41. Diamond Theory: Symmetry In Binary Spaces Plato tells how Socrates helped Meno's slave boy remember the geometry of a diamond. Twentyfour centuries later, this geometry has a new theorem. http://m759.freeservers.com/

Related sites: The 16 Puzzle Bibliography On the author Diamond Theory by Steven H. Cullinane Plato's Diamond Motto of Plato's Academy Abstract: Symmetry in Finite Geometry Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous. Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of non continuous (and a symmetric) transformations. (These might be called noncontinuous groups, as opposed to so-called discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4x4 array. ( Details By embedding the 4x4 array in a 4x6 array, then embedding A in a supergroup that acts in a natural way on the larger array, one can, as R. T. Curtis discovered, construct the Mathieu group M which is, according to J. H. Conway, the "most remarkable of all finite groups."

42. Interactive Application: Projective Conics These pages were developed by Mathew Frank, a student in the 1995 Summer Institute held at the geometry Center. conics and projective geometry, and he developed some programs to illustrate a classical result called Pascal's theorem. http://www.geom.umn.edu/apps/conics

Up: Gallery of Interactive Geometry Projective Conics: These pages were developed by Mathew Frank , a student in the 1995 Summer Institute held at the Geometry Center. Mathew was interested in conics and projective geometry, and he developed some programs to illustrate a classical result called Pascal's theorem. As part of his project, Mathew wrote the following pages describing the theorem and some related material, as well as the interactive application available from the button below: Conics and Hexagons Pascal's Theorem Brianchon's Theorem Some Stronger Theorems Some Special Cases ... An Explanation of What You Will See Note: There are no subscripts or superscripts in these pages. Instead, superscripts are denoted by "^", so " x squared" is written " x ^2". Similarly, subscripts are indicated by "_", so " x sub 1" is written " x _1". The notation AB will refer to the line connecting points A and B , and AB.CD will refer to the point of intersection of lines AB and CD Up: Gallery of Interactive Geometry The Geometry Center Home Page Comments to: webmaster@geom.umn.edu

43. The Pythagorean TheoremThe Pythagorean Theorem Home Page If You Do Not Have The Dole was learning geometry), came up with one of the most famous theorems ever, the Pythagorean theorem. It says in http://www.geom.umn.edu/~demo5337/Group3

The Pythagorean Theorem Home Page If you do not have the Geometer's Sketchpad software on your computer, Click here to download a demo copy of this software to use the Sketchpad activities contained in this web site. Go to the Main Menu Do you have questions to send one of the authors of this web site? Send a message to one of the authors Go to the Geometry Center Home Page

44. Citations: Realization Of A Geometry Theorem -proving Machine - Gelernter (Resea Similar pages More results from citeseer.nj.nec.com TETRA/geometry/theorem/4COLOR/PROOFSELECTED IDEAS OF BUCKMINSTER FULLER. TETRAHEDRA. geometry. 4-COLOR theoremPROOF. Polygonally all spherical surface systems are maximally http://citeseer.nj.nec.com/context/39961/0

34 citations found. Retrieving documents... H. Gelernter. Realization of a geometry-theorem proving machine . In E.A. Feigenbaum and J. Feldman, editors, Computers and Thought, pages 134152. McGraw-Hill Book Company, New York, 1963. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: First 50 documents Circumscription and Generic Mathematical Objects - Bertossi (1992) (1 citation) (Correct) ....would be considered generic. Some computational systems for mechanical theorem proving in geometry use auxiliary diagrams to guide the proofs of theorems. These diagrams are constructed with genericity in mind. As P. C Gilmore puts it in his logical reconstruction of Gelernter s Geometry Machine : One more qualification is generally added in order not to admit as a diagram a set of points with special relationships that are not part of the premises . 5, page 179] Our concern in this paper is with formalizing the concept of a generic mathematical object. Specifically , we explore ....

