81. Non-Euclidean Geometry NonEuclidean geometry. Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html

Non-Euclidean geometry Geometry and topology index History Topics Index In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems: To draw a straight line from any point to any other. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance. That all right angles are equal to each other. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. It is clear that the fifth postulate is different from the other four. It did not satisfy Euclid and he tried to avoid its use as long as possible - in fact the first 28 propositions of The Elements are proved without using it. Another comment worth making at this point is that Euclid , and many that were to follow him, assumed that straight lines were infinite. Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that

82. Piguy 's Math Javascripts Page This page has dozens of math calculators, making all sorts of math, from algebra to arithmetic to geometry, easier. Also, 50,000 decimals of pi and a logarithm table. http://www.geocities.com/CapeCanaveral/Hall/1216/

Click above for more info about making money surfing... by the way, I have PROOF that you actually get paid piguy's Math Page Search this page! This search engine hasn't been working that well, so if you don't get good results, you aren't alone. It will be fixed soon (hopefully). Howdy! Thanks for coming to my web page. I am working on getting more stuff, but it is a slow process. JavaScript Things Logarithm Stuff Links Math Forum ... Webrings If you have any kind of complaint, comment, or hint, Please and let me know. Don't forget to sign my guestbook! Have a good Day! Take my poll! Click here to vote for other things. My website quality check How good is this website Totally, 100% AWESOME !!! If I liked Math, It would be pretty cool Decent, not great, but pretty good Not very good, needs work, probably in bottom 30% of websites Where in the world did you learn how to make a website?!?! Current Results FastCounter by LinkExchange people have visited this Math page since Sept. 18th, 1998 Sorry, you do not have a javascript compatible browser. You will not be able to see the beauty of this website. Find out about me and my family by clicking here

83. Bubble Geometry - Science Museum Of Minnesota Thinking Fountain image (17k), Bubble geometry. Have you ever seen a square bubble? Experiment with bubbles. Create bubble wands out http://www.smm.org/sln/tf/b/bubblegeometry/bubblegeometry.html

Bubble Geometry Have you ever seen a square bubble? Experiment with bubbles. Create bubble wands out of found objects (straws, pipe cleaners, strawberry baskets and coathangers) and have your own bubble festival. How can you catch a bubble? The secret is the soap solution. Try catching a bubble with a dry hand versus a wet hand. Which lasts longer? This activity is included in the Shapes cluster developed with K-2 teachers at the Ross School in San Francisco. What could you use to make lots of tiny bubbles? How could you measure a bubble? Why do bubbles fall towards the ground? Bubbles Soap bubbles are so fun! Make your own bubble prints Gathered by topic Connected together Index of ideas Try something new Science Learning Network

84. Notes On Differential Geometry By B. Csikós Notes by Bal¡zs Csik³s. Chapters in PostScript. 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.

85. GANG | Geometry Analysis Numerics Graphics The GANG Gallery of Constant Mean Curvature Surfaces. The GANG Gallery of Willmore Surfaces. The GANG Gallery of Minimal Surfaces. http://www.gang.umass.edu/

The GANG Gallery of Constant Mean Curvature Surfaces The GANG Gallery of Willmore Surfaces The GANG Gallery of Minimal Surfaces The GANG Gallery of Pseudospherical Surfaces Summer REU program at GANG

86. Geometry Of Sri Yantra Artistic and Historical Background. Historical Methods of Duplication. Modern Experiments in Construction. http://alumni.cse.ucsc.edu/~mikel/sriyantra/sriyantra.html

Artistic and Historical Background A curiosity: The mathematics of sriyantra by Joseph The geometry of Sri-yantra by Bolton and Macleod Art of Sri Yantra A connection with Indian ragas by Mookerjee Historical Methods of Duplication Construction by Order of Destruction Construction by Order of Creation Modern Experiments in Construction Construction by Checkerboard Construction by Golden Angle Complexity Measure A note on Computational Complexity of Sri Yantra by Kulaichev Bibliography Links please send your feedback to "mikel_maron [at] yahoo [dot] com"

87. Connected Geometry Home Page Connected geometry. Project Information. Connected topics Habits of Mind An Introduction to geometry; A Perfect Match Congruence and Proof; http://www.edc.org/LTT/ConnGeo/

