Squaring the Circle in Elliptic Geometry
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Lesson 3: Rectangles Inscribed in Circles
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection . -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
And Are Congruent Chords, So the Corresponding Arcs RS and ST Are Congruent
9-3 Arcs and Chords ALGEBRA Find the value of x. 3. SOLUTION: 1. In the same circle or in congruent circles, two minor SOLUTION: arcs are congruent if and only if their corresponding Arc ST is a minor arc, so m(arc ST) is equal to the chords are congruent. Since m(arc AB) = m(arc CD) measure of its related central angle or 93. = 127, arc AB arc CD and . and are congruent chords, so the corresponding arcs RS and ST are congruent. m(arc RS) = m(arc ST) and by substitution, x = 93. ANSWER: 93 ANSWER: 3 In , JK = 10 and . Find each measure. Round to the nearest hundredth. 2. SOLUTION: Since HG = 4 and FG = 4, and are 4. congruent chords and the corresponding arcs HG and FG are congruent. SOLUTION: m(arc HG) = m(arc FG) = x Radius is perpendicular to chord . So, by Arc HG, arc GF, and arc FH are adjacent arcs that Theorem 10.3, bisects arc JKL. Therefore, m(arc form the circle, so the sum of their measures is 360. JL) = m(arc LK). By substitution, m(arc JL) = or 67. ANSWER: 67 ANSWER: 70 eSolutions Manual - Powered by Cognero Page 1 9-3 Arcs and Chords 5. PQ ALGEBRA Find the value of x. SOLUTION: Draw radius and create right triangle PJQ. PM = 6 and since all radii of a circle are congruent, PJ = 6. Since the radius is perpendicular to , bisects by Theorem 10.3. So, JQ = (10) or 5. 7. Use the Pythagorean Theorem to find PQ. -
Elliptic Geometry Hawraa Abbas Almurieb Spherical Geometry Axioms of Incidence
Elliptic Geometry Hawraa Abbas Almurieb Spherical Geometry Axioms of Incidence • Ax1. For every pair of antipodal point P and P’ and for every pair of antipodal point Q and Q’ such that P≠Q and P’≠Q’, there exists a unique circle incident with both pairs of points. • Ax2. For every great circle c, there exist at least two distinct pairs of antipodal points incident with c. • Ax3. There exist three distinct pairs of antipodal points with the property that no great circle is incident with all three of them. Betweenness Axioms Betweenness fails on circles What is the relation among points? • (A,B/C,D)= points A and B separates C and D 2. Axioms of Separation • Ax4. If (A,B/C,D), then points A, B, C, and D are collinear and distinct. In other words, non-collinear points cannot separate one another. • Ax5. If (A,B/C,D), then (B,A/C,D) and (C,D/A,B) • Ax6. If (A,B/C,D), then not (A,C/B,D) • Ax7. If points A, B, C, and D are collinear and distinct then (A,B/C,D) , (A,C/B,D) or (A,D/B,C). • Ax8. If points A, B, and C are collinear and distinct then there exists a point D such that(A,B/C,D) • Ax9. For any five distinct collinear points, A, B, C, D, and E, if (A,B/E,D), then either (A,B/C,D) or (A,B/C,E) Definition • Let l and m be any two lines and let O be a point not on either of them. -
+ 2R - P = I + 2R - P + 2, Where P, I, P Are the Invariants P and Those of Zeuthen-Segre for F
232 THE FOURTH DIMENSION. [Feb., N - 4p - (m - 2) + 2r - p = I + 2r - p + 2, where p, I, p are the invariants p and those of Zeuthen-Segre for F. If I is the same invariant for F, since I — p = I — p, we have I = 2V — 4p — m = 2s — 4p — m. Hence 2s is equal to the "equivalence" in nodes of the point (a, b, c) in the evaluation of the invariant I for F by means of the pencil Cy. This property can be shown directly for the following cases: 1°. F has only ordinary nodes. 2°. In the vicinity of any of these nodes there lie other nodes, or ordinary infinitesimal multiple curves. In these cases it is easy to show that in the vicinity of the nodes all the numbers such as h are equal to 2. It would be interesting to know if such is always the case, but the preceding investigation shows that for the applications this does not matter. UNIVERSITY OF KANSAS, October 17, 1914. THE FOURTH DIMENSION. Geometry of Four Dimensions. By HENRY PARKER MANNING, Ph.D. New York, The Macmillan Company, 1914. 8vo. 348 pp. EVERY professional mathematician must hold himself at all times in readiness to answer certain standard questions of perennial interest to the layman. "What is, or what are quaternions ?" "What are least squares?" but especially, "Well, have you discovered the fourth dimension yet?" This last query is the most difficult of the three, suggesting forcibly the sophists' riddle "Have you ceased beating your mother?" The fact is that there is no common locus standi for questioner and questioned. -
20. Geometry of the Circle (SC)
20. GEOMETRY OF THE CIRCLE PARTS OF THE CIRCLE Segments When we speak of a circle we may be referring to the plane figure itself or the boundary of the shape, called the circumference. In solving problems involving the circle, we must be familiar with several theorems. In order to understand these theorems, we review the names given to parts of a circle. Diameter and chord The region that is encompassed between an arc and a chord is called a segment. The region between the chord and the minor arc is called the minor segment. The region between the chord and the major arc is called the major segment. If the chord is a diameter, then both segments are equal and are called semi-circles. The straight line joining any two points on the circle is called a chord. Sectors A diameter is a chord that passes through the center of the circle. It is, therefore, the longest possible chord of a circle. In the diagram, O is the center of the circle, AB is a diameter and PQ is also a chord. Arcs The region that is enclosed by any two radii and an arc is called a sector. If the region is bounded by the two radii and a minor arc, then it is called the minor sector. www.faspassmaths.comIf the region is bounded by two radii and the major arc, it is called the major sector. An arc of a circle is the part of the circumference of the circle that is cut off by a chord. -
Geometry Course Outline
GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid -
Symmetry of Graphs. Circles
Symmetry of graphs. Circles Symmetry of graphs. Circles 1 / 10 Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Symmetry of graphs. Circles 2 / 10 Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. -
Chapter 14 Hyperbolic Geometry Math 4520, Fall 2017
Chapter 14 Hyperbolic geometry Math 4520, Fall 2017 So far we have talked mostly about the incidence structure of points, lines and circles. But geometry is concerned about the metric, the way things are measured. We also mentioned in the beginning of the course about Euclid's Fifth Postulate. Can it be proven from the the other Euclidean axioms? This brings up the subject of hyperbolic geometry. In the hyperbolic plane the parallel postulate is false. If a proof in Euclidean geometry could be found that proved the parallel postulate from the others, then the same proof could be applied to the hyperbolic plane to show that the parallel postulate is true, a contradiction. The existence of the hyperbolic plane shows that the Fifth Postulate cannot be proven from the others. Assuming that Mathematics itself (or at least Euclidean geometry) is consistent, then there is no proof of the parallel postulate in Euclidean geometry. Our purpose in this chapter is to show that THE HYPERBOLIC PLANE EXISTS. 14.1 A quick history with commentary In the first half of the nineteenth century people began to realize that that a geometry with the Fifth postulate denied might exist. N. I. Lobachevski and J. Bolyai essentially devoted their lives to the study of hyperbolic geometry. They wrote books about hyperbolic geometry, and showed that there there were many strange properties that held. If you assumed that one of these strange properties did not hold in the geometry, then the Fifth postulate could be proved from the others. But this just amounted to replacing one axiom with another equivalent one. -
Pappus of Alexandria: Book 4 of the Collection
Pappus of Alexandria: Book 4 of the Collection For other titles published in this series, go to http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences Managing Editor J.Z. Buchwald Associate Editors J.L. Berggren and J. Lützen Advisory Board C. Fraser, T. Sauer, A. Shapiro Pappus of Alexandria: Book 4 of the Collection Edited With Translation and Commentary by Heike Sefrin-Weis Heike Sefrin-Weis Department of Philosophy University of South Carolina Columbia SC USA [email protected] Sources Managing Editor: Jed Z. Buchwald California Institute of Technology Division of the Humanities and Social Sciences MC 101–40 Pasadena, CA 91125 USA Associate Editors: J.L. Berggren Jesper Lützen Simon Fraser University University of Copenhagen Department of Mathematics Institute of Mathematics University Drive 8888 Universitetsparken 5 V5A 1S6 Burnaby, BC 2100 Koebenhaven Canada Denmark ISBN 978-1-84996-004-5 e-ISBN 978-1-84996-005-2 DOI 10.1007/978-1-84996-005-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009942260 Mathematics Classification Number (2010) 00A05, 00A30, 03A05, 01A05, 01A20, 01A85, 03-03, 51-03 and 97-03 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. -
Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica
Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica John B. Little Department of Mathematics and CS College of the Holy Cross June 29, 2018 Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions I’ve always been interested in the history of mathematics (in addition to my nominal specialty in algebraic geometry/computational methods/coding theory, etc.) Want to be able to engage with original texts on their own terms – you might recall the talks on Apollonius’s Conics I gave at the last Clavius Group meeting at Holy Cross (two years ago) So, I’ve been taking Greek and Latin language courses in HC’s Classics department The subject for today relates to a Latin-to-English translation project I have recently begun – working with the Geometria Practica of Christopher Clavius, S.J. (1538 - 1612, CE) Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Overview 1 Introduction – Clavius and Geometria Practica 2 Book 6 and Greek approaches to duplication of the cube 3 Book 7 and squaring the circle via the quadratrix 4 Conclusions Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Clavius’ Principal Mathematical Textbooks Euclidis Elementorum, Libri XV (first ed. -
Math 109: Mathematics for Design
MATH 109: MATHEMATICS FOR DESIGN SUMMARY OF GOALS FOR THE COURSE In our contemporary culture the dialogue between math and art, while sometimes strained by Course details misunderstandings, is a dynamic and living one. Art Time: MWF 10:20 – 11:30am continues to inspire and inform mathematical Place: Mon, Wed PPHAC 235 thinking, and mathematics helps artists develop Fri PPHAC 112 additional insight when reasoning about the content and structure of their work. The tools of mathematics Instructor: Kevin Hartshorn also aid in the construction of conceptual Office: PPHAC 215 frameworks that are useful in all aspects of life. Hours: Mon/Wed 2:30-3:30pm, Tue/Thu 8:30-9:30am, This course will introduce students to ideas in or by appointment mathematical thinking that are related to artistic considerations. Students will need to show e-mail: [email protected] proficiency with some mathematical ideas and then Web: https://sites.google.com/a/moravian.edu/ apply those ideas in creating their own works of art. math-109-spring-2015 In the process, students will also be called to analyze Text: Squaring the Circle: Geometry in Art existing artwork with a mathematical eye. In this and Architecture, by Paul Calter way, students will be provided a new tool to use in their approach to art and aesthetics. KEY IDEAS FOR THIS COURSE Each assignment and class discussion will be aimed at expanding on these key notions: • Mathematics and mathematical thinking involve a creative effort, not just rote memorization. • There is a rich and complex connection between mathematics and art. • Very basic mathematical concepts can be used to solve seemingly complex real-world problems.