61. 1001newsgroups.com Sci.math Date, Sujet, Auteur. 21.02. Re Elementary geometry Question. Ken Pledger. 21.02.Excel. Jim. 21.02. Re pascals triangle. Dave Rusin. 21.02. Elementary Topology http://1001newsgroups.com/lmess.php?name=sci.math

62. The 'Phinest' Source To The Golden Section, Golden Mean, Divine Proportion, Fibo The Golden Section, Ratio or Mean, Geometric construction of phi in the goldensection and pascals triangle Phi s Phormula Pi, Phi Fibo s http://goldennumber.net/

GoldenNumber.net Home Contact Meet the Phi Guy "Phriends" in Phi Site Search ... News Dedicated to providing you with the phinest information on The Golden Section , Ratio or Mean The Divine Proportion The Fibonacci Series and the golden number, Phi ( NEW PhiMatrix Graphic Design Software Free Download Site Sections: Overview Design/Composition Life Mathematics ... Other Resources Welcome to GoldenNumber.Net - The Phi Source GoldenNumber.Net exists to share information on the pervasive appearance of Phi in life and the universe. Its goal is to present a broad sampling of phi related topics in an engaging and easy-to-understand format and to provide an online community (aka The Phi Nest), in which new findings about Phi can be shared. Select an area of interest below or START HERE . Enjoy the 'phi nomenon'! SITE CONTENTS Overview Design/Composition Life Mathematics ... Universe News: The Da Vinci Code Dan Brown's book "

63. The CTK Exchange Forums Pythagorean problem the math question is geometry show that a triangle with Pascalstriangle link What is the link between pascal s triangle and polygonal http://www.cut-the-knot.org/cgi-bin/dcforum/ctk.cgi?az=list&forum=DCForumID6&mm=

64. Biopasca In 1632, the pascals left Clermont for Paris, where Blaise s father took it uponhimself to Blaise Pascal taught himself geometry at the age of 12. http://www.andrews.edu/~calkins/math/biograph/biopasca.htm

Back to the Table of Contents Biography of A Mathematician: Pascal Table of Contents: Background An Early Achiever The Famous Triangle Working with Fermat ... Summary: Important Points Background Blaise Pascal, the only son of Etienne Pascal, was born on June 19, 1623 in what was Clermont (now Clermont-Ferrand), Auvergne, France. In 1632, the Pascals left Clermont for Paris, where Blaise's father took it upon himself to educate the family. Thus, Pascal was not allowed to study mathematics until the age of 15, and all math texts were removed from the house. Despite all this, Blaise's curiosity grew and he began to work on geometry himself at the age of 12. After discovering that the sum of the angles of a triangle is two right angles, his father relented and gave him a copy of a Euclidian geometry textbook. An Early Achiever Blaise Pascal made many discoveries between the ages of fourteen and twenty-four. At fourteen, he attended his father's geometry meetings, and at 16, he composed an essay on conic sections, which was published in 1640. Between the ages of 18 and 22, he invented a digital calculator, called a Pascaline, to assist his father in collecting taxes.

65. Biopasca fig. 2, Pascal s triangle, in table form. Working With Fermat Blaise Pascaltaught himself geometry at the age of 12. Pascal s works include Pensées, http://www.andrews.edu/~calkins/math/biograph/199899/biopasca.htm

Back to the Table of Contents Blaise Pascal: 1623-1662 fig. 1, Blaise Pascal, 1623-1662 Table of Contents: Background An Early Achiever The Famous Triangle Working with Fermat ... Summary: Important Points Background Blaise Pascal, the only son of Etienne Pascal, was born on June 19, 1623 in what was Clermont (now Clermont-Ferrand), Auvergne, France. In 1632, the Pascals left Clermont for Paris, where Blaise's father took it upon himself to educate the family. Thus, Pascal was not allowed to study mathematics until the age of 15, and all math texts were removed from the house. Despite all this, Blaise's curiosity grew and he began to work on geometry himself at the age of 12. After discovering that the sum of the angles of a triangle is two right angles, his father relented and gave him a copy of a Euclidian geometry textbook. An Early Achiever Blaise Pascal made many discoveries between the ages of fourteen and twenty-four. At fourteen, he attended his father's geometry meetings, and at 16, he composed an essay on conic sections, which was published in 1640. Between the ages of 18 and 22, he invented a digital calculator, called a Pascaline, to assist his father in collecting taxes.

