41. Human Form From Sacred Geometry became very interested in the three dimensional representation of this geometry and I to find that by multiplying the inner sphere s diameters by pi gives the http://saturn.vcu.edu/~chenry/

SACRED GEOMETRY New Discoveries Linking The Great Pyramid to the Human Form Professor, Department of Sculpture Virginia Commonwealth University Richmond, Virginia This site is best viewed on Microsoft Internet Explorer 4.0 or higher with screen set to 1024 X 768 pixels, 24 bit ...16 million colors. Set ... View/Text Size ... to Meduim Click on thumbnails to view larger images. For more than twenty years, I have been studying the image generating properties of reflective spheres stacked in 52 degree angle pyramids. The 52 (51.827) degree angle slope of the sides of The Great Pyramid in Cairo, Egypt embodies the Golden Mean which is the ratio that is used in Nature to generate growth patterns in space. Sacred Geometry studies such primal systems which reveal the unity of the cosmos by representing the relationships between numbers geometrically. The Vesica Piscis is one of the most fundamental geometrical forms of this ancient discipline and it reveals the relationship between the The Great Pyramid and the 2 dimensional expansion of a circle of one unit radius R as shown in Figure 1. This relationship is more completely described in The New View Over Atlantis by John Michell published by Thames and Hudson. Figure 1 Vesica Piscis in 2 Dimensions In the early 1970s, I became very interested in the three dimensional representation of this geometry and I visualized this as a three dimensional pyramid inside two intersecting spheres shown in Figure 2.

42. D&M Pyramid - Geometry (sqrt 3)/2 = 0.866025 e/pi = 0.865256. It is this ambiguity that is resolved by the geometry of a circumscribed tetrahedron. The http://users.starpower.net/etorun/pyramid/geometry.html

Evaluation The D&M Pyramid appears to be positioned with architectural alignment to other enigmatic objects nearby that have also been studied as possibly artificial. The main axis of the D&M as illustrated above points at the Face in Cydonia. Henceforth we will refer to this direction as the "front" of the pyramid. The front of the D&M Pyramid has three edges, spaced 60 degrees apart. As noted above, the center axis points to the Face. The edge on the left of this axis points toward the center of a feature that has been nicknamed the "City" by the Cydonia investigators. The edge on the right of the center axis points toward the apex of a dome-like structure known as the "Tholus". The five-sidedness, bilateral symmetry, and primary alignments were first observed by Richard Hoagland after studying quality digital enlargements prepared in 1984 by SRI International from negatives of images processed by DiPietro and Molenaar. These events are documented in detail by Hoagland and Pozos Turning back to the reconstructed geometry, we will now consider the internal symmetries of this object.

43. 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

44. Knowledge-Mathematic-Geometry-Pi.page pi. What is pi? Meh, still to busy. pi = 3.1415926535897932384626433832795028841971693993751 05820974944592307816406286208998280348253421170679 http://www.geocities.com/ultrastupidneal/Knowledge-Mathematic-Geometry-Pi.html

Pi What is pi? Meh, still to busy. I'll explain some other day. Pi = 3.1415926535897932384626433832795028841971693993751 Unfortunately those are the first 100 digits. I can put more, when I get a little done with all my fields. I don't think I'll ever want to put more than a thousand in the history of my site, as when some of you as my site dependers really need at least that much, you will have to E-mail me at LonelyNoone@Hotmail.com . As of now, I plan on putting in to 200, or maybe 100 at a time. Geometry Well, here's the first 1,000 digits of pi, and I made it in a larger font. Q:Yo SiteMaster, your site about pi is crappy, I could easily find even more digits of pi anywhere! What? You ave the first 10, 20, 50, and even a hundred, as well as a thousand? I could easily find 2 thousand, 5,000 and even 10 thousand anywhere! Your site's last on my list! A: Link

45. Knowledge-Mathematic-Geometry-Pi-X.page 3. 141592653589793238462643383279502884197169399375105820974944 592307816406286208998628034825342117067982148086513282306647 http://www.geocities.com/ultrastupidneal/Knowledge-Mathematic-Geometry-Pi-X.html

If you find this chart to be useful, and you would like this chart to be in a much larger font, things can be arranged if you want to contact me via message board. Try not to e-mail, I'm a very busy guy. And if anyone can guess how many digits this whole chart is, take a wild guess what it is!