45. 1. Introduction To Geometry - Theorem 1 1. Introduction to geometry theorem 1. Introduces plane, or two-dimensional,geometry; Pi, lines, line-segments, angles, parallelograms http://www.chiptaylor.com/ttlmnp0962-.html

1. Introduction to Geometry - Theorem 1 Introduces plane, or two-dimensional, Geometry; Pi, lines, line-segments, angles, parallelograms, and proofs; also Theorem 1 looks at vertically opposite angles. 98/03DR JSCA 15 min. Home

46. Geometry Of The Gauss-Markov Theorem geometry of the GaussMarkov theorem. We have already described asa plane. In the following sections, we explain the significance http://emlab.berkeley.edu/GMTheorem/node9.html

Next: Spherical Distributions and Up: The Geometry of the Previous: Projections Geometry of the Gauss-Markov Theorem We have already described as a plane. In the following sections, we explain the significance of the sphere and the cylinder. The sphere represents the variance-covariance matrix of y . The cylinder helps to illustrate a nonorthogonal projection of the variance sphere onto Spherical Distributions and Variance Spheres Projections of the Variance Sphere and Variance Ellipsoids ruud@econ.Berkeley.EDU

47. Interactive Mathematics Miscellany And Puzzles, Geometry Interactive Mathematics Miscellany and Puzzles. geometry Page. Angle TrisectionJava; Angle Trisectors on Circumcircle Java; An Old Japanese theorem; http://www.cut-the-knot.com/geometry.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Interactive Mathematics Miscellany and Puzzles Geometry Page 3 Utilities Puzzle 4 Travelers problem 9-point Circle as a Locus of Concurrency [Java] A Case of Similarity [Java] A Geometric Limit A problem with equilateral triangles [Java] About a Line and a Triangle [Java] Altitudes [Java] Altitudes and the Euler Line [Java] Angle Bisectors [Java] Angle Bisectors in a Quadrilateral [Java] Angle Preservation Property [Java] Angle Trisection [Java] Angle Trisectors on Circumcircle [Java] An Old Japanese Theorem Apollonian Gasket [Java] Apollonius Problem [Java] Archimedes' Method Area of Parallelogram [Java] Arithmetic-Geometric Mean Inequality Assimilation Illusion [Java] Asymmetric Propeller [Java] Barbier, The Theorem of [Java] Barycentric coordinates Barycentric coordinates and Geometric Probability Bender: A Visual Illusion [Java] Bisecting arcs Bisecting a shape Bounded Distance Brahmagupta's Theorem [Java] Brianchon's theorem [Java] Bulging lines illusion [Java] Butterfly Theorem Two Butterflies Theorem [Java] Cantor Set and Function Cantor sets [Java] C - C Carnot's Theorem Carnot's Theorem (Generalization of Wallace's theorem) [Java] Centroids in a Polygon [Java] Ceva's Theorem Ceva's Theorem and Fibonacci Bamboozlement [Java] Chain of Inscribed Circles [Java] Chaos, Emergence of

48. Pappus' Theorem The Duality Principle is a handy feature of Projective geometry you prove one theoremand get another one for free. The principle is quite simple to prove. http://www.cut-the-knot.com/pythagoras/Pappus.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Pappus' Theorem The word Geometry is of the Greek and Latin origin. In Latin, geo- ge- means earth, while metron is measure. Originally, the subject of Geometry was earth measurement. With time, however, both the subject and the method of geometry have changed. From the time of Euclid's Elements rd century B.C.), Geometry was considered as the epitome of the axiomatic method which itself underwent a fundamental revolution in the 19 th century. Revolutionary in many other aspects, the 19th century also witnessed metamorphosis of a single science - Geometry - into several related disciplines The subject of Projective Geometry , for one, is the incidence of geometric objects : points, lines, planes. Incidence (a point on aline, a line through a point) is preserved by projective transformations, but measurements are not. Thus in Projective Geometry, the notion of measurement is completely avoided, which makes the term - Projective Geometry - an oxymoron. In Projective Geometry