66. Sierpinski Gasket A geometric method of creating the gasket is to start with a triangle and cut out Another way to create the Sierpinski gasket is via pascals triangle. http://astronomy.swin.edu.au/~pbourke/fractals/gasket/

Sierpinski Gasket Written by Paul Bourke March 1993 See also: Photographs by Gayla Chandler PovRay rendering by Angelo Pesce Geometric models of the Menger sponge level 3 plain level 4 efficient level 3 crumpled Introduction The following is an attempt to acquaint the reader with a fractal object called the Sierpinski gasket. The gasket was originally described in two dimensions but represents a family of objects in other dimensions. This family of objects will be discussed in dimensions 1, 2, 3, and an attempt will be made to visualise it in the 4th dimension. Cantor set or Dust The nineteenth century mathematician Georg Cantor became fascinated by the infinite number of points on a line segment. The set of points described here has been attributed to Cantor because of his attempts to imagine what happens when an infinite number of line segments are removed from an initial line interval. To generate the Cantor set start with a closed interval [0,1] (includes the points and 1). On the first iteration replace the interval with 3 equal length pieces and remove the middle third, or ]1/3, 2/3[ (excludes the points 1/3 and 2/3) Subsequent iterations involve removing the middle portion of the remaining line segments.. The gap removed each time is usually called a trema from the Latin tremes = termite.

67. Blaise Pascal (1623 - 1662) His early essay on the geometry of conics, written in 1639, but not published Pascal s arithmetical triangle, to any required order, is got by drawing a http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html

Blaise Pascal (1623 - 1662) From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball. Among the contemporaries of Descartes none displayed greater natural genius than Pascal, but his mathematical reputation rests more on what he might have done than on what he actually effected, as during a considerable part of his life he deemed it his duty to devote his whole time to religious exercises. Blaise Pascal Elements , a book which Pascal read with avidity and soon mastered. In 1650, when in the midst of these researches, Pascal suddenly abandoned his favourite pursuits to study religion, or, as he says in his , ``contemplate the greatness and the misery of man''; and about the same time he persuaded the younger of his two sisters to enter the Port Royal society. His famous Provincial Letters directed against the Jesuits, and his , were written towards the close of his life, and are the first example of that finished form which is characteristic of the best French literature. The only mathematical work that he produced after retiring to Port Royal was the essay on the cycloid in 1658. He was suffering from sleeplessness and toothache when the idea occurred to him, and to his surprise his teeth immediately ceased to ache. Regarding this as a divine intimation to proceed with the problem, he worked incessantly for eight days at it, and completed a tolerably full account of the geometry of the cycloid. I now proceed to consider his mathematical works in rather greater detail.

68. Blaise Pascal -- Facts, Info, And Encyclopedia Article At age 16, Pascal produced a treatise on ((geometry) a curve generated by (Click link for more info and facts about Pascal s triangle) Pascal s triangle http://www.absoluteastronomy.com/encyclopedia/b/bl/blaise_pascal.htm

Blaise Pascal [Categories: 1662 deaths, 1623 births, Jansenism, French physicists, French theologians, French mathematicians, French philosophers, Christian philosophers] Blaise Pascal (The Romance language spoken in France and in countries colonized by France) French (A person skilled in mathematics) mathematician (A scientist trained in physics) physicist , and religious (A specialist in philosophy) philosopher . Important contributions by Pascal to the natural sciences include the construction of mechanical calculators, considerations on (The branch of applied mathematics that deals with probabilities) probability theory , the study of fluids, and clarification of concepts such as (The force applied to a unit area of surface; measured in pascals (SI unit) or in dynes (cgs unit)) pressure and (An electrical home appliance that cleans by suction) vacuum . Following a (Click link for more info and facts about mystical) mystical experience in 1654, he fell away from mathematics and physics and devoted himself to reflection and writing about philosophy and (The rational and systematic study of religion and its influences and of the nature of religious truth) theology . He suffered from ill-health throughout his life and died two months after his 39th birthday. Family Born in (Click link for more info and facts about Clermont) Clermont , in the (A region in central France) Auvergne region of (A republic in western Europe; the largest country wholly in Europe)