46. Anti-Grain Geometry - AntiGrain geometry - Version 2.1 cover_mask //-cover_full }; //-pi const double pi = 3 http://www.antigrain.com/__code/include/agg_basics.h.html

// Anti-Grain Geometry - Version 2.1 // Permission to copy, use, modify, sell and distribute this software // This software is provided "as is" without express or implied // warranty, and with no claim as to its suitability for any purpose. // Contact: mcseem@antigrain.com // mcseemagg@yahoo.com // http://www.antigrain.com #ifndef #define namespace agg typedef signed char typedef unsigned char typedef signed short typedef unsigned short typedef signed int typedef unsigned int typedef unsigned char enum pi const double pi inline double double deg return deg pi inline double double rad return rad pi template class T struct typedef T T x1 T y1 T x2 T y2 T x1_ T y1_ T x2_ T y2_ const normalize T t if t t if t t return this bool clip const r if r r if r r if r r if r r return bool const return template class Rect Rect const Rect const Rect Rect r if r r if r r if r r if r r return r template class Rect Rect const Rect const Rect Rect r if r r if r r if r r if r r return r typedef int rect rect typedef double enum enum inline bool unsigned c return c c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c c c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c inline bool unsigned c return c inline unsigned unsigned c return c inline unsigned unsigned c return c inline unsigned unsigned c return c inline unsigned unsigned

47. Anti-Grain Geometry - AntiGrain geometry - Version 2.1 static unsigned dimension() { return 6; } static double calc_weight(double x) { if(x 0.0) return 1.0; x *= pi; double x3 http://www.antigrain.com/__code/include/agg_image_filters.h.html

// Anti-Grain Geometry - Version 2.1 // Permission to copy, use, modify, sell and distribute this software // This software is provided "as is" without express or implied // warranty, and with no claim as to its suitability for any purpose. // Contact: mcseem@antigrain.com // mcseemagg@yahoo.com // http://www.antigrain.com // Image transformation filters, // Filtering classes ( // Basic filter shape classes: #ifndef #define #include math h #include " agg_basics.h " namespace agg // See Implementation agg_image_filters.cpp enum class public unsigned dimension unsigned dimension const return int start const return const double const return const int const return protected void weight unsigned idx double val double unsigned idx const void normalize private const const operator const unsigned int double int template class FilterF class public public FilterF::dimension unsigned i unsigned dim dimension for i i dim i weight i i normalize private : FilterF m_filter_function class public static unsigned dimension return static double double x return x ? x

48. Urban Legends Reference Pages: Religion (Alabama's Slice Of Pi) pi is merely an artifact of Euclidean geometry. Humbleys is working on a theory which he says will prove that pi is determined by the geometry of three http://www.snopes.com/religion/pi.htm

Alabama's Slice of Pi Claim: Responding to pressure from religious groups, Alabama's state legislature redefined the value of pi from to 3 in order to bring it in line with Biblical precepts. Status: False. Example: [Collected on the Internet, 1998] introduced without fanfare by Leonard Lee Lawson (R, Crossville), and rapidly gained support after a letter-writing campaign by members of the Solomon Society, a traditional values group. Governor Guy Hunt says he will sign it into law on Wednesday. The law took the state's engineering community by surprise. "It would have been nice if they had consulted with someone who actually uses pi," said Marshall Bergman, a manager at the Ballistic Missile Defense Organization. According to Bergman, pi is a Greek letter that signifies the ratio of the circumference of a circle to its diameter. It is often used by engineers to calculate missile trajectories. Prof. Kim Johanson, a mathematician from University of Alabama, said that pi is a universal constant, and cannot arbitrarily be changed by lawmakers. Johanson explained that pi is an irrational number, which means that it has an infinite number of digits after the decimal point and can never be known exactly. Nevertheless, she said, pi is precisly defined by mathematics to be "3.14159, plus as many more digits as you have time to calculate". "I think that it is the mathematicians that are being irrational, and it is time for them to admit it," said Lawson. "The Bible very clearly says in

