61. Geometry Classes geometry Classes. Working in groups, students were asked to find solutions tothe problems presented at seven stations. Back to the pi Day Page. http://www.nvnet.org/nvhs/dept/math/pi/geometry.html

Geometry Classes Working in groups, students were asked to find solutions to the problems presented at seven stations. They spent approximately ten minutes at each station over a period of two days. The stations were positioned in the classroom so that students could move from station to station in a clockwise fashion. Station 1 - Pool Problem Students are given the area of a circular pool and a distance from the edge where a circular fence will be constructed. They are to find the amount of fencing needed. Station 2 - National Park Problem Students are given sufficient information to find the circumference of a tree. They are to find the diameter of that tree. Station 3 - Shaded Region Problem Students are given three sketches involving the same square with a different number of circles within the square. They must determine which sketch has more shaded area. Station 4 - Windshield Wiper Problem Students are given the length of a windshield wiper, the length of a rubber wiping blade and the central angle of the sector. They must determine how much area is covered by the blade. Station 5 - Pizza Problem Students are given the diameter and the calories, per square unit, of a pizza. They must determine the measure of the central angle of a slice, given a restriction of number of calories per slice.

62. Comp.Graphics.Algorithms FAQ, Section 6 See Amenta s list of computational geometry software. Because the cube tosphere volume ratio is pi/6, the average number of iterations before an 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.

63. Math: Geometry > Circles In Geometry Circles in geometry Grades 36. Overview Grade school geometry doesn t have toget into a Then measure the circumference as well. Calculate for pi. http://www.teachnet.com/lesson/math/geometry/circlesingeo.html

Front page Lesson Plans Math Geometry Circles in Geometry Circles in Geometry Overview: Grade school geometry doesn't have to get into a detailed lesson on Pi to communicate the basics of this constant. Teacher Preparation: flexible tape measure. Procedure Ideas: Break into groups, giving each group something circular to measure, both the diameter and the circumference. Then divide the circumference by the diameter to get a number. When all groups are finished, have each group read off the answer to the division problem. Use this as a lead-in to your further discussion of circle properties. Ideas of circular items to measure: basketballs, softballs, globes, hoola-hoops. A whole-class activity could be: have one student stand in the middle of the gymnasium, holding one end of a known length of string. Then walk the other end of the string around to form a circle, placing students evenly on the imaginary circumference. When finished, you should have your students representing a fairly good circle, with one in the middle. Use the tape measure to find the diameter, using the center student for accuracy in measuring through the circle's center. Then measure the circumference as well. Calculate for Pi. Discuss real-world applications for knowing Pi to estimate circumferences of objects or areas: How long a piece of paper must be to cover a round container for an art project; finding the actual diameter of the earth by using a globe with a distance scale to first determine the circumference.

64. Harvard Summer School Mathematics explore a wide variety of methods for computing pi, using geometry and calculusto prove MATH S302 Theory and Practice of Teaching geometry (31756) http://www.summer.harvard.edu/2005/courses/math.jsp

65. Human Form From Sacred Geometry Sacred geometry studies such primal systems which reveal the unity of the The expansion by pi reinforced my suspicion that this 10 sphere cluster is a http://www.people.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.

66. ROCO Resonance: Geometry Any time aN (equipped with a lone pair) lies next to a pi system, a symmetricgeometry provides maximum pi electron delocalization and stabilization? http://academic.reed.edu/chemistry/roco/Resonance/geometry.html

CHEM 201/202 HOME ROCO HOME Resonance Hybrid ... Everything Resonance affects molecular geometry Electron delocalization frequently reveals itself through a distorted molecular geometry. The most frequently cited situation is a symmetric set of bond distances in a molecule that does not have a symmetric Lewis structure, but delocalization can also create other kinds of geometrical distortions. The first two sections in this essay examine distortions in bond distances and bond angles. The last section looks at recent research on the "real" factors that make benzene (and possibly many other molecules) symmetric. Bond Distances Bond Angles Benzene Bond distances Ozone, O , is an excellent example of a symmetrical resonance hybrid with distorted bond distances. If ozone was a normal molecule with localized bonds, that is, if it was adequately described by Lewis structure I, we might expect to find one short OO bond of ~122 pm (this distance is observed in O=O ) and one long bond of ~147 pm (this distance is observed in HO-OH).

