Parallel Postulate

Total Page：16

File Type：pdf, Size：1020Kb

Homework 7 - Parallel postulate Cristina Hernandez October 7, 2015 1. Read the excerpts "History of the Parallel Postulate" and the "Discovery of Hy- perbolic Gemometry", from Euclidean and Non-Euclidean Geometries: Development and History by Greenberg. 2. P roof: A Saccheri quadrilateral is a quadrilateral whose base angles are right angles and whose base-adjacent sides are congruent to each other. The summit angles can either be right, acute, or obtuse. By I1 through I3 we can draw diagonals. With the diagonals we have created two triangles ACD and BDC that have right angles at the base of the quadrilateral that by C6, the two triangles are congruent and have congruent interior angles. Therefore the diagonals AD and BC are congruent. There are two pairs of congruent angles ACD and BDC and then BCD and ADC. Therefore angle ACB and angle BDA are congruent. The triangles ACD and BDC and the triangles CBA and BAB give us the congruent angles CBA and DAB, so the summit angles are congruent. Thus, the summit angles of a Saccheri quadrilateral are congruent. 3. Case 3 cannot happen in Neutral Geometry, it would lead to a contradiction. Accord- ing to the excerpt "History Of The Parallel Postulate", Greenberg says that "if the summit angles were obtuse, the angle sum of the quadrilateral would be more than 360 degrees, contradicting Corollary 2 to the Saccheri-Legendre Theorem." The Saccheri-Legendre The- 1 orem says that the angle sim of a triangle cannot exceed 180. Therefore, the summit angles of a Saccheri quadrilateral cannot be obtuse. 4. Case 1 In this case we get a regular rectangle. If you create diagonals in this case you get two congruent trianlges and their interior angles add up to 180 degrees. It will hold also hold for Euclid's 5th postulate. Case 2 2 In this case the summit angles are acute so if you were to make triangles with the diagonals, they would add up to be less that 180 degrees. We read that this later led to the discovery of Hyperbolid Geometry. 5. The difference between Saccheri and Lambert Squares. A Lambert Quadrilateral has three right angles and two equal sides. A Saccheri Quadri- lateral has to right angles at the base and we proved that the two summit angles can be acute or right as well. My first encounter with these two was that I didn't think that quadrilaterals could have non-straight lines. Now that we have learned about different geometries, we know that polygons can come in many forms depending on which geometry we are working with. In homework 2 we constructed a square from a given line segment. This square turned 3 out to be a right saccheri square, a right lambert square, and a right angle rhombus. Here are some examples. Here we have an acute Lambert Square. Here we have an acute Saccheri Square. 4 Bibliography Greenberg. (n.d.). History Of The Parallel Postulate. In Euclidean and Non-Euclidean Geometries (pp. 148-163). 5.

Copy Link

Recommended publications

Downloaded from Bookstore.Ams.Org 30-60-90 Triangle, 190, 233 36-72
Index 30-60-90 triangle, 190, 233 intersects interior of a side, 144 36-72-72 triangle, 226 to the base of an isosceles triangle, 145 360 theorem, 96, 97 to the hypotenuse, 144 45-45-90 triangle, 190, 233 to the longest side, 144 60-60-60 triangle, 189 Amtrak model, 29 and (logical conjunction), 385 AA congruence theorem for asymptotic angle, 83 triangles, 353 acute, 88 AA similarity theorem, 216 included between two sides, 104 AAA congruence theorem in hyperbolic inscribed in a semicircle, 257 geometry, 338 inscribed in an arc, 257 AAA construction theorem, 191 obtuse, 88 AAASA congruence, 197, 354 of a polygon, 156 AAS congruence theorem, 119 of a triangle, 103 AASAS congruence, 179 of an asymptotic triangle, 351 ABCD property of rigid motions, 441 on a side of a line, 149 absolute value, 434 opposite a side, 104 acute angle, 88 proper, 84 acute triangle, 105 right, 88 adapted coordinate function, 72 straight, 84 adjacency lemma, 98 zero, 84 adjacent angles, 90, 91 angle addition theorem, 90 adjacent edges of a polygon, 156 angle bisector, 100, 147 adjacent interior angle, 113 angle bisector concurrence theorem, 268 admissible decomposition, 201 angle bisector proportion theorem, 219 algebraic number, 317 angle bisector theorem, 147 all-or-nothing theorem, 333 converse, 149 alternate interior angles, 150 angle construction theorem, 88 alternate interior angles postulate, 323 angle criterion for convexity, 160 alternate interior angles theorem, 150 angle measure, 54, 85 converse, 185, 323 between two lines, 357 altitude concurrence theorem,
[Show full text]
Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used
Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the ﬁrst 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Deﬁnitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent.