49. LucyTuning*LucyScaleDevelopments*LucyTuned Lullabies*Pi Tuning*John Longitude Ha geometry is a mapping system of space. Music is a mapping system of sound and other vibrating patterns. The Ancient Greeks can not claim the discovery of pi, http://www.lucytune.com/academic/music_and_geometry.html

Search This Site The Web for Home New to LucyTuning? Academic & Research, John "Longitude" Harrison LucyTuned Lullabies ... info@lucytune.com MusicAsEasyAsPi - Music, Ancient Greek Ratios, Geometry and Speculations. Once upon a time some two and a half thousand years ago the Ancient Greek philosophers thought about space, time and musical sounds. Pythagoras became best know in our era for his concept that the sum of the squares of the adjacent sides of a right angle triangle are equal to the square of the hypotenuse. (a^2 + b^2 = c^2) Stated like this, some people find it difficult to visualize, but a simple drawing makes it abundantly clear. Since the original drawings were probably marked with a stick in sand my untidy proportions are comparable to the first attempts. A picture as with music, a sound is worth a million words. What does all this have to do with music? You can justifiably ask.

50. POV-Ray: A Tool For Creating Engaging Visualisation Of Geometry A more interesting geometric primitive is a parametric surface, to illustrate this consider the parametric { function { cos(2*pi*u pi/2)*cos(2*pi*(-u+v)+pi/2 http://astronomy.swin.edu.au/~pbourke/povray/representation/

POV-Ray: A Tool for Creating Engaging Visualisation of Geometry Written by Paul Bourke January 2004 Abstract Computers are now a standard tool for creating, exploring, and presenting geometric form and mathematics. Finding the right software tools can be difficult, especially so when high quality and visually appealing images are required. This paper will discuss one particular package (POV-Ray) used with great success by the author. A general discussion of the desirable features will be presented along with examples based around the familiar tetrahedral form. Introduction The mathematician and geometer often needs to represent equations or geometry visually, both for their own insight (visualisation) and as a way of conveying information to a wider audience. There are a number of software packages that have been designed to meet this need but most concentrate on the former goal and as such may be able to create informative images for the expert but tend to be limited in their ability to create higher quality images that may be more informative and attractive to a general audience. There are a number of consideration when choosing software for any task and there are some others that are relevant to the presentation of geometry.

51. Themepark - Utah Education Network What did the little acorn say when he grew up? geometry . pi Day Page http//planetpi.8m.com/ Start making plans now to celebrate your own pi Day http://www.uen.org/themepark/html/patterns/geometry.html

Themepark has moved. Please update your bookmarks. You are being redirected. Go to Themepark

52. History Of Mathematics: Egyptian Math, Pi, Magic Squares, Chinese Arithmetic, Me the value of pi Archimedes method of exhaustion, Leibniz series, Machin formula using tangents, and others. THALES, FOUNDER OF GREEK geometry (585 BCE) The http://nunic.nu.edu/~frosamon/history/bc3000.html

HISTORY OF NUMERAL SYSTEMS (4700 B.C.E.-1500 C.E.) A timeline and brief history of numeral systems were indicated from 4700 B.C. to 1500 A.D. Many cultures through-out the world had developed numeral systems for their own community technological advancement. HISTORYCAL CREATORS OF MATHEMATICAL GAMES AND THEIR BIOGRAPHIES (1850 B.C.E.-Present) Mathematical games and recreations started around 1850 BC and continued on to the present by famous mathematicians. The biographies of mathematicians who invented the games are reported including pictures and graphs in this web site. MAGIC SQUARES (2200 B.C.E.) The magic square has been studied for a long period of time. It shows how a magic square is formed and who studied the magic squares. ARISTOTLE-DEDUCTIVE LOGIC (340 B.C.E.) Aristotle wrote a book called "TOPICS" which started out with a discussion of deductive logic. The whole world reestablished this book starting with the Islamic translation on through time. HISTORY OF PI (287 B.C.E. to present time) There are a several different methods of estimating the value of pi: Archimedes' method of exhaustion, Leibniz series, Machin formula using tangents, and others. THALES, FOUNDER OF GREEK GEOMETRY (585 B.C.E.)

