DOCSLIB.ORG

Circle Conjectures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Circle Conjectures Project AMP Dr. Antonio R. Quesada – Director, Project AMP Circle Conjectures Lab Summary: This lab will deal with certain properties of circles. In particular, eight conjectures will be covered in this lab. As each problem of the lab is completed, the student should discover certain properties that hold for circles. These properties relate to tangents lines, inscribed angles, and chords of circles. Upon completion if this lab, the student should be familiar with all these concepts. The lesson plan I have implemented involves conjectures of circles. These conjectures are certain properties on circles that the students will discover through working with the lab. Included in this lesson plan is: · A statement of the objectives this lesson will cover, that is, the conjectures to be discovered by the students. Each conjecture is defined and a diagram of each is provided. This is the knowledge the students are to have learned after lesson is complete. The teacher can use this to review the conjectures once the students have completed the labs to ensure that the students have a firm grasp of the material. This can also be used as a guide when grading the labs. · A list of definitions needed in order to complete the lab. These definitions are meant to be a review of material the student should have seen already. If needed, the teacher may want to briefly go over each of these terms to ensure that the students have a firm grasp on this material before starting the lab. The teacher could also provide each student with a copy of the definitions to be used as a reference when working on the lab. · A lab in which the students will explore through Cabri the conjectures to be covered in this lesson. The lab will help the students to discover for themselves each conjecture that is to be covered. By completing each lab problem, the student will learn the conjecture covered by that problem. The lab is a lesson plan in and of itself, in that, in order to complete the lab, the student must discover each conjecture. Therefore, it is up to the teacher to decide how much review is needed before starting the lab, and how much explanation is needed after the lab is completed to ensure the concepts have been grasped. Two extension questions are also provided in order to test and challenge what the students have learned from the lab. Evaluating their lab work will provide an evaluation of the students’ understanding of the material. Successful completion of the lab should result in the students having a firm grasp of the material. Key Words: Circles, tangents, radius, inscribed, perpendicular, chords, tangent segments Project AMP Dr. Antonio R. Quesada – Director, Project AMP Background knowledge: This lesson will concentrate on certain properties found in circles. The lesson is implemented through a lab in which the student will discover each property through experimentation using Cabri. Specifically, the conjectures covered will be: · Tangent Conjecture I: Any tangent line to a circle is perpendicular to the radius drawn to the point of tangency. · Tangent Conjecture II: Tangent segments to a circle from a point outside the circle are equal in length. · Inscribed Angles Conjecture I: In a circle, the measure of an inscribed angle is half the measure of the central angle with the same intercepted arc. Project AMP Dr. Antonio R. Quesada – Director, Project AMP · Inscribed Angles Conjecture II: In a circle, two inscribed angles with the same intercepted arc are congruent. · Inscribed Angles Conjecture III: Any angle inscribed in a semi-circle is a right angle. · Congruent Chords Conjecture: If two chords are congruent, then the following properties hold: 1. They determine central angles that are equal in measurement. 2. Their intercepted arcs are congruent. Also, the chords are equal distance from the center. · Perpendicular Bisector of a Chord Conjecture: The perpendicular bisector of a chord in a circle passes through the center of the circle. Project AMP Dr. Antonio R. Quesada – Director, Project AMP Learning Objectives: 1. Students will discover the eight conjectures covered in background knowledge. Materials: Geometry software Worksheets Procedures: Each of these conjectures is covered as a separate problem in the lab that follows. The student should be able to discover these conjectures themselves by working out the problems using Cabri. Two extension questions are given which challenge the student to put to use certain conjectures discovered earlier in order to come up with a solution. Extension I requires students to use the perpendicular bisector of a chord property in order to find the center of any given circle or arc. To do this, the student should construct two chords in the given circle or arc and bisect each with a perpendicular line. Where the two perpendicular lines intersect is the center of the circle. Extension II requires students to use the method found in Extension I to, given any arc, find the circumference of the circle the arc is a part of. To do this student should find the center of the circle using the method discovered in Extension I. Once the center is found, construct a radius and measure it. Use this measurement to find the circumference of the circle. Assessment: Completed worksheets Project AMP Dr. Antonio R. Quesada – Director, Project AMP Circle Conjectures Activity One: Tangent and Radii Team members’ names: __________________________________________________ File name: _____________________________________________________________ Goal: Investigate the angle formed by the tangent and the radius of a circle. a. Construct a circle A with radius AB . [use circle tool/segment tool] b. Construct a point, C on the circle and draw a secant that runs through B and C (see Fig. 1). [use point/line tool] c. Measure the angle Ð ABC that is formed. [use angle tool] suur d. Now grab and move point C around the circle until the line BC is tangent to the circle. What is the measurement of angle Ð ABC now? What conjecture can you make about a tangent and a radius to the point of tangency? e. Use this conjecture to construct a tangent line to a circle at a given point P. Fig. 1 Project AMP Dr. Antonio R. Quesada – Director, Project AMP Circle Conjectures Activity Two: Tangent Segments Team members’ names: __________________________________________________ File name: _____________________________________________________________ Goal: Investigate the relationship between tangent segments. a. Construct a circle A with radius AB . [use circle and segment tools] b. Construct a tangent line, t through B. [use perpendicular line tool] c. Construct another radius AD and a tangent line, s through C(see Fig2). [use point, segment, and line tools] d. Construct and label the point where lines t and s intersect as point D. [use intersection points and label tool] e. Measure the