Connected Geometry Project Information Connected Geometry is a curriculum development project, funded by the National Science Foundation, designed to help teachers and students engage in meaningful mathematical activity by offering students a chance to understand and appreciate the connections and unifying themes within mathematics, and to build on the connections between students' backgrounds and mathematics. Project Description Information on Availability Book Descriptions The Connected Geometry student book contains six chapters, allowing students to explore one idea in depth, seeing how it connects to many different mathematical topics: Habits of Mind: An Introduction to Geometry A Perfect Match: Congruence and Proof The Cutting Edge: Investigations in Dissection and Area A Matter of Scale: Pathways to Similarity and Trigonometry ... Optimization: A Geometric Approach Teacher materials include a complete Teacher's Guide (with suggested assessments), an extensive Solution and Problem Solving Resource guide (containing a complete solution to every problem), a book of Assessment Resources, and a book of Teaching Resources (masters). Along with these materials, teachers will receive a CD-ROM containing six modules (corresponding to the chapters above) with additional student lessons, solutions, and teaching notes. If you have any comments on the Connected Geometry WWW pages please contact us at: hlebowitz@edc.org

88. [gr-qc/9911051] Complex Geometry Of Nature And General Relativity A paper by Giampiero Esposito attempting to give a selfcontained introduction to holomorphic ideas in general relativity. The main topics are complex manifolds, spinor and twistor methods, heaven spaces. http://arxiv.org/abs/gr-qc/9911051

General Relativity and Quantum Cosmology, abstract gr-qc/9911051 From: [ view email ] Date: Mon, 15 Nov 1999 11:06:50 GMT (124kb) Complex Geometry of Nature and General Relativity Author: Giampiero Esposito Comments: 229 pages, plain Tex Report-no: DSF preprint 99/38 An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven spaces. Full-text: PostScript PDF , or Other formats References and citations for this submission: SLAC-SPIRES HEP (refers to , cited by , arXiv reformatted); CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv gr-qc find abs

89. Dynamic Geometry Home Page Dynamic geometry. Dynamic geometry is a twoyear research project, funded by the National Science Foundation, housed at Education Development Center, Inc. http://www.edc.org/LTT/DG/

Dynamic Geometry Dynamic Geometry is a two-year research project, funded by the National Science Foundation, housed at Education Development Center, Inc. in Newton. From this Web site, you can read: A short project description Some interesting quotes from our interviews Interview Data The project produced over thirty hours of videotaped data. There is, of course, more to be seen in these tapes than we had originally aimed to find, and more than we can analyze. We provide the transcripts in the hopes they will be of value to others doing this kind of research. Each transcript has been edited to add essential figures and labels. Read about and download sketches used in the interviews. Get student interview transcripts. Project Papers The Nature and Growth of Students' Understanding of the Notion of Function in Dynamic Geometry Environments (abstract available here) Dynamic Geometry as a Bridge from Euclidean Geometry to Analysis (abstract available here) Lifting the Curtain: A History of The Geometer's Sketchpad (abstract available here) Problem Solving and Proof in the Age of Dynamic Geometry (abstract available here) Tripping Over the Cosine Law While on the Lookout for Invariants (abstract available here) Dynamic Geometry and the Sculptures of Arthur Ganson (abstract available here) Mathematics of Finance (outline and sketches available here) Questions and comments are welcome: Paul Goldenberg

90. Geometry Main Page Basic overviews, graphics and quizzes to help students review geometry concepts. http://www.mcwdn.org/geometry/geoframe.html

GEOMETRY This geometry review page will help master the concepts learned in class. Ideas will be reviewed with the assistance of graphics and sound to help you learn these important ideas. Even the titles contain clues as to the attributes of the geometric shape. After many of the pages are quizzes to help reinforce the ideas learned. Simply click on the topic you need to review in the index and you are on your way. For questions or comments on this site: calfieri@rochester.rr.com A site Geometry Home Coordinates Lines Polygons ... Geometry Activities

91. Geometry Solution We are excited and proud to introduce GoMath s geometry Solutions. geometry Solutions is a sophisticated calculator that calculates http://www.gomath.com/geometrycal.html