53. Solution For /geometry/hole.in.sphere Solution to the /geometry/hole.in.sphere problem. The It is pi * (D/2)^2. The same area as a circle with that diameter. Proof big http://rec-puzzles.org/sol.pl/geometry/hole.in.sphere

Solution to the /geometry/hole.in.sphere problem The volume of the leftover material is equal to the volume of a 6" sphere. First, lets look at the 2 dimensional equivalent of this problem. Two concentric circles where the chord of the outer circle that is tangent to the inner circle has length D. What is the annular area between the circles? It is pi * (D/2)^2. The same area as a circle with that diameter. Proof: big circle radius is R little circle radius is r 2 2 area of donut = pi * R - pi * r 2 2 = pi * (R - r ) Draw a right triangle and apply the Pythagorean Theorem to see that 2 2 2 R - r = (D/2) so the area is 2 = pi * (D/2) Take a general plane at height h above (or below) the center of the solids. The radius of the circle of intersection on the sphere is radius = srqt(3^2 - h^2) so the area is pi * ( 3^2 - h^2 ) For the ring, once again we are looking at the area between two concentric circles. The outer circle has radius sqrt(R^2 - h^2), The area of the outer circle is therefore pi (R^2 - h^2) The inner circle has radius sqrt(R^2 - 3^2). So the area of the inner circle is

54. MSN Encarta - Geometry Archimedes, one of the greatest Greek scientists, made a number of important contributions to geometry during the 3rd century bc. See pi. http://encarta.msn.com/encyclopedia_761569706_3/Geometry.html

MSN Home My MSN Hotmail Shopping ... Money Web Search: logoImg('http://sc.msn.com'); Encarta Subscriber Sign In Help Home ... Upgrade to Encarta Premium Search Encarta Tasks Find in this article Print Preview Send us feedback Related Items Mathematics Plane Geometry, geometric figures in two dimensions more... Magazines Search the Encarta Magazine Center for magazine and news articles about this topic Further Reading Editors' Picks Geometry News Search MSNBC for news about Geometry Internet Search Search Encarta about Geometry Search MSN for Web sites about Geometry Also on Encarta Encarta guide: The Reagan legacy Compare top online degrees Proud papas: Famous dads with famous kids Also on MSN Father's Day present ideas on MSN Shopping Breaking news on MSNBC Switch to MSN in 3 easy steps Our Partners Capella University: Online degrees LearnitToday: Computer courses CollegeBound Network: ReadySetGo Kaplan Test Prep and Admissions Encyclopedia Article from Encarta Advertisement Page 3 of 3 Geometry Multimedia 6 items Article Outline Introduction Methodology Euclidean Geometry Analytic Geometry ... History of Geometry D Conic Sections Conic sections are curves formed by the intersection of a plane with the surface of a cone. (When discussing conic sections

55. Geometry 101 Example #5 Inspection of the geometry indicates a solution between pi/4 and pi/3 where the upper bound is determined from the equation x=acos(a/2r) when a = r. The http://home.netcom.com/~bled/geometry.html

