61. Chinese Pi Discs The geometry of Chinese pi Discs by Michael S. Schneider M.Ed. Mathematics. Quite many ancient cultures understood mathematics as 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

62. Excel Geometry Functions Computes the arc sine function. arcsin = pi / 2 arccos(x). End Function. segment defined by the points (x1,y1) and (x2,y2) using plane geometry. http://nmml.afsc.noaa.gov/Software/ExcelGeoFunctions/excelgeofunc.htm

If you see this message you have JavaScript disabled. JavaScript is used for the navigation in this site. To enable JavaScript in Internet Explorer, click here here National Marine Mammal Laboratory Research Programs What is NMML Search New at NMML ... Auke Bay Laboratory Excel Geometry Functions Download self extracting zip file here ). Functions within EXCEL such as Solver and wizards are add-in files. You can use this add-in file by simply copying the file to the appropriate XLSTART sub-directory or in the AddIns sub-directory and use Add-in under the Tools menu item. For complete instructions consult EXCEL help. Each time EXCEL loads, it will load the Geofunc.xla file and all of the functions it contains will be available. The functions can be used like any other EXCEL built-in function by typing them into a formula with the appropriate arguments. If you use functions through the f x icon, the functions in Geofunc will be listed in alphabetical order under the User-defined category. The Visual Basic code for each function is listed below with comments that describe what the function does and the assumed input and output measurement units. The functions are organized alphabetically as follows:

63. Riverdeep | Tangible Math | Geometry Inventor | Circles And Pi Unit, geometry Inventor. Activity, Circles and pi, http://www.riverdeep.net/math/tangible_math/tm_activity_pages/geometry_inventor/

Home Login Store About Us ... Top Products Site Search By Grade Elementary (PreK-6) Middle School (6-9) High School (9-12) Teacher Support Destination Math Tangible Math Correlations edConnect To find support materials for Tangible Math activities, including lesson plans and student handouts, you will first need to select a Tangible Math unit using the tabs below and then a specific activity listed underneath. Product Tangible Math Unit Geometry Inventor Activity Circles and Pi Overview Students: Show that a circle is the path of a point moving so that it is a fixed distance from a fixed point. Show that a circle divides a plane into three regions: inside, on, and outside the circle. Show that pi is the ratio circumference/diameter and that it is a constant equal to ~3.14. User Name: Password: Support About Us Contact Us Become a Riverdeep Reseller ... Terms and Conditions Questions? Call

64. Geometric Figures Fast Facts Pythagorean Theorem, Trigonometric ratios, circles, coordinate geometry, lines, quadrilaterals diameter divided by two; Circumference equals pi times diameter http://www.mccc.edu/~kelld/page1400.html

GEOMETRIC FIGURES FAST FACTS: Take me to an interactive tutorial involving circumference and area of circles. Take me to an interactive tutorial involving perimeter and area of polygons : squares, rectangles, parallelograms, triangles, and trapezoids. Take me to a site that will give me some nice reference materials and a few memory aids. How about a site that is a comprehensive study guide for geometry-related topics ranging from: right triangles, similar polygons, ratios in special right triangles, Pythagorean Theorem, Trigonometric ratios, circles, coordinate geometry, lines, quadrilaterals, polygons, and more. A rectangle : Perimeter equals two times the length plus two times the width; Area equals the length times the width. P = 2L + 2W A = LW A square : Perimeter equals four times the side; Area equals the side squared. P = 4S A = S A circle : Diameter equals two times the radius; Radius equals diameter divided by two; Circumference equals pi times diameter or two times the radius times pi. Area is pi times the square of the radius. (pi equals approximately 3.14)

65. Volume Calculations For Cylinder Shaped Balloons -- Geometry Of Scaling The perimeter is the most elegant geometric measurement. its circumference, is divided by its diameter, the result is 3.1416 , or pi. http://www.overflite.com/volume.html