[Show full text]
The Algebra of Projective Spheres on Plane, Sphere and Hemisphere
Journal of Applied Mathematics and Physics, 2020, 8, 2286-2333 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 The Algebra of Projective Spheres on Plane, Sphere and Hemisphere István Lénárt Eötvös Loránd University, Budapest, Hungary How to cite this paper: Lénárt, I. (2020) Abstract The Algebra of Projective Spheres on Plane, Sphere and Hemisphere. Journal of Applied Numerous authors studied polarities in incidence structures or algebrization Mathematics and Physics, 8, 2286-2333. of projective geometry [1] [2]. The purpose of the present work is to establish https://doi.org/10.4236/jamp.2020.810171 an algebraic system based on elementary concepts of spherical geometry, ex- tended to hyperbolic and plane geometry. The guiding principle is: “The Received: July 17, 2020 Accepted: October 27, 2020 point and the straight line are one and the same”. Points and straight lines are Published: October 30, 2020 not treated as dual elements in two separate sets, but identical elements with- in a single set endowed with a binary operation and appropriate axioms. It Copyright © 2020 by author(s) and consists of three sections. In Section 1 I build an algebraic system based on Scientific Research Publishing Inc. This work is licensed under the Creative spherical constructions with two axioms: ab= ba and (ab)( ac) = a , pro- Commons Attribution International viding finite and infinite models and proving classical theorems that are License (CC BY 4.0). adapted to the new system. In Section Two I arrange hyperbolic points and http://creativecommons.org/licenses/by/4.0/ straight lines into a model of a projective sphere, show the connection be- Open Access tween the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry.
[Show full text]
Saccheri and Lambert Quadrilateral in Hyperbolic Geometry
MA 408, Computer Lab Three Saccheri and Lambert Quadrilaterals in Hyperbolic Geometry Name: Score: Instructions: For this lab you will be using the applet, NonEuclid, developed by Castel- lanos, Austin, Darnell, & Estrada. You can either download it from Vista (Go to Labs, click on NonEuclid), or from the following web address: http://www.cs.unm.edu/∼joel/NonEuclid/NonEuclid.html. (Click on Download NonEuclid.Jar). Work through this lab independently or in pairs. You will have the opportunity to finish anything you don't get done in lab at home. Please, write the solutions of homework problems on a separate sheets of paper and staple them to the lab. 1 Introduction In the previous lab you observed that it is impossible to construct a rectangle in the Poincaré disk model of hyperbolic geometry. In this lab you will study two types of quadrilaterals that exist in the hyperbolic geometry and have many properties of a rectangle. A Saccheri quadrilateral has two right angles adjacent to one of the sides, called the base. Two sides that are perpendicular to the base are of equal length. A Lambert quadrilateral is a quadrilateral with three right angles. In the Euclidean geometry a Saccheri or a Lambert quadrilateral has to be a rectangle, but the hyperbolic world is different ... 2 Saccheri Quadrilaterals Definition: A Saccheri quadrilateral is a quadrilateral ABCD such that \ABC and \DAB are right angles and AD ∼= BC. The segment AB is called the base of the Saccheri quadrilateral and the segment CD is called the summit. The two right angles are called the base angles of the Saccheri quadrilateral, and the angles \CDA and \BCD are called the summit angles of the Saccheri quadrilateral.