49. StudyWorks! Online : Interactive Geometry get a handson feel for some of the fundamental principles of geometry. Link toPythagorean theorem (1) Pythagorean theorem — Proof 1 This applet shows the http://www.studyworksonline.com/cda/explorations/main/0,,NAV2-21,00.html

Algebra Explorations Astronomy Biology Chemistry ... Weather Center Interactive Geometry The activities in this section will help you get a "hands-on" feel for some of the fundamental principles of geometry. Try them all to help understand theorems and proofs. Note: These activities are all based on Java applets which may take a few moments to download if you are connecting by modem. Please be patient. Alternate Angles When a transversal intersects two parallel lines, the alternate interior and exterior angles are congruent. Angle Trisector See how to trisect an angle in this activity. Angles of a Triangle Here's a visual guide to finding the sum of the angles of a triangle. Changing Border Line Keeping areas constant when a common boundary is changed. Congruent Triangles (1) Prove that two triangles are congruent. Congruent Triangles (2) Prove that two triangles are congruent. Congruent Triangles (3) Prove that two triangles are congruent. Congruent Triangles (4) Prove that two triangles are congruent. Conservation of Area Which has a larger area, a rectangle or a parallelogram? Corresponding Angles When a transversal crosses two parallel lines, the corresponding angles are congruent.

50. NRICH | Secondary Topics | Geometry-Euclidean | Pythagoras Theorem Coordinate + geometryCoordinate - geometry-Euclidean + 3D + Angle Properties + Polygons+ Proof + Properties of Shapes - Pythagoras theorem + 3D + Application http://www.nrich.maths.org.uk/topic_tree/Geometry-Euclidean/Pythagoras_Theorem/

NRICH Prime NRICH Club ... Get Printable Page March 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level Algebra Analysis Calculus Combinatorics Complex Numbers Geometry Geometry-Cartesian Geometry-Coordinate Geometry-Coordinate Geometry-Euclidean Angle Properties Circles Polygons Proof Properties of Shapes Pythagoras Theorem Application Enlargements Other shapes Pythagorean triples Ratio and Proportion Similarity Squares Triangles polygons proof transformations Incircles ( July 2001 ) Incircles ( January 2002 ) Graph Theory Groups Investigation Investigations Logic Measures Mechanics Number Pre-calculus Probability Programs Properties of Shapes Pythagoras Sequences Statistics Symmetry Trigonometry Unclassified algebra number

51. NRICH | Secondary Topics | Measures | Euclidean Geometry | Pythagoras Theorem geometryCoordinate + geometry-Euclidean + Graph Theory + Groups + Investigation+ Logic - Measures + Area - Euclidean geometry - Pythagoras theorem Are you http://www.nrich.maths.org.uk/topic_tree/Measures/Euclidean_Geometry/Pythagoras_

NRICH Prime NRICH Club ... Get Printable Page March 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level Algebra Analysis Calculus Combinatorics Complex Numbers Geometry Geometry-Cartesian Geometry-Coordinate Geometry-Coordinate Geometry-Euclidean Graph Theory Groups Investigation Investigations Logic Measures Area Euclidean Geometry Pythagoras Theorem Are you kidding?

52. Notes On Differential Geometry By B. Csikós Notes by Balázs Csikós. Chapters in PostScript.Category Science Math Publications Online Texts...... translation, symmetric connections, Riemannian manifolds, compatibility with a Riemannianmetric, the fundamental theorem of Riemannian geometry, LeviCivita http://www.cs.elte.hu/geometry/csikos/dif/dif.html