Volume Calculations for Cylinder Shaped Model Hot air Balloons Homemade Plastic Bag Model Hot Air Balloons are generally shaped like pillowcases. To calculate volumes, they can be compared with classic cylinders. Here a balloon is imagined as a stack of circles, or ovals, with a top. The volume is equal to the average area of the ovals, multiplied by an "Effective Height," which accounts for the material at the top of the balloon. The simplest way to calculate areas for circles and ovals is to compare them with squares. The perimeter is the most elegant geometric measurement. The quarter-perimeter though is usually more practical to use. NOTE: If the Perimeter of a circle, ie. its circumference, is divided by its diameter, the result is 3.1416..., or Pi. So, the Perimeter / 2 Pi equals the radius. Similarly, 2 * Quarter-Perimeter / Pi also equals the radius. Area of a Square = Quarter-Perimeter ^ 2 (ie. the square of one of the sides) Area of a Square = Perimeter ^2 / 16 Area of a Circle = Pi * Radius ^2 = Pi * ( 2 * Quarter-Perimeter / Pi ) ^2 = 1.273 * Quarter-Perimeter ^ 2

66. Geometric Formulas - Colby Community College Mathematics Department Information Geometric Formulas. (pi = = 3.141592 ). Areas. square = a 2. rectangle = ab. parallelogram = bh. trapezoid = h/2 (b 1 + b 2 ). circle = pi r 2. ellipse = pi r 1 r 2. http://colbycc.edu/www/math/geometry/areasvols.htm

Geometric Formulas pi Areas square = a rectangle = ab parallelogram = bh trapezoid = h/2 (b + b circle = pi r ellipse = pi r r triangle = (1/2) b h equilateral triangle = [ (3)/2] a (3/4) a triangle given SAS = (1/2) a b sin C triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula) Volumes cube = a rectangular prism = a b c irregular prism = b h cylinder = b h = pi r h pyramid = (1/3) b h cone = (1/3) b h = 1/3 pi r h sphere = (4/3) pi r ellipsoid = (4/3) pi r r r Surface Area cube = 6 a prism: (lateral area) = perimeter( b ) L (total area) = perimeter( b ) L + 2 b sphere = 4 pi r Summary: A collection of math related tables, facts, information and formulas ... with the main title and index page located at: http://www.colbycc.org/www/math/math.htm . These pages are maintained by Colby Community College and the Colby Community College Mathematics Department. If you have any comments, suggestions or a page that you've constructed (that you believe would be valuable and appropriate to include on/in/at our math site), please contact our CCC Math Department Faculty Page Keywords: Mathematics, Mathematic, Mathematical, Math, Mathematically, Reference, Table, Tables, Formula, Formulas, Fact, Facts, Information, Data, Computation, Computational, Algebra, Geometry, Trigonometry, Trig, Analysis, Calculus, Calc, Addition, Subtraction, Multiplication, Division, Powers, Roots, Exponents, Identities, Chart, Graph, List, Java, On-line, Games, Puzzles, Listing, Equations, Equation, Graphs, Graph, Conversions, Convert, Number, Numerical, Factorial, Sequence, Series, Conic, Integral, Differential, Mensuration, Measure, Function, Statistics, Stat, Math Reference.

67. AVirtualSpaceTimeTravelMachine : DETERMINISTIC FRACTAL GEOMETRY PICTURE (IMAGE D 2 pi rotation about Y and Z axes of a quaternionic Julia set rotation de 2 pi autour des axes Y et Zd un ensemble de Julia dans le corps des quaternions. http://www.lactamme.polytechnique.fr/Mosaic/images/JU.b1.16.D/display.html

2 pi rotation about Y and Z axes of a quaternionic Julia set [ rotation de 2 pi autour des axes Y et Z d'un ensemble de Julia dans le corps des quaternions (mpg). Some beautiful points of view from this rotation: (this picture was created on 05/23/1995) (this page -belonging to the CMAP28 site- was last updated on 06/01/2004 19:41:18 -CEST-) [Go back to AVirtualSpaceTimeTravelMachine [ Retour a AVirtualSpaceTimeTravelMachine The Y2K bug [ Le bug de l'an 2000 ... Courrier