67. Efg's Reference Library: Delphi Graphics Algorithms -- Math/Geometry Note Normally, ArcTan2 returns a value from pi to pi. geometry Library,geometry.ZIP. This unit contains types, functions and procedures for http://www.efg2.com/Lab/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);

68. Chinese Pi Discs The geometry of Chinese pi Discs by Michael S. Schneider M.Ed. Mathematics.Quite many ancient cultures understood mathematics as a divine language, http://www.constructingtheuniverse.com/Pi Discs.html

The Geometry of Chinese Pi Discs by Michael S. Schneider M.Ed. Mathematics Quite many ancient cultures understood mathematics as a divine language, and ritually applied it to the designs of their sacred arts, crafts and architecture. The ancient Chinese were very deeply interested in this and, in fact, designed and ruled their civilization through many centuries with mathematics at the core of their culture, including in religion, mythology, fashion and statecraft. One example of the deliberate use of mathematics in Chinese "art", if we can call it merely that, is found in the jade Pi (or P'i or Bi ) discs. The Pi disc was the highest emblem of Chinese noble status. Among other ritual appearances, it was used to guide a deceased spirit to heaven through the Pole Star, symbolized by the hole at the disc's center. Click here to read about Pi discs and Chinese jade art. Here is a sample of three different Pi discs to examine and learn from: Pi discs, like other ritual items, were designed deliberately and made in various sizes and patterns. Their schemes are very straightforward if you know how to approach them. Please don't try to understand their dimensions using measurements, unless you use the Chinese measures of the time and are familiar with the proportions of various geometric polygons. And please don't use the dreadful, artificial modern metric system, for you are sure to get lost when exploring the designs of antiquity. But a knowledge of simple geometric constructions will solve the mystery of their proportions. Above all, you need to realize that the small circular hole at the center of each

69. Geometry -- Cardinal Stritch University Basic geometry Formulas. You should know and be able to use basic area, perimeter, If an angle is pi/4 radians, what is its measure in degrees? http://faculty.stritch.edu/breynolds/mt322_03/mt322sgB1.html

Mt 322 Topics in Geometry - Fall 2002 CARDINAL STRITCH UNIVERSITY Sr. Barbara E. Reynolds, Ph.D. Study Guide for Benchmark Test #1 This benchmark test will contribute 5% to your final course grade. The test will be available September 16. The window-of-opportunity for Benchmark #1 is September 16 - 26. You may arrange with the instructor to take this test outside of class any time during this window-of-opportunity. This test will have ten problems on material which is prerequisite for this course. The sample benchmark questions give an idea of the difficulty level and type of question to expect on this test. Although the test is not timed, many students in the past have found that they can complete this Benchmark test in 30 - 40 minutes. To pass this benchmark, a student must get at least nine problems correct, with no partial credits. Although the passing score for the Benchmarks is very high, you do have an opportunity to retake this test if you don't pass it on the first attempt. Since there is a requirement that you study between two attempts of this test, you cannot take (and retake) the test twice on the same day. If you want to take advantage of being able to retake this benchmark, you should attempt it early in this window-of-opportunity. You may NOT use a calculator or computer as you work on the benchmark test.

70. TITLE Discovering Pi AUTHOR Jack Eckley, Sunset Elem., Cody, WY without understanding, formulas that we use in geometry or other mathematicareas. This particular activity allows students to discover why pi works in http://www.col-ed.org/cur/math/math23.txt