We are excited and proud to introduce GoMath's Geometry Solutions. Geometry Solutions is a sophisticated calculator that calculates the perimeter, lateral and surface areas, and volume of plane and solid geometric figures. For example, when you type in the angle, side, or the hypotenuse of a right angle, Geometry Solutions will calculate the perimeter and area based on the information you entered. The formulas used to calculate the perimeter and area of each kind of geometric figure is also given. It is very user friendly and easy to use. If you have difficulties in calculating the area of geometric angles, simply type the problem or similar problem into GoMath's Geometry Solutions. Click on "calculate" and an answer will instantly appear on your screen. Use the given formulas for the step by step calculation and the answer as a guide to the final solution. Geometry Solutions and geometric formulas when used together will strengthen your knowledge and abilities in geometry. Triangle Parallelogram Trapezoid Rectangle ... Back to GoMath

92. Dipartimento Di Matematica Applicata G. Sansone Editors' page. http://www.jgp.unifi.it/

93. The Geometry Of War Online catalogue for an exhibit at the Museum of the History of Science, Cambridge, England relating http://www.mhs.ox.ac.uk/geometry/content.htm

Introduction Summaries Essay Catalogue Introduction Summaries Essay Catalogue ... Museum Home Page

94. Mathematik.com Individual pages on different topics in Mathematics. Examples group theory, dynamical systems theory, geometry or number theory. http://www.mathematik.com/

Mathematik.com Gradus Suavitatis Turing Billiard Bifurcation ... Feedback

95. Geometry geometry. Galileo (1623). The Ribbed Sphere Do plans and elevations describe unique geometry? Quadric equations Surfaces in x and y of degree 2. Fitting puzzle. http://astronomy.swin.edu.au/~pbourke/geometry/

G e o m e t r y Index Geometry Formats Curves Surfaces Polyhedra Fractals, Chaos Projection Stereographics Rendering Radiance PovRay OpenGL Modelling Terrain Colour Textures Other Data Formats Analysis Fun Puzzles Old Stuff Papers Surfaces Curves Fractals ... Polyhedra, platonic solids, polytopes Philosophy is written in this grand book - I mean universe - which stands continuously open to our gaze, but which cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. Galileo (1623) The Ribbed Sphere Do plans and elevations describe unique geometry? Quadric equations Surfaces in x and y of degree 2. Fitting puzzle Slicing a torus How many ways can a torus be cut (with a single plane) so that the resulting cross sections are perfect circles? Interpretation illusion How simple geometry can be interpreted in many ways.

96. Index Of /~cramer/RelViz/text/exhib1 An exhibition on relativistic computer dynamics used to present the theory of black holes. http://www.astro.ku.dk/~cramer/RelViz/text/exhib1

Index of /~cramer/RelViz/text/exhib1 Name Last modified Size Description ... Parent Directory 22-Sep-1999 08:23 - contents.pl 14-Jun-1996 23:34 1k exhib1.css 14-Jun-1996 23:34 1k exhib1.html 15-Jun-1996 01:41 4k images.aux 14-Jun-1996 23:34 1k images.log 14-Jun-1996 23:34 3k images.pl 14-Jun-1996 23:34 1k images.tex 14-Jun-1996 23:34 1k img1.gif 14-Jun-1996 23:34 1k img2.gif 14-Jun-1996 22:15 1k img3.gif 14-Jun-1996 22:15 1k img4.gif 14-Jun-1996 22:15 1k img5.gif 14-Jun-1996 23:07 1k img6.gif 14-Jun-1996 23:07 1k img7.gif 14-Jun-1996 23:07 1k img8.gif 14-Jun-1996 22:16 1k img9.old 14-Jun-1996 22:15 1k node1.html 14-Jun-1996 23:34 2k sections.pl 14-Jun-1996 23:34 1k test.gif 14-Jun-1996 23:23 1k Apache/1.3.9 Server at www.astro.ku.dk Port 80

97. Topology And Geometry Jeff Weeks . Topology and geometry Software. Fun and Games for ages 10 and up. Torus and Klein Bottle games (online), Kali (Windows, Macintosh http://geometrygames.org/

Jeff Weeks' Topology and Geometry Software Fun and Games for ages 10 and up Torus and Klein Bottle games (online) Kali (Windows, Macintosh) KaleidoTile (Windows, Macintosh) Classroom Materials for teachers grades 6-10 Exploring the Shape of Space Curved Spaces for software developers Computer Graphics in Curved Spaces (OpenGL, Direct3D) Research Software for mathematicians SnapPea (Linux, Macintosh, Windows) Comments? Problems? Suggestions? Contact Jeff Weeks awards and links