68. AVirtualSpaceTimeTravelMachine : DETERMINISTIC FRACTAL GEOMETRY PICTURE (IMAGE D artistic view of a 2 pi rotation about the Y axis of a quaternionic Julia set vue artistique de la rotation de 2 pi autour de l axe Yd un ensemble de Julia http://www.lactamme.polytechnique.fr/Mosaic/images/JU.T1.m.16.D/display.html

artistic view of a 2 pi rotation about the Y axis of a quaternionic Julia set [ vue artistique de la rotation de 2 pi autour de l'axe Y d'un ensemble de Julia dans le corps des quaternions (mpg). (this picture was created on 04/14/1995) (this page -belonging to the CMAP28 site- was last updated on 06/01/2004 19:39:44 -CEST-) [Go back to AVirtualSpaceTimeTravelMachine [ Retour a AVirtualSpaceTimeTravelMachine The Y2K bug [ Le bug de l'an 2000 ... Courrier

69. Squaring The Circle But we know that it is impossible to ever get the square root of pi from rational numbers in this way. Up geometry Forum Articles http://www.geom.uiuc.edu/docs/forum/square_circle/

Up: Geometry Forum Articles Squaring the Circle Article: 125 of geometry.college Newsgroups: geometry.college From: sander@geom.umn.edu (Evelyn Sander) Subject: Squaring the Circle 1 Organization: University of Minnesota, Twin Cities Date: Tue, 21 Dec 1993 20:26:40 GMT Lines: 143 The following is a proof that given an arbitrary circle, it is impossible to construct a square of the same area using only straight edge and compass. People started to try to solve the classical Greek problem of squaring a circle by construction around 200 BC. It was finally proven impossible in 1882, when Lindemann proved that pi was trancendental. (I will omit the proof of this here.) Here are the basic ideas of the proof, following closely the discussion by Courant and Robbins in "What Is Mathematics." Given an arbitrary circle, let us define our unit of measure to be the radius of the circle. That means that the circle has area pi. Therefore, in order to construct a square of equal area, we need to construct the side of the square, which must have length square root of pi. I show that this is impossible. First, I must explain what it actually means to say that pi is transendental. This means that pi does not satisfy any rational polynomial equation. In other words, there is no equation of the form a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0)=0 with a's all rational, which holds for x=pi. In particular, this means we cannot find pi by a finite number of applications of the operations of addition, subtraction, multiplication, division, and taking nth roots of rational numbers. Note that it is also impossible to retrieve sqrt(pi) in this way. Now I show that these operations are the only ones available using construction.

70. N-Dimensional Volumes Thus the volume of this ball is 4/3*pi*(1/2)^3=pi/6. Paul Burchard, a postdoc at the geometry Center, showed me how to extend these results to n dimensions http://www.geom.uiuc.edu/docs/forum/ndvolumes/

Up: Geometry Forum Articles Volumes in nD Using Basic High School Geometry Article: 90 of geometry.college Xref: news1.cis.umn.edu geometry.pre-college:146 geometry.college:90 Newsgroups: geometry.pre-college,geometry.college From: sander@geom.umn.edu (Evelyn Sander) Subject: Volumes in nD Using Basic High School Geometry Organization: University of Minnesota, Twin Cities Date: Wed, 25 Aug 1993 20:17:37 GMT Lines: 108 This is a description of a geometric means to calculate the volume of a n-ball inscribed in a n dimensional hypercube. In two dimensions we know that the disk inscribed in the unit square has radius 1/2 and therefore area pi*(1/2)^2=pi/4. In three dimensions, the ball inscribed in the unit cube again has radius 1/2, where sqrt means take the square root. Thus the volume of this ball is 4/3*pi*(1/2)^3=pi/6. Paul Burchard, a postdoc at the Geometry Center, showed me how to extend these results to n dimensions (denoted R^n) without using more than basic high school geometry and a few pictures. The extension turns out to be a recursive relation based on the two and three dimensional results: the n-ball inscribed in the unit hypercube has volume equal to pi/(2*n) times the volume of the (n-2)-ball. Perhaps this gives a good way to introduce a high school geometry course to higher dimensional spaces. In the course of reading the article, please look at the associated figures. We can think of the cone over the (n-1)-sphere as a union of cones over infinitesimal n-1 dimensional cubes. Note that the volume of a cone over a flat object only depends on the distance from the plane of the object to the origin.