TITLE: Discovering Pi AUTHOR: Jack Eckley, Sunset Elem., Cody, WY GRADE LEVEL/SUBJECT: 5-7, geometry OVERVIEW: Many students tend to memorize, without understanding, formulas that we use in geometry or other mathematic areas. This particular activity allows students to discover why pi works in solving problems dealing with finding circumference. OBJECTIVES: The students will: 1. Measure the circumference of an object to the nearest millimeter. 2. Measure the diameter of an object to the nearest millimeter. 3. Explain how the number 3.14 for pi was determined. 4. Demonstrate that by dividing the circumference of an object by its diameter you end up with pi. 5. Discover the formula for finding circumference using pi, and demonstrate it. RESOURCES/MATERIALS: round objects such as jars, lids, etc., measuring tapes, or string and rulers, paper, pencil, calculator ACTIVITIES AND PROCEDURES: 1. Divide class into groups of two. 2. Give materials to student teams. 3. Have student teams make a table or chart that shows name or number of object, circumference, diameter, and ?. 4. Have students measure and record each object's circumference and diameter, then divide the circumference by the diameter and record result in the ? column. 5. Have students find the average for the ? column and compare to other groups in the class to determine a pattern. Students can then find the average number for the class. 6. Explain to the students that they have just discovered pi, which is very important in finding the circumference of an object. (You may wish to give some historical information about pi at this time or have students research the information.) 7. Have students come up with a formula to find the circumference of an object knowing only the diameter of that object, and the number that represents pi. Students must prove their formula works by demonstration and measuring to check their results. TYING IT ALL TOGETHER: 1. Have students write their conclusions for the activities they have just done. Students may also share what they have learned with other members of the class. 2. Give students three problems listing only the diameter of each object and have them find the circumference. 3. Encourage students to share learned knowledge with parents.

71. MSN Encarta - Related Items - Geometry Solid geometry, geometric figures in three dimensions pi, Greek letter (p)used in mathematics as the symbol for the ratio of the circumference of a http://encarta.msn.com/related_761569706_25/pi.html

var fSendSelectEvents = true; var fSendExpandCollapseEvents = true; var fCallDisplayUAText = false; MSN Home My MSN Hotmail Shopping ... Sign In Web Search: Encarta Subscriber Sign In Help Home ... Upgrade your Encarta Experience Search Encarta Related Items from Encarta Geometry Math Homework Help Mathematics Plane Geometry, geometric figures in two dimensions Solid Geometry, geometric figures in three dimensions ... ) used in mathematics as the symbol for the ratio of the circumference of a circle to its diameter. This ratio is a universal ... View article © 2005 Microsoft MSN Privacy Legal Advertise Feedback ... Help

72. Enumerative Real Algebraic Geometry: Bibliography pi, M. piERI, Sul problema degli spazi secanti, Rend. So8, _,Some real and unreal enumerative geometry for flag manifolds, Mich. Math. http://www.math.tamu.edu/~sottile/pages/ERAG/bibliography.html

Up: Table of Contents Bibliography [Be] D. N. B ERNSTEIN The number of roots of a system of equations , Funct. Anal. Appl., 9 (1975), pp. 183-185. [BGG] I. N. B ERNSTEIN, I. M. G ELFAND, AND S. I. G ELFAND Schubert cells and cohomology of the spaces G P , Russian Mathematical Surveys, 28 (1973), pp. 1-26. [Ber] A. B ERTRAM Quantum Schubert calculus , Adv. Math., 128 (1997), pp. 289-305. [Br] R. B RICARD [BCS] P. B URGISSER, M. C LAUSEN, AND M. S HOKROLLAHI Algebraic Complexity Theory , Springer-Verlag, 1997. [COGP] P. C ANDELAS, X. C. DE LA O SSA, P. S. G REEN, AND L. P ARKES A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory , Nuclear Phys. B, 359 (1991), pp. 21-74. [Ca] G. C ASTELNUOVO Numero delle involuzioni razionali gaicenti sopra una curva di dato genere , Rendi. R. Accad. Lineci, 4 (1889), pp. 130-133. [CE-C] R. C HIAVACCI AND J. E SCAMILLA- C ASTILLO Schubert calculus and enumerative problems , Bollettino Un. Math. Ital., 7 (1988), pp. 119-126. [Cl] J. C

73. Crockett Johnson Homepage: Paintings By Crockett Johnson According to his article On the Mathematics of geometry in My Abstract Fraction of pi (to .0000003) in a Square of One; Fraction of pi (to .0000003. http://www.ksu.edu/english/nelp/purple/art.html