[Show full text]
The Role of Definitions in Geometry for Prospective Middle and High School Teachers
2018 HAWAII UNIVERSITY INTERNATIONAL CONFERENCES STEAM - SCIENCE, TECHNOLOGY & ENGINEERING, ARTS, MATHEMATICS & EDUCATION JUNE 6 - 8, 2018 PRINCE WAIKIKI, HONOLULU, HAWAII THE ROLE OF DEFINITIONS IN GEOMETRY FOR PROSPECTIVE MIDDLE AND HIGH SCHOOL TEACHERS SZABO, TAMAS DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN WHITEWATER, WISCONSIN Dr. Tamas Szabo Department of Mathematics University of Wisconsin Whitewater, Wisconsin THE ROLE OF DEFINITIONS IN GEOMETRY FOR PROSPECTIVE MIDDLE AND HIGH SCHOOL TEACHERS ABSTRACT. Does verbalizing a key definition before solving a related problem help students solving the problem? Perhaps surprisingly, the answers turns out to be NO, based on this yearlong experiment conducted with mathematics education majors and minors in two geometry classes. This article describes the experiment, the results, suggests possible explanations, and derives some peda-gogical conclusions that are useful for teachers of any mathematics or science course. Keywords: concept definition, concept image, proof writing, problem solving, secondary mathematics teachers. 1. INTRODUCTION Definitions play a very important role in mathematics, the learning of mathematics, and the teaching of mathematics alike. However, knowing a definition is not equivalent with knowing a concept. To become effective problem solvers, students need to develop an accurate concept image. The formal definition is only a small part of the concept image, which also includes examples and non-examples, properties and connections to other concepts. The distinction between concept definition and concept image has been examined by many research studies (e.g., Edwards, 1997; Vinner and Dreyfus, 1989; Tall, 1992; Vinner, 1991). Another group of articles analyze, argue for, and provide examples of engaging students in the construction of definitions (e.g., Zandieh and Rasmussen, 2010; Herbst et al., 2005; Johnson et al., 2014; Zaslavski and Shir, 2005).
[Show full text]
Volume 6 (2006) 1–16
FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University b bbb FORUM GEOM Volume 6 2006 http://forumgeom.fau.edu ISSN 1534-1178 Editorial Board Advisors: John H. Conway Princeton, New Jersey, USA Julio Gonzalez Cabillon Montevideo, Uruguay Richard Guy Calgary, Alberta, Canada Clark Kimberling Evansville, Indiana, USA Kee Yuen Lam Vancouver, British Columbia, Canada Tsit Yuen Lam Berkeley, California, USA Fred Richman Boca Raton, Florida, USA Editor-in-chief: Paul Yiu Boca Raton, Florida, USA Editors: Clayton Dodge Orono, Maine, USA Roland Eddy St. John’s, Newfoundland, Canada Jean-Pierre Ehrmann Paris, France Chris Fisher Regina, Saskatchewan, Canada Rudolf Fritsch Munich, Germany Bernard Gibert St Etiene, France Antreas P. Hatzipolakis Athens, Greece Michael Lambrou Crete, Greece Floor van Lamoen Goes, Netherlands Fred Pui Fai Leung Singapore, Singapore Daniel B. Shapiro Columbus, Ohio, USA Steve Sigur Atlanta, Georgia, USA Man Keung Siu Hong Kong, China Peter Woo La Mirada, California, USA Technical Editors: Yuandan Lin Boca Raton, Florida, USA Aaron Meyerowitz Boca Raton, Florida, USA Xiao-Dong Zhang Boca Raton, Florida, USA Consultants: Frederick Hoffman Boca Raton, Floirda, USA Stephen Locke Boca Raton, Florida, USA Heinrich Niederhausen Boca Raton, Florida, USA Table of Contents Khoa Lu Nguyen and Juan Carlos Salazar, On the mixtilinear incircles and excircles,1 Juan Rodr´ıguez, Paula Manuel and Paulo Semi˜ao, A conic associated with the Euler line,17 Charles Thas, A note on the Droz-Farny theorem,25 Paris Pamﬁlos, The cyclic complex of a cyclic quadrilateral,29 Bernard Gibert, Isocubics with concurrent normals,47 Mowaffaq Hajja and Margarita Spirova, A characterization of the centroid using June Lester’s shape function,53 Christopher J.