71. Geometry Review geometry Review. We have used several simple facts here A triangle inscribed in a semicircle, as shown below, is a right triangle. http://personal.bgsu.edu/~carother/pi/geometry.html

Geometry Review We have used several simple facts here: A triangle inscribed in a semicircle, as shown below, is a right triangle. [Proof] [Return to Archimedes' method] Given triangle inscribed in semicircle , as shown below, the central angle is twice the angle [Proof] [Return to Archimedes' method] [Main Index] [Pi Index] ... Neal Carothers - carother@bgnet.bgsu.edu

72. Geometry Review geometry Review, Part II. Given triangle DAC inscribed in semicircle , as shown below, the central angle is twice the angle . To http://personal.bgsu.edu/~carother/pi/geometry-p2.html

Geometry Review, Part II Given triangle DAC inscribed in semicircle , as shown below, the central angle is twice the angle To prove this, we use the fact that is a right angle and the fact that the angles in any triangle sum to Summing the angles in triangle BDC , we get , or Since triangle ADB is isosceles (two of its legs are radii of the circle), the missing angle must be equal to [Return to geometry review] [Return to Archimedes's method] [Main Index] [Pi Index] ... Neal Carothers - carother@bgnet.bgsu.edu

73. Perplexus.info :: Geometry : What Can You Prove? Home Shapes geometry Assuming AC=1, CD=2, DB=1 Angle CDB 89 Diagonal CB B3 2.220403201 (=SQRT(1+44*COS($E$2*pi()/180))) Angle BCD B4 26.76293713 (=ASIN(SIN http://perplexus.info/show.php?pid=1380&cid=11061

74. Perplexus.info :: Geometry : Like Clockwork Home Shapes geometry Like Clockwork (Posted on 200402-27), I started with theta_M(t)=pi/2 - 2pi t and theta_H(t)=pi/2 - (pi/6)t where t is in hours. http://perplexus.info/show.php?pid=1565&cid=12486

75. Cynthia Lanius' Lessons: The History Of Geometry Egyptians, c. 2000 500 BC. Ancient Egyptians demonstrated a practical knowledge of geometry through surveying and In the Rhind Papyrus, pi is approximated. http://math.rice.edu/~lanius/Geom/his.html

Cynthia Lanius Thanks to PBS for permission to use the Pyramid photo. History of Geometry Egyptians c. 2000 - 500 B.C. Ancient Egyptians demonstrated a practical knowledge of geometry through surveying and construction projects. The Nile River overflowed its banks every year, and the river banks would have to be re-surveyed. See a PBS Nova unit on those big pointy buildings. In the Rhind Papyrus, pi is approximated. Babylonians c. 2000 - 500 B.C. Ancient clay tablets reveal that the Babylonians knew the Pythagorean relationships. One clay tablet reads 4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth. Greeks c. 750-250 B.C. Ancient Greeks practiced centuries of experimental geometry like Egypt and Babylonia had, and they absorbed the experimental geometry of both of those cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic. Euclid's (400BC) important geometry book The Elements formed the basis for most of the geometry studied in schools ever since.

76. Enumerative Real Algebraic Geometry: Bibliography P. PEDERSEN AND B. STURMFELS, Mixed monomial bases, in Algorithms in Algebraic geometry and Applications pi, M. piERI, Sul problema degli spazi secanti, Rend. http://www.math.umass.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

77. Schwarzschild Geometry diagram represents a 3dimensional spatial sphere of circumference 2 pi r. Dark The Schwarzschild spacetime geometry appears ill-behaved at the horizon, the http://casa.colorado.edu/~ajsh/schwp.html

More about the Schwarzschild Geometry Back to Dive into the Black Hole Forward to White Holes and Wormholes Andrew Hamilton's Homepage Other Relativity and Black Hole links index movies approach orbit singularity dive ... links Schwarzschild geometry A description of this embedding diagram appears below. Try John Walker's Orbit's in Strongly Curved Spacetime for a Java applet which allows you to play around with orbits in the Schwarzschild geometry. Schwarzschild radius One of the remarkable predictions of Schwarzschild's geometry was that if a mass M were compressed inside a critical radius r s , nowadays called the Schwarzschild radius, then its gravity would become so strong that not even light could escape. The Schwarzschild radius r s of a mass M is given by r s where G is Newton's gravitational constant , and c is the speed of light . For a 30 solar mass object, like the black hole in the fictional star system here, the Schwarzschild radius is about 100 kilometers. Curiously, the Schwarzschild radius had already been derived (with the correct result, but an incorrect theory) by John Michell in 1783 (this reference is from Erk's Relativity Pages ) in the context of Newtonian gravity and the corpuscular theory of light. Michel derived the critical radius by setting the gravitational escape velocity