Differential Geometry Budapest Semesters in Mathematics Lecture Notes by Balázs Csikós FAQ: How to read these files? Answer: The files below are postscript files compressed with gzip . First decompress them by gunzip , then you can print them on any postscript printer, or you can use ghostview to view them and print selected (or all) pages on any printer. CONTENTS Unit 1. Basic Structures on R n , Length of Curves. Addition of vectors and multiplication by scalars, vector spaces over R, linear combinations, linear independence, basis, dimension, linear and affine linear subspaces, tangent space at a point, tangent bundle; dot product, length of vectors, the standard metric on R n ; balls, open subsets, the standard topology on R n , continuous maps and homeomorphisms; simple arcs and parameterized continuous curves, reparameterization, length of curves, integral formula for differentiable curves, parameterization by arc length. Unit 2. Curvatures of a Curve Convergence of k-planes, the osculating k-plane, curves of general type in R n , the osculating flag, vector fields, moving frames and Frenet frames along a curve, orientation of a vector space, the standard orientation of R n , the distinguished Frenet frame, Gram-Schmidt orthogonalization process, Frenet formulas, curvatures, invariance theorems, curves with prescribed curvatures.

53. Golenor's Geometry Gala Theorem Page GOLENOR'S GALA theorem PAGE The measure of an exterior angle of a triangle is equalto the sum of the two nonsupplementary angles. Isosceles Triangle theorem http://teacherlink.org/content/math/interactive/geometrygala/theorem_page.html

RETURN TO HOME RETURN TO TRIANGLES RETURN TO CIRCLES RETURN TO QUADRILATERALS ... RETURN TO AUTHOR PAGE G O L E N O R' S G A L A T H E O R E M P A G E TRIANGLE THEOREMS CIRCLE THEOREMS QUADRILATERAL THEOREMS LINES... THEOREMS TRIANGLE THEOREMS Exterior Angle of a Triangle Return to Movie The measure of an exterior angle of a triangle is equal to the sum of the two non-supplementary angles. Isosceles Triangle Theorem Return to Movie If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Triangle Inequality Theorem Return to Movie The sum of any two sides of a triangle must be strictly larger than the third side. Hinge Theorem Return to Movie If two sides of triangle A are congruent to two sides of triangle B and the angle between the sides of A is greater than the angle of B, then the third side of A is larger than the third side of B. Acute Angles of a Right Triangle Return to Movie The acute angles of a right triangle are complementary.

54. 51M04: Elementary Euclidean Geometry (2-dimensional) (long use of analytic geometry and then Pick's theorem (Area of a lattice triangledetermined by number of interior lattice points) statement, citations http://www.math.niu.edu/~rusin/known-math/index/51M04.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 51M04: Elementary Euclidean geometry (2-dimensional) Introduction Ordinary plane geometry (such as is studied in US secondary schools) holds an irresistible appeal, although many results derive what appear to be unimaginative conclusions from tortured premises. Nonetheless, from time to time something catches our eye and gets us to think about ordinary triangles and circles. History Applications and related fields Constructibility with compass and straightedge is dealt with elsewhere Tilings and packings in the plane are part of Convex Geometry Many topics regarding polygons (e.g. decompositions into triangles and so on) are treated as part of polyhedral geometry Subfields Parent field: 51M - Real and Complex Geometry Textbooks, reference works, and tutorials Software and tables A compendium of plane curves For computational geometry see 68U05: Computer Graphics Other web sites with this focus The Geometry Junkyard has a "pile" for planar geometry (and other related topics of interest!)

55. Brianchon's Theorem -- From MathWorld 146147, 1888. Coxeter, H. S. M. and Greitzer, S. L. Brianchon's theorem. §3.9in geometry Revisited. Washington, DC Math. Assoc. Amer., pp. 77-79, 1967. http://mathworld.wolfram.com/BrianchonsTheorem.html

Geometry Curves Plane Curves Conic Sections ... Barile Brianchon's Theorem The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section , the lines joining opposite polygon vertices polygon diagonals ) meet in a single point. )-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line. Duality Principle Pascal's Theorem References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Coxeter, H. S. M. and Greitzer, S. L. "Brianchon's Theorem." §3.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 77-79, 1967. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "Extensions of Pascal's and Brianchon's Theorems." Ch. 2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8-30, 1974. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 261, 1930.