[Show full text]
Tightening Curves on Surfaces Monotonically with Applications
Tightening Curves on Surfaces Monotonically with Applications † Hsien-Chih Chang∗ Arnaud de Mesmay March 3, 2020 Abstract We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of de Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and ∆Y -transformations for arbitrary integer k. Previous algorithms only existed in the planar setting when k 4, and all of them rely on extensive case-by-case analysis based on different values of k. Our algorithm≤ makes use of the connection between electrical transformations and homotopy moves, and thus solves the problem in a unified fashion.
[Show full text]
Saccheri Quadrilaterals Definition: Let Be Any Line Segment, and Erect Two
Saccheri Quadrilaterals Definition: Let be any line segment, and erect two perpendiculars at the endpoints A and B. Mark off points C and D on these perpendiculars so that C and D lie on the same side of the line , and BC = AD. Join C and D. The resulting quadrilateral is a Saccheri Quadrilateral. Side is called the base, and the legs, and side the summit. The angles at C and D are called the summit angles. Lemma: A Saccheri Quadrilateral is convex. ~ By construction, D and C are on the same side of the line , and by PSP, will be as well. If intersected at a point F, one of the triangles ªADF or ªBFC would have two angles of at least measure 90, a contradiction (one of the linear pair of angles at F must be obtuse or right). Finally, if and met at a point E then ªABE would be a triangle with two right angles. Thus and must lie entirely on one side of each other. So, GABCD is convex. Theorem: The summit angles of a Saccheri Quadrilateral are congruent. ~ The SASAS version of using SAS to prove the base angles of an isosceles triangle are congruent. GDABC GCBAD, by SASAS, so pD pC. Corollaries: • The diagonals of a Saccheri Quadrilateral are congruent. (Proof: ªABC ªBAD by SAS; CPCF gives AC = BD.) • The line joining the midpoints of the base and summit of a quadrilateral is the perpendicular bisector of both the base and summit. (Proof: Let N and M be the midpoints of summit and base, respectively.
[Show full text]
A Phenomenon in Geometric Analysis M
23 Indian Journal of Science and Technology Vol.2 No 4 (Apr. 2009) ISSN: 0974- 6846 A phenomenon in geometric analysis M. Sivasubramanian Department of Mathematics, Dr.Mahalingam College of Engg. & Technology, Pollachi, Tamil Nadu 642003, India [email protected] Abstract: It has been established that it is impossible to He recognized that three possibilities arose from omitting deduce Euclid V from Euclid I, II, III, and IV. The Euclid's Fifth; if two perpendiculars to one line cross investigations devoted to the parallel postulate gave rise another line, judicious choice of the last can make the to a number of equivalent propositions to this problem. internal angles where it meets the two perpendiculars Also, while attempting to prove this statement as a equal (it is then parallel to the first line). If those equal special theorem Gauss, Bolyai and Lobachevsky internal angles are right angles, we get Euclid's Fifth; independently found a consistent model of first non- otherwise, they must be either acute or obtuse. He Euclidean geometry namely hyperbolic geometry. persuaded himself that the acute and obtuse cases lead Gauss’s student Riemann developed another branch of to contradiction, but had made a tacit assumption non-Euclidean geometry which is known as Riemannian equivalent to the fifth to get there. geometry. The formulae of Lobachevskyian geometry Nasir al-Din al-Tusi (1201-1274), in his Al-risala al- widely used tot sty the properties of atomic objects in shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion quantum physics. Einstein’s general theory of relativity is Which Removes Doubt about Parallel Lines) (1250), nothing but beautiful application of Riemannian geometry.