69. Connect-ME - Weblinks This site investigate the properties, geometry and art of spirolaterals. Palindrome Pascal s triangle Venn Diagrams http://educ.queensu.ca/connectme/weblinks/strands.htm

Classroom Resources for Teachers Curriculum Strands Measurement Magic Squares - Math Forum http://mathforum.org/alejandre/magic.square.html Late Primary - Intermediate The Math Page - Skill In Arithmetic http://www.themathpage.com/ARITH/arithmetic.htm Multiplication: An Adventure in Number Sense http://naturalmath.com/mult/index.html This site illustrates patterns in multiplying 2 numbers up to 10 x 10. By using patterns and and 'twin facts' a 10 by 10 multiplication table is reduced to only 13 facts to be memorized. Numbers: Facts, Figures and Fiction http://richardphillips.org.uk/number/ Primary/Junior Numbers: Facts, Figures and Fiction Place Value http://www.sedl.org/scimath/compass/v03n02/place.html Primary The activities described in this lesson reinforce students' understanding of place value by using rhythm, physical action, and introspection. Simply Number Sense http://library.thinkquest.org/17888/opening.shtml

70. Sum Of Combinations - Physics Help And Math Help - Physics Forums Look at pascals triangle. What is the value of the sum of all the numbers in And I think the geometric proof is more useful in the sense that you have a http://physicsforums.com/showthread.php?t=83309

71. Blaise Pascal Early Life And Achievements At the age of twelve, Pascal started to study geometry himself; he discovered thatall the angles of a triangle add up to the sum of two ninety degree angles. http://cranfordschools.org/chs/scholars/2004/17c/backiel.html

72. Xah: Special Plane Curves: Conic Sections From right triangle PQA, we have PQ PA Cosa. Pascal s theorem (and itsdual) are important in projective geometry, which in turn is a fundamental http://www.xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.htm

If you spend more than 30 minutes on this site, please send $1 to me. Go to http://paypal.com/ and make a payment to xah@xahlee.org. Or send to: P. O. Box 390595, Mountain View, CA 94042-0290, USA. ☠Back to Table of Contents Conic Sections Intersections of parallel planes and a double cone, forming ellipses parabolas , and hyperbolas respectively. graphics code cone_cut.gcf Mathematica Notebook for This Page History ... Related Web Sites History Appollonius was the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola. A brief explanation of the naming can be found in Howard Eves, An Introduction to the History of Math. 6th ed. page 172. (also see J.H.Conway's newsgroup message, link at the bottom) In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, espcially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic sections is a rich classic topic that has spurred many developments in the history of mathematics. Description Hyperbola ellipse , and parabola are together known as conic sections, or just conics. So called because they are the intersection of a right circular cone and a plane.

73. Untitled Now it is a property of ``Pascal s triangle that. displaymath334 Sluse madea thorough study of Cavalieri and Torricelli on the geometry of the http://www.math.tamu.edu/~don.allen/history/precalc/precalc.html