56. TheMathPage  Some Theorems Of Plane Geometry Some theorems of Plane geometry. HERE ARE THE FEW theoremS that any student oftrigonometry should know. To begin with, a theorem is a statement that can be http://www.themathpage.com/aTrig/theorems-of-geometry.htm

The Topics Home Some Theorems of Plane Geometry H ERE ARE THE FEW THEOREMS that any student of trigonometry should know. To begin with, a theorem is a statement that can be proved. We shall not prove the theorems, however, but rather, we will present each one with its enunciation and its specification . The enunciation states the theorem in general terms. The specification restates the theorem with respect to a specific figure. (See Theorem 1 below.) First, though, here are some basic definitions. 1. An angle is the inclination to one another of two straight lines that meet. 2. The point at which two lines meet is called the vertex of the angle. 3. If a straight line standing on another straight line makes the adjacent angles equal to one another, then each of those angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it. 4. An acute angle is less than a right angle. An obtuse angle is greater than a right angle. 5. Angles are complementary (or complements of one another) if, together, they equal a right angle. Angles are

57. Projective Conics: Pascal's Theorem Pascal line too far off the diagram to be seen. Next Brianchon's TheoremUp Projective Conics HOME The geometry Center Home Page. http://www.geom.umn.edu/apps/conics/conic1.html

Next: Brianchon's Theorem Up: Projective Conics Pascal's Theorem We use this diagram to construct the points on a point conic: We are given five points P P' Q R , and S , and can show that the conic lying on these five points was given by the locus of blue points. Now let us define N as the intersection of x and z . We see in the diagram that N is on the conic, and can verify that our construction would send PN to P'N . We can state this as a theorem: If PR.QN RP'.NS , and P'Q.SP are collinear, then N lies on the conic determined by PP'QRS Rather than saying that N lies on the conic determined by PP'QRS , we could simply say that NPP'QRS lie on a conic. It will also simplify things to speak about the hexagon PRP'QNS ; then the points lie on a conic if and only if the hexagon is inscribed in that conic. Making these modifications and some changes of labelling, we have the theorem: If opposite sides of a hexagon (ABCDEF) intersect in three points (AB.DE, BC.EF, CD.FA) which are collinear, then the hexagon may be inscribed in a conic. This is known as the converse of Pascal's theorem So Pascal's theorem says: If a hexagon (ABCDEF) is inscribed in a conic, then opposite sides intersect in three points

58. Math5337: Technology In The Geometry Classroom This lab uses Sketchpad to explore the geometry of circles, culminating in the discoveryof a famous result of the nineteenth century called Monge's theorem. http://www.geom.umn.edu/locate/math5337/

Next: Sample syllabus Up: The Geometry Center Technology in the Geometry Classroom Course Materials These materials were developed at the Geometry Center and are used for teaching pre- and in-service teachers of high-school geometry who are interested in using technology in their classrooms. See the sample syllabus for more information on the course. The following table of contents links directly to all course materials. The materials are divided into four self-contained parts: Internet Skills, Classical Geometry, Dynamical Systems, and Symmetries and Patterns. Within these parts, the materials build on each other. For example, Introductory Questions for Geometer's Sketchpad assumes less knowledge than Monge's Theorem, which in turn assumes less sophistication than Peaucellier's Linkage. Table of Contents Internet Skills Introduction to the World-Wide Web Technical Information A compilation of pages giving the technical details of internet communication on the Macintosh. Topics include using email and how to prepare documents for the Web on a Unix Web server. Classical Geometry Introductory Questions for Geometer's Sketchpad A set of geometry problems using Sketchpad. Although not difficult, the problems are not just a set of instructions leading you to the answer.

59. Geometry Butterfly Theorem Proof - Antonio Gutierrez Butterfly theorem. geometry Step by Step with animation. Antonio Gutierrez http://agutie.homestead.com/files/GeometryButterfly.html

60. Step 0 Geometry Eyeball Theorem Esta página usa marcos, pero su explorador no los admite. http://agutie.homestead.com/files/Eyeball0.htm

Esta página usa marcos, pero su explorador no los admite.

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