98. Topological Aspects Of Real Algebraic Geometry Research semester, MSRI , Berkeley, CA, USA; 2 January 14 May 2004. http://zeta.msri.org/calendar/programs/ProgramInfo/120/show_program

Calendar Topological Aspects of Real Algebraic Geometry January 5, 2004 to May 14, 2004 at the Mathematical Sciences Research Institute, Berkeley, California Organized by: Selman Akbulut Grisha Mikhalkin Victoria Powers Boris Shapiro ... Frank Sottile (chair), and Oleg Viro Real algebraic geometry the geometry of varieties defined by systems of real polynomial equations is a classical subject presently encompassing many distinct lines of inquiry. This program will cover modern developments in real algebraic geometry and its applications emphasizing topological aspects of this subject and its relations to other fields of mathematics. These relations arise as real algebraic varieties appear naturally in various mathematical contexts and, in particular, in applied mathematics, and there continue to be important interactions with these subjects. (One interaction with solving equations.) Besides the traditional directions of topological classification of real algebraic varieties, we mean to focus on enumerative problems and relations to convex geometry via the theory of amoebas and tropical geometry. This will include many recent and notable advances in real algebraic geometry, as well as some of its most important open problems. Of particular emphasis will be the following topics. Real algebraic curves. (The pictures below are of two constructions of real plane curves exploiting convexity.)

99. Arithmetic Algebraic Geometry A European network of 12 working groups from 6 countries. http://www.maths.univ-rennes1.fr/arithgeom/

A Research Training Network of the European Union Overview Partners Programme Positions Activities Project overview Developing powerful methods taken from geometry to study the arithmetical properties of algebraic equations Algebraic equations and their arithmetical properties have interested mankind since antiquity. One has only to think of the works of Pythagoras and Diophantus, which were a milestone in their time. For many centuries such problems have fascinated both serious mathematicians (Fermat, Gauss, ...) and amateurs alike. However, developments in recent years have transformed the subject into one of the central areas of mathematical research, which has relations with, or applications to, virtually every mathematical field, as well as an impact to contemporary everyday life (for example, the use of prime numbers and factorisation for encoding "smart" cards). The classical treatment of equations by analysis and geometry in the realm of complex numbers in this century has found a counterpart, in the similar theories over finite and p -adic fields, which have particular significance for arithmetic questions. The study of certain functions encoding arithmetic information and generalising the Riemann zeta-function (

100. Sacred Geometry Sacred geometry. The universe is created by a consciousness SACRED geometry IN ARCHITECTURE. The term sacred geometry is used by http://www.crystalinks.com/sacred_geometry.html

Sacred Geometry The universe is created by a consciousness which manifests in physical reality through a geometric blueprint that we call Sacred Geometry which repeats over and over giving the illusion of linear time. GEOMETRIC PATTERNS Sacred Geometry Home Page Bruce Rawles Sacred Geometry Discovery Vincent Beall Dodecahedron and Other Shapes John Martineau Dan Winter Meru Foundation Stan Tenen Secret of the Hebrew Letters Stan Tennen GOLDEN RATIO - GOLD=ALCHEMY - CONSCIOUSNESS - PHI RATIO - 1.61803398874989484820 GOLDEN MEAN AND PHI GOLDEN MEAN SPIRAL GOLDEN RATIO GOLDEN RATIO - PHI ... FUSION ANOMALY - GOLDEN MEAN SACRED GEOMETRY AND THE GREAT PYRAMID Sacred Geometry Charles Henry Linking The Great Pyramid to the Human Form The Great Pyramid, The Golden Ratio and The Royal Cubit Geometry of the Great Pyramid SACRED GEOMETRY IN ARCHITECTURE The term "sacred geometry" is used by archaeologists, anthropologists, and geometricians to encompass the religious, philosohical, and spiritual beliefs that have sprung up around geometry in various cultures during the course of human history. It is a catch-all term covering Pythagorean geometry and neo-Platonic geometry, as well as the perceived relationships between organic curves and logarithmic curves. Armenian Architecture Native American Geometry Rennes le Chateau Sacred Geometry in Building ... The Prehistoric Alignment of World Wonders SACRED GEOMETRY IN DANCE SACRED GEOMETRY AND THE MEVLEVI SUFI ROUND DANCE RELIGIOUS DANCING OF THE SUFI MUSLIMS SACRED GEOMETRY IN ART

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