Next: About this document April 2, 1997 Early Calculus I Albert Girard (1595-1632) - Theory of Equations Jan de Witt (1623-1672) - Analytic Geometry Marin Mersenne (1588-1648) - Scientific Journal/Society Girard Desargues (1591-1661) - Projective Geometry Frans von Schooten (1615-1660) - Analytic Geometry Christian Huygens (1629-1695) - Probability Johann Hudde Early Probability Early serious attempts at probability had already been attempted by Cardano and Tartaglia. They desired a better understanding of gambling odds. Some study about dice date even earlier. There are recorded attempts to understand odds dating back to Roman times. Cardano published Liber de Ludo Alea (Book on Games of Chance) in 1526. He discusses dice as well stakes games. He then computes fair stakes based on the number of outcomes. He was also aware of independent events and the multiplication rule: if A and B are independent events then Cardano discussed this problem: How many throws must be allowed to provide even odds for attaining two sixes on a pair of dice? Cardano reasoned it should be 18. He also argued that with a single dice, three rolls are required for even odds of rolling a 2. He was wrong. This type problem still challenges undergraduate math majors to this day.

74. Mathematics Class Syllabus Sequence Geometric Sequence Arithmetic Series Geometric Series Sigma Notation Sums of Series Binomial Expansion by pascals triangle Binomial Expansion by http://www.fuchsmizrachi.org/mrs.dyer/Syllabus.htm

Mathematics Class Syllabus This site includes Mrs. Dyer's current classes. ALGEBRA I COURSE CURRICULUM (PART 1) Introduction to Algebra Variables Grouping Symbols Equations Translating Words into Symbols Translating Sentences into Equations Translating Problems into Equations A Problem Solving Plan Number Lines Opposites and Absolute Value Rules of Real Numbers Basic Laws, Properties or Assumptions Addition on a Number Line Addition Rules for Real Numbers Subtraction Rules for Real Numbers The Distributive Law Multiplication Rules for Real Numbers Problem Solving w/ Consecutive Integers The Reciprocal of a Real Number Division Rules for Real Numbers Solving Equations Transforming Equations: Addition/Subtraction Transforming Equations: Multiplication/Division Using Several Transformations Equations w/Variables on Both Sides Word Problems Word Problems with Cost (Chart Setup) Rules of Polynomials Exponents Adding and Subtracting Polynomials Multiplying Monomials Powers of Monomials Multiplying Polynomials by Monomials Multiplying two Polynomials Transforming Formulas Word Problems (D=r x t) Word Problems with Area Word Problems with no Solution Factoring Polynomials Factoring Integers and GCF Dividing Monomials Monomials Factors of Polynomials Multiplying Binomials Mentally Factoring with Differences of two Squares

75. 142 Queries Inequality 64 Queries Symmetric 50 Queries triangle series7 solvable 7 queries limits 7 queries differential geometry 7 queries http://aux.planetmath.org/misc/pmsearches_200205230243.txt

76. Blaise Pascal According to his sister Gilberte, Pascal ``discovered geometry on his own. They taught the pascals about Jansenism and Blaise, who found Jansenist http://math.berkeley.edu/~robin/Pascal/

Accomplishments Pascal's Theorem Bibliography Back to the front page Blaise Pascal - Life Julia Chew Elements and from this time on allowed him to continue his studies in mathematics. (Bishop) Pascal's father then brought him into the society of mathematicians with whom he was associated with. The met every week to discuss current topics in science and math. (Bishop) Members of this group, headed by Mersenne, included other reknowned mathematicians such as Desargue, Roberval, Fermat and Descartes. (Davidson) At these meetings, Pascal was introduced to the latest developments in math. Soon he was making his own discoveries and publishing his own results. By the age of sixteen, he published his Essai pour les Coniques (1640) In the same year, the family moved to Rouen. Two years later, Pascal began working on his calculating machine which was completed in 1644. (Krailsheimer) The same year, Pascal found a new interest in physics. A family friend introduced the Pascals to Torricelli's experimet involving a tube of mercury turned upside down in a bowl also filled with mercury. They found that the mercury fell to a certain point in the tube and stopped. Pascal continued to conduct the experiment many times with variations. The results of his experiments and his conclusions were published in 1651 as Traite du vide (Treatise on the vacuum). (Davidson).