78. Film Ideas: Geometry Perimeter, Circumference pi This video presents a general explanation of the content of the perimeter of a geometric figure, including circumference of a http://www.filmideas.com/geometry.html

Geometry "Now I See It" Geometry Video Series Angles This module gives an historical overview of the development of measurement including an explanation of the various units of measure in common usage, stresses the need for making accurate measurements and demonstrates various instruments for measuring, with emphasis on measuring angles. Geometric Constructions This video module demonstrates a variety of geometric constructions with a compass and straight edge, including bisection of a line, bisection of an angle, perpendicular from a point to line. Students worksheets provide reinforcement and practice. Triangles This video module explains the classification of triangles by their sides (isosceles, scalene, equilateral) and by their angles (acute, obtuse, right). Student worksheets give students practice in classifying triangles in a variety of ways. This video presents a general explanation of the content of the perimeter of a geometric figure, including circumference of a circle. Worksheets provide practice in finding perimeter and circum-ference and a wide variety of geometric figures. Area of Common Geometric Figures This video module develops the concept of area measurement and uses it to develop some formulas for the area of geometric figures. Student worksheets provide practice in finding the area of geometric figures both regular and irregular shaped.

79. Radical Pi Talk MATH CLUB TALK The Discovery of NonEuclidean geometry Professor Susan Goldstine. Wednesday, November 7 at 500 in MW 724. Euclid s http://www.math.ohio-state.edu/~goldstin/noneuclidean.html

MATH CLUB TALK The Discovery of Non-Euclidean Geometry Professor Susan Goldstine Wednesday, November 7 at 5:00 in MW 724 Euclid's Fifth Postulate (paraphrased): Given a line and a point not on that line, there is exactly one line through the given point parallel to the given line. Euclid's Parallel Postulate spawned the longest-standing controversy in the history of mathematics. Could it really be that the fifth postulate does not follow from the other four? Why does Euclid prove the converse of his fifth postulate but not the fifth postulate itself? Only after two thousand years of failed attempts to demonstrate the Parallel Postulate was it finally established that the statement cannot be proven or disproven from Euclid's other postulates, and that assuming its opposite yields a new and radically different form of plane geometry. The talk will give an overview of the events leading to this discovery, which heralded a revolution in modern mathematics, and a description of the properties of the non-Euclidean plane. Technical prerequisites: basic Euclidean geometry (e.g., the sum of the angles in a triangle is 180 degrees) and a little bit of trigonometry, although the trigonometry is not essential.

80. Hybridization same as. The triple bond consists of one sigma bond and two pi bonds. The geometry around each carbon is linear with a bond angle of 180 o . http://www.towson.edu/~ladon/carbon.html

HYBRIDIZATION OF CARBON The element, carbon, is one of the most versatile elements on the periodic table in terms of the number of compounds it may form. It may form virtually an infinite number of compounds. This is largely due to the types of bonds it can form and the number of different elements it can join in bonding. Carbon may form single, double and triple bonds. The hybridization of carbon involved in each of these bonds will be investigated in this handout. Bonding in any element will take place with only the valence shell electrons. The valence shell electrons are found in the incomplete, outermost shell. By looking at the electron configuration, one is able to identify these valence electrons. Let's look at the electron configuration of ground state (lowest energy state) carbon: From the ground state electron configuration, one can see that carbon has four valence electrons, two in the 2s subshell and two in the 2p subshell. The 1s electrons are considered to be core electrons and are not available for bonding. There are two unpaired electrons in the 2p subshell, so if carbon were to hybridize from this ground state, it would be able to form at most two bonds. Recall that energy is released when bonds form, so it would be to carbon's benefit to try to maximize the number of bonds it can form. For this reason, carbon will form an excited state by promoting one of its 2s electrons into its empty 2p orbital and hybridize from the excited state. By forming this excited state, carbon will be able to form four bonds. The excited state configuration is shown below:

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

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