[Show full text]
Arxiv:1808.05573V2 [Math.GT] 13 May 2020 A.K.A
THE MAXIMAL INJECTIVITY RADIUS OF HYPERBOLIC SURFACES WITH GEODESIC BOUNDARY JASON DEBLOIS AND KIM ROMANELLI Abstract. We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function. The main results of this paper relate to maximal injectivity radius among hyperbolic surfaces with geodesic boundary. For a point p in the interior of a hyperbolic surface F , by the injectivity radius of F at p we mean the supremum injrad p(F ) of all r > 0 such that there is a locally isometric embedding of an open metric neighborhood { a disk { of radius r into F that takes 1 the disk's center to p. If F is complete and without boundary then injrad p(F ) = 2 sysp(F ) at p, where sysp(F ) is the systole of loops at p, the minimal length of a non-constant geodesic arc in F with both endpoints at p. But if F has boundary then injrad p(F ) is bounded above by the distance from p to the boundary, so it approaches 0 as p approaches @F (and we extend it continuously to @F as 0). On the other hand, sysp(F ) does not approach 0 as p @F .
[Show full text]
A Brief Survey of Elliptic Geometry
A Brief Survey of Elliptic Geometry By Justine May Advisor: Dr. Loribeth Alvin An Undergraduate Proseminar In Partial Fulfillment of the Degree of Bachelor of Science in Mathematical Sciences The University of West Florida April 2012 i APPROVAL PAGE The Proseminar of Justine May is approved: ________________________________ ______________ Lori Alvin, Ph.D., Proseminar Advisor Date ________________________________ ______________ Josaphat Uvah, Ph.D., Committee Chair Date Accepted for the Department: ________________________________ ______________ Jaromy Kuhl, Ph.D., Chair Date ii Abstract There are three fundamental branches of geometry: Euclidean, hyperbolic and elliptic, each characterized by its postulate concerning parallelism. Euclidean and hyperbolic geometries adhere to the all of axioms of neutral geometry and, additionally, each adheres to its own parallel postulate. Elliptic geometry is distinguished by its departure from the axioms that define neutral geometry and its own unique parallel postulate. We survey the distinctive rules that govern elliptic geometry, and some of the related consequences. iii Table of Contents Page Title Page ……………………………………………………………………….……………………… i Approval Page ……………………………………………………………………………………… ii Abstract ………………………………………………………………………………………………. iii Table of Contents ………………………………………………………………………………… iv Chapter 1: Introduction ………………………………………………………………………... 1 I. Problem Statement ……….…………………………………………………………………………. 1 II. Relevance ………………………………………………………………………………………………... 2 III. Literature Review …………………………………………………………………………………….
[Show full text]
Chapter 4 Euclidean Geometry
Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles De¯nition 4.1 Two distinct lines ` and m are said to be parallel ( and we write `km) i® they lie in the same plane and do not meet. Terminologies: 1. Transversal: a line intersecting two other lines. 2. Alternate interior angles 3. Corresponding angles 4. Interior angles on the same side of transversal 56 Yi Wang Chapter 4. Euclidean Geometry 57 Theorem 4.2 (Parallelism in absolute geometry) If two lines in the same plane are cut by a transversal to that a pair of alternate interior angles are congruent, the lines are parallel. Remark: Although this theorem involves parallel lines, it does not use the parallel postulate and is valid in absolute geometry. Proof: Assume to the contrary that the two lines meet, then use Exterior Angle Inequality to draw a contradiction. 2 The converse of above theorem is the Euclidean Parallel Postulate. Euclid's Fifth Postulate of Parallels If two lines in the same plane are cut by a transversal so that the sum of the measures of a pair of interior angles on the same side of the transversal is less than 180, the lines will meet on that side of the transversal. In e®ect, this says If m\1 + m\2 6= 180; then ` is not parallel to m Yi Wang Chapter 4. Euclidean Geometry 58 It's contrapositive is If `km; then m\1 + m\2 = 180( or m\2 = m\3): Three possible notions of parallelism Consider in a single ¯xed plane a line ` and a point P not on it.
[Show full text]