Geometry 101 Previous Example Next Example This academic exercise is a static problem which goes something like this: A friendly neighbor gave Johnny permission to tie his goat on the perimeter of his circular pasture provided that no more than half of the pasture is grazed. How long should that rope be? pi*a^2*(x/pi) + [pi*r^2*(y/pi) - a*r*sin(x)] = pi*r^2/2 (a/r)^2*x + (pi - 2x) - (a/r)sin(x) = pi/2 2xcos(x)^2 - sin(x)cos(x) - x = -pi/4 The first equation defines the desired surface area of the overlapping circles contoured by the radius r and chord a respectively. A bit of algebra and the angular identity y = pi - 2x produces the second equation. The final equation follows a substitution for the ratio a/r = 2cos(x) which calculates the rope length once the angle x is known. Solving the nonlinear equation for x with, say, Newton-Raphson method, is a quick option for a programmable calculator or a canned PC math package. Here however, we highlight the simple mechanism with which the "solver" interacts with the system to be solved. call nlsq (isol

56. B.U. Center For Polymer Studies:JAVA Applets Our study of the geometry tells us this ratio is pi/4. So, now we can estimate pi as. pi = 4 x (Number of Darts in Circle) / (Number of Darts in Square). http://polymer.bu.edu/java/java/montepi/MontePi.html

Monte Carlo Estimation for Pi Monte Pi Java Applet The Monte Carlo method is used in modeling a wide-range of physical systems at the forefront of scientific research today. Monte Carlo simulations are statistical models based on a series of random numbers. Let's consider the problem of estimating Pi by utilizing the Monte Carlo method. Suppose you have a circle inscribed in a square (as in the figure). The experiment simply consists of throwing darts on this figure completely at random (meaning that every point on the dartboard has an equal chance of being hit by the dart). How can we use this experiment to estimate Pi? The answer lies discovering the relationship between the geometry of the figure and the statistical outcome of throwing the darts. Let's first look at the geometry of the figure. Let's assume the radius of the circle is R , then the Area of the circle = Pi R and the Area of the square = 4 R Now if we divide the area of the circle by the area of the square we get Pi / 4 But, how do we estimate Pi by simulation? In the simulation, you keep throwing darts at random onto the dartboard. All of the darts fall within the square, but not all of them fall within the circle. Here's the key. If you throw darts completely at random, this experiment estimate the ratio of the area of the circle to the area of the square, by counting the number of darts in each. Our study of the geometry tells us this ratio is Pi/4 . So, now we can estimate Pi as

57. Geometry Question Re geometry question Posted Tuesday, May 11, 2004 759 PM (CST y = (float)Math.Cos(xRot * Math.pi / 180.0f) * y (float)Math.Sin(xRot * Math.pi / 180.0f) * z http://www.randyridge.com/Tao/Discussions/Help/1776.aspx

Home Weblog Tao Links ... About Search: Go Login Register Home ... Help Help Author Thread: Geometry question kalme Geometry question Posted: Tuesday, May 11, 2004 2:57 AM (CST) Hallo, my problem is not directly TAO related, and furthermore this may be a rather stupid question. But I'm stuck for some time now and I hope someone can give me a clue with this. What I want to do is transforming some coordinates into the current OpenGL coordinate system ("by hand"). The situation is something like this. I have done some rotations and transformations like Gl.glRotatef(fXRot, 1.0f, 0.0f, 0.0f); Gl.glRotatef(fYRot, 0.0f, 1.0f, 0.0f); Gl.glTranslatef(-fXTrans, 0.0f, -fZTrans); // [x'] = [1 ] * [x] // [y'] = [0 cos -sin] * [y] // [z'] = [0 sin cos] * [z] y = (float)Math.Cos(fXRot * Math.PI / 180.0f) * y - (float)Math.Sin(fXRot * Math.PI / 180.0f) * z; z = (float)Math.Sin(fXRot * Math.PI / 180.0f) * y + (float)Math.Cos(fXRot * Math.PI / 180.0f) * z; // [x'] = [cos sin] * [x] // [y'] = [0 1 ] * [y] // [z'] = [-sin cos] * [z] x = (float)Math.Cos(fYRot * Math.PI / 180.0f) * x

58. Easy As Pi EASY AS pi. Three wonderful things happen when you use literature to enhance your existing mathematics program. Back to Ideas. geometry. http://www.todaysteacher.com/EasyAsPi.htm

EASY AS Pi Three wonderful things happen when you use literature to enhance your existing mathematics program. One, curiosity is generated from the literature being used. Two, the literature creates questions that propel your students’ mathematical understandings. Three, students develop a relationship between the real world and mathematics through the literature. Children of all ages, yes, even in middle school, love to be read to. Use this to your advantage and start spicing up your math lessons. Literature Ideas COUNTING Patterns and Functions Estimation / Comparison Measurement ... Money COUNTING The Twelve Circus Rings , Seymour Chwast - Students count the performers in the circus as it is read. Have students write their own versions of the story, which follows the rhythm of The Twelve Days of Christmas. The Hundred Penny Box , Sharon Bell Mathis - Students can count, measure, and weigh pennies. Write “Penny Autobiographies” writing a paragraph about each year of their life on a penny shaped piece of paper, creating a book shaped like a penny.

59. Comp.Graphics.Algorithms FAQ, Section 6 Ti not incident to pi. The areas of these triangles Ti are proportional to the barycentric coordinates ti of p. Reference Coxeter, Intro. to geometry, p.217 http://exaflop.org/docs/cgafaq/cga6.html

Comp.Graphics.Algorithms Frequently Asked Questions Section 6. Geometric Structures and Mathematics (C) 1998 Joseph O'Rourke. Subject 6.01: Where can I get source for Voronoi/Delaunay triangulation? For 2-d Delaunay triangulation, try Shewchuk's triangle program. It includes options for constrained triangulation and quality mesh generation. It uses exact arithmetic. The Delaunay triangulation is equivalent to computing the convex hull of the points lifted to a paraboloid. For n-d Delaunay triangulation try Clarkson's hull program (exact arithmetic) or Barber and Huhdanpaa's Qhull program (floating point arithmetic). The hull program also computes Voronoi volumes and alpha shapes. The Qhull program also computes 2-d Voronoi diagrams and n-d Voronoi vertices. The output of both programs may be visualized with Geomview. There are many other codes for Delaunay triangulation and Voronoi diagrams. See Amenta's list of computational geometry software. The Delaunay triangulation satisfies the following property: the circumcircle of each triangle is empty. The Voronoi diagram is the closest-point map, i.e., each Voronoi cell identifies the points that are closest to an input site. The Voronoi diagram is the dual of the Delaunay triangulation. Both structures are defined for general dimension. Delaunay triangulation is an important part of mesh generation.

60. Efg's Reference Library: Delphi Graphics Algorithms -- Math/Geometry angle = 180 * (1 + ArcTan2(jTextjTarget, iTarget-iText) / pi); IF angle = 360.0 THEN angle = angle - 360.0;. geometry, geometry Library, geometry.ZIP. http://homepages.borland.com/efg2lab/Library/Delphi/Graphics/Math.htm

US UK DE US UK ... Delphi Graphics : Algorithms A. General Graphics B. Color C. Image Processing D. Mathematics/Geometry in look for Delphi Graphics Delphi Graphics Image Processing ... Mathematics GraphicsMathPrimitives. 2D/3D Clipping, 3D-to-2D Projections. See 2D/3D vector graphics examples on efg's Graphics projects page. Angle USES Math; // ArcTan2 // Given points(iText,jText) and (iTarget,jTarget). The angle between // the x-axis and the vector (iTarget-iText, jTarget-jText) ranges in the // open interval [0.0, 360.0) degrees. (0 degrees is the +X axis). // Note: Normally, ArcTan2 returns a value from -PI to PI. angle := 180 * (1 + ArcTan2(jText-jTarget, iTarget-iText) / PI); THEN angle := angle - 360.0; Clock Angles www.delphiforfun.com/Programs/clock_angle.htm Arcs The TCanvas.Arc method is is not that easy to use for some applications. The DrawArcs demo shows how to define a bounding rectangle and then draw the specified type of arc between opposite corners of the rectangle. TYPE TArcOrientation = (aoSouthWest, aoSouthEast, aoNorthEast, aoNorthWest);

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