: Graphs are among the most basic structures in mathematics. They consist of a set of points, the vertices of the graph, and a set of edges which link certain pairs of vertices. By their very simplicity it is therefore not surprising that graphs are important in many part of mathematics, computing and the sciences. This course is designed as an introduction to the theory and applications of graphs. Thus we first develop the basic notions of connectivity and matchings. Graphs appear also in topology and these aspects will be investigated in a section on planarity. This is the question if a graph can be embedded in the plane in such a fashion that edges do not intersect. We shall aim to prove a famous theorem of Kuratowski which gives exact conditions for the planarity of a graph. Of further interest will be graph colourings. These are assignments of colours to the edges of the graphs such that different edges of the same colour never meet at a vertex. There are also vertex colourings where vertices of the same colour may not be joined by an edge. Such questions lead to many beautiful results and interesting applications. One of the best known theorems in graph theory is the Four Colour Theorem. This result is not within our reach. However we shall prove a weakening of it, and this is Five Colour Theorem.

91. The Geometry Of The Sphere 5
In the next section we use Girard s theorem to give a proof of Euler s famoustheorem relating the numbers of vertices, sides, and edges of polyhedron.
http://math.rice.edu/~pcmi/sphere/gos5.html
##### Consequences of Girard's Theorem
Exercise: Distortion in maps. Everyone knows that a map from even a small portion of the sphere to the plane must involve some distortion. However Girard's Theorem makes that statement more precise. Let's call a map ideal if does two things.
• It maps great circles to straight lines. It preserves angles.
Does an ideal map exist? Are there maps that have one of the propeties? It does not seem to be too much too ask for a map to be ideal. Think about these two questions. You can find the answers here. Exercise: Similarity. Suppose we have a triangle on the sphere with angles A B , and C . Can we find a larger triangle with the same angles? In other words, do similar triangles exist on the sphere? The answer is no! By the area formula, any triangle with these angles must have the same area, and therefore cannot be larger. Exercise: Congruence theorems. In the plane there are a number of theorems aobut the congruence of triangles. They are usualy referred to by their acronyms, i.e., SSS, SAS, AAS, and ASA. Can you modify the proofs of the planar theorems to prove these theorems on the sphere? It might be necessary to modify the statements of the theorems to take care of some special cases. (You should be warned that some of these are difficult to prove.)

92. Geometry
Previews by Thumbshots Morley s Miracle A discussion of Morley sfamous theorem and the research of which it was a tiny part.
http://www.oobdoo.com/directory/Science/Math/Geometry/
 World Wide Search Engine and Portal to the Best Sites on the Internet Over 15million sites and over 550,000 categories Top Science Math : Geometry (488) See Also: Science: Math: Algebra: Geometric Algebra Science: Math: Recreations: Specific Numbers Problem of the Minimum Rotation Surface - Java applet demonstrating the catenary. Geometry in Action - Includes collections from various areas in which ideas from discrete and computational geometry meet real world applications. The Geometry Junkyard - Usenet clippings, web pointers, lecture notes, research excerpts, papers, abstracts, programs, problems, and other stuff related to discrete and computational geometry. Erich's Packing Center - Graphics and links for various packing, tiling and covering problems. Tim Lister's Hyperbolic Geometry - Cabri constructions for the demonstration of the basic concepts of hyperbolic geometry in the Poincare disc model. Bette Veteto's Homepage - History of Mathematics, very strong on geometry.

93. From Euclid To Ptolemy In English Crop Circles
the note A(2). Here the mathem atical level required application of Ptolemy s famoustheorem of chords to confirm the A(2) ratio of exactly 10/3. The chords
http://www.aas.org/publications/baas/v29n5/aas191/abs/S035006.html
Previous abstract Next abstract Session 35 - HAD III: From Hamlet to Crop Circles.
Oral session, Wednesday, January 07
Monroe,
##### [35.06] From Euclid to Ptolemy in English Crop Circles
G. S. Hawkins (Boston University Research) The late Lord Soli Zuckerman, science advisor to several British governments, encouraged the author, an astronomer, to test the theory that all crop circles were made by hoaxers. Within the hundreds of formations in Southern England he saw a thread of surprising historical content at the intellectual level of College Dons. One diagram in celestial mechanics involved triple conjunctions of Mercury, Venus and Mars every 67 2/3 years. Ptolemy's fourth musical scale, tense diatonic, occurred in the circles during the period 1978-88. Starting on E, Ptolemaic ratios make our perfect diatonic scale of white notes on the keyboard of the piano or church organ. For separated circles the ratio was given by diameters, and for concentric circles it was diameters squared. A series of rotationally symmetric figures began in 1988 which combined Ptolemy's ratios with Euclid's theorems. In his last plane theorem, Euclid (Elements 13,12) proved that the square on the side of an equilateral triangle is 3 times the square on the circumcircle radius diatonic note G. From the 1988 figure one can prove the square on the side is 16/3 times the square on the semi-altitude, giving note F(3). Later rotational figures over the next 5 years led to diatonic ratios for the hexagon, square and triangle. They gave with the exactness of Euclidean theorems the notes F, C(2) and E(2), and they are the only regular polygons to do so. Although these 4 crop theorems derive from Euclid, they were previously unknown as a set in the literature, nor had the Ptolemaic connection been published.

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