77. Triangle If one starts with Pascal s triangle with 2^n rows and colors the even numbers A triangle is one of the basic shapes of geometry a twodimensional http://www.websters-online-dictionary.org/definition/english/tr/triangle.html

Philip M. Parker, INSEAD. Triangle Definition: Triangle Triangle Noun . A three-sided polygon. . Any of various triangular drafting instruments used to draw straight lines at specified angles. . A percussion instrument consisting of a metal bar bent in the shape of an open triangle. Source: WordNet 1.7.1 Date "triangle" was first used in popular English literature: sometime before 1321. ( references Etymology: references Specialty Definition: Triangle Domain Definition Dream Interpretation To dream of a triangle , foretells separation from friends, and love affairs will terminate in disagreements. Source: Ten Thousand Dreams Interpreted ... Fine Arts Steel rods bent into the form of equilateral triangles; they are sounded with an iron rod. Source: European Union. references Hold the tripod feet in sockets and thus prevent them from sliding apart on smooth surfaces. Source: European Union. references Source: compiled by the editor from various references ; see credits. Top Specialty Definition: Sierpinski triangle (From Wikipedia , the free Encyclopedia) The Sierpinski triangle , also called the Sierpinski gasket , is a fractal, named after Waclaw Sierpinski.

78. IMACS factorials, binomial coefficients, Pascal s triangle, the product rule; Development of first order theories for the incidence geometry of the plane http://imacs.org/IMACSWeb/default.aspx?page=Mathematics

79. Math Research Project PASCAL S triangle; PERMUTATIONS; PI; PLATONIC SOLIDS. POLYHEDRONS; POLYNOMIALS Art geometry a study in space intuitions See pp. 8794. http://www.smithlib.org/page_young_adult_math_research_p.html

Math Research Project PARADOXES PASCAL'S TRIANGLE PERMUTATIONS PI ... PYTHAGOREAN THEOREM PARADOXES Agostini, Franco. Math and logic games See pp.91-104. Gardner, Martin. Knotted doughnuts and other mathematical entertainments See pp. 155-172. Time travel and other mathematical bewilderments See pp. 55-71. Gibilisco, Stan. Puzzles, paradoxes, and brain teasers See pp. 1-18. Hofstadter, Douglas R. Godel, Escher, Bach: an eternal golden braid Newman, James Roy. The world of mathematics: a small library of the literature of mathematics from A'h-mose the scribe to Albert Einstein See pp. 2369-84. Weaver, Warren. Lady luck: the theory of probability See pp. 163-9. PASCAL'S TRIANGLE Subject Headings: Pascals Triangle (see also Math Research Project Vertical File) Beckmann, Petr. A history of pi See pp. 121-3. Bell, Eric Temple. Men of mathematics See pp. 73-89. Boyer, Carl B. A history of mathematics See pp. 395-402. Eves, Howard Whitley. An introduction to the history of mathematics See pp. 259-63. Ivins, William Mills. See pp. 87-94. Jacobs, Harold R.

80. Triangle A triangle is one of the basic shapes of geometry a twodimensional figure withthree vertices and three sides which are straight line segments. http://www.fidosrevenge.com/auto/triangle.htm

Triangle Car Areas: Automobiles travel car audio automobile manufacturers ... road safety Car Types: internal combustion engine microcar convertible hatchback ... armored car Car Parts: engine carburetor fuel pump engine configuration ... axle Pages Related to: Triangle geometry radian polygon proportional ... synthetic geometry ''This article is about the geometric shape; for the musical instrument , see triangle (instrument) ; for the Raleigh Durham Chapel Hill region of North Carolina , see Research Triangle A triangle is one of the basic shapes of geometry : a two-dimensional figure with three vertices and three sides which are straight line segments. Table of contents 1 Types of triangles 2 Basic facts 3 Points, lines and circles associated with a triangle 4 Computing the area of a triangle ... 6 External links Types of triangles Triangles can be classified according to their side lengths. These classifications are: equilateral if all sides have the same length. If a triangle is equilateral then it is also equiangular (with all angles equal). isosceles if two sides have equivalent length. If a triangle is isosceles, then it will have the same number of equivalent angles as it has equivalent sides.

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