Crockett Johnson Homepage Paintings by Crockett Johnson Crockett Johnson Homepage: Paintings Division of a One-by-Two Rectangle by Conic Rectangles Similar Triangles Transcendental Curve Heptagon from Its Seven Sides ... Square Divided by Conic Rectangles During the last decade of his life (1965-1975), Crockett Johnson devoted his time to creating abstract geometrical paintings, all of them based on mathematical theorems. According to his article "On the Mathematics of Geometry in My Abstract Paintings" (1972), Johnson began this work in 1961 "upon belatedly discovering aesthetic values in the Pythagorean right triangle and Euclidian geometry" (97). In all, he painted as many as 100 canvases, at least 60 of which are held by the Smithsonian Institution's National Museum of American History, Division of Information Technology and Society. Of the remaining paintings, some are privately held and others have been lost. On this page, you'll find a few of Johnson's paintings; for a more complete bibliographic listing, see the " Art " section of the Bibliography . To see a larger image of most paintings below, please click on them. Excepting

74. Ethnomathematics Digital Library (EDL) Other terms calculus, pi, geometry. (Includes 28 references). Subject CulturalContext Cultural Perspectives on Mathematics, Mathematics History, http://www.ethnomath.org/search/browseResources.asp?type=subject&id=445

75. Comparative Geometric Analysis Of The Great Pyramid And The Pyramid Of The Sun B As a general rule, philosophical geometry considers pi to be in the realm of thecircle, phi in that of the square, roots and rectangles in that of the http://www.nexusjournal.com/Reynolds.html

Abstract. Mark Reynolds examines the Pyramid of the Sun at Teotihuacan and the Great Pyramid of Khufu from the point of view of geometry, uncovering similarities between them and their relationships to the Golden Section and pi A Comparative Geometric Analysis of the Heights and Bases of the Great Pyramid of Khufu and the Pyramid of the Sun at Teotihuacan Mark Reynolds 667 Miller Avenue Mill Valley, CA 94941 USA INTRODUCTION L ooking back into the murky mysteries of ancient times, there are reminders of past glories in the art, architecture, and design of our ancestors, and, in the number systems they employed in those designs. These number systems were clearly expressed in the geometry they used. Among these works are the mammoth pyramids that dot the Earth's surface. Accurate in their placement as geodetic markers and mechanically sophisticated as astronomical observatories, these wonders of ancient science stand as reminders that our brethren of antiquity may well have known more than we think. In our attempts to understand and decode these objects of awe, we realize that the winds of time and the ignorant hands of humanity have eroded the precise measurements and canons that were infused into these monuments. By their sheer magnitude [ the pyramids tell us that their builders clearly wanted future civilizations to not only notice them, but to also investigate them in an attempt to find out what knowledge these masons and priests possessed regarding the world and the universe; and although the precision of the structures may be missing, we can still see the intentions through the geometry that remains. This study was undertaken with that in mind.

76. Geometry - Patterns - Themepark Locate formulas for area and volume of geometric shapes. The pi Pagehttp//www.ballandclaw.com/upi/pi.html This site refers to pi as the world s most http://www.uen.org/themepark/patterns/geometry.shtml

Tessellations General Math Fractions/Decimals Geometry ... Patterns Geometry Geometry is the branch of mathematics that involves studying the shape, size, and position of geometric figures. These figures include plane (flat) figures, such as circles, triangles, and rectangles, and solid (three-dimensional) figures, such as cubes, cones, and spheres. The name geometry comes from two Greek words meaning earth and to measure. The world is full of geometric shapes and patterns. Sample some of the following activities to learn more about geometry. Places To Go People To See Things To Do Teacher Resources ... Bibliography Places To Go The following are places to go (some real and some virtual) to find out about geometry. Geometry Center http://www.scienceu.com/geometry/ Visit the Geometry Center. It has interactive activities, geometry articles, and classroom help. Illuminations: Geometry Resources This site features links to 120+ sites that deal with geometric concepts. Each site has been reviewed by math professionals to ensure its academic value. People To See Euclid of Alexandria http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html

77. POV-Ray: A Tool For Creating Engaging Visualisation Of Geometry It has a very powerful language for describing geometry which includes a programming 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.

78. Determining If A Point Lies On The Interior Of A Polygon if (ABS(angle) pi) return(FALSE); else return(TRUE); } /* Return the angle positive anticlockwise The result is between pi - pi */ double http://astronomy.swin.edu.au/~pbourke/geometry/insidepoly/

Determining if a point lies on the interior of a polygon Written by Paul Bourke November 1987 Solution 1 (2D) The following is a simple solution to the problem often encountered in computer graphics, determining whether or not a point (x,y) lies inside or outside a 2D polygonally bounded plane. This is necessary for example in applications such as polygon filling on raster devices. hatching in drafting software, and determining the intersection of multiple polygons. Consider a polygon made up of N vertices (x i ,y i ) where i ranges from to N-1. The last vertex (x N ,y N ) is assumed to be the same as the first vertex (x ,y ), that is, the polygon is closed. To determine the status of a point (x p ,y p ) consider a horizontal ray emanating from (x p ,y p ) and to the right. If the number of times this ray intersects the line segments making up the polygon is even then the point is outside the polygon. Whereas if the number of intersections is odd then the point (x p ,y p ) lies inside the polygon. The following shows the ray for some sample points and should make the technique clear. Note: for the purposes of this discussion will be considered even, the test for even or odd will be based on modulus 2, that is, if the number of intersections modulus 2 is then the number is even, if it is 1 then it is odd.

79. Mathematics Metadata Working Group Rather, there is pi in the context of geometry and pi in the context of number . For example, geometry is a broadening of pi because geometry is a http://mathmetadata.org/ammtf/taxonomies/ttexplanation.html

EXPLANATION OF TAXONOMIC TABLES (Written by Robby Robson : Send comments to him!) [OPTION: Spare me the theory and explanations and just show me what's what in the tables Table of Contents Basic Structure Narrowing Broadening Additional Structure ... Towards Formal Definitions (refers to Taxonomies for School and College Mathematics AMMTF Announcements About ... [IMMTF BASIC STRUCTURE The Level I and Level II taxonomies generated by the American Mathematics Metadata Task Force start with words and phrases. These words and phrases are intended to describe the content of mathematical resources. They are NOT keywords that necessarily appear in the resources themselves. Words and phrases in the taxonomy can be combined in several ways to select a desired set of resources from a collection of available resources in different ways. One way is to combine words and phrases using and, or, and not . For example, if we wanted resources that discussed proofs and discussed geometry, we could look for resources simultaneously labeled with proof and geometry.

80. Overview Of The Level II Classification For example, geometry.pi refers to pi as a geometric ratio whereas number andoperation.number.pi refers to pi as a number. Software products will be able http://mathmetadata.org/ammtf/taxonomies/overview.html

American Mathematics Metadata Task Force Overview of the Level II Classification and Tables CONTENTS: WHAT THE CLASSIFICATION DOES HOW THE CLASSIFCIATION WORK STATUS OF THE CLASSIFICATION WHAT THE CLASSIFICATION DOES Students, teachers and the general public have available to them an increasinlgy large amount of online resources. To be useful, they need to be able to locate resources on the right topic at the right educational level. This requires giving a structure to the set of resources, much as the library of congress system gives structure to a library. The American Mathematical Metadata Task Force is attempting to do this. Resources are classified according to three levels (I, II, and III) of which level II covers mathematics generally taught in high school (after the introduction of variables) and the first years of college (before proofs become commonplace). Within each level is a subject classification. The structure of the subject classification for level II is explained below. The classification will be used for many purposes, the most immediate of which are cataloging and discovery. Cataloging refers to digital libraries. Discovery refers to searching, but it is important to realize that the intended audience will not necessarily have an overview or accurate understanding of the subject. The classification of an online resource will, in most cases, not be immediately visible to the user (unlike the classification of a book in a library) but can nonetheless be submitted to an appropriate search engine and used to find similar, more special, or more general resources. As the user becomes more sophisticated, the classification system can also be used in the more familiar modes of navigating through a catalogue directly or successively submitting search strings and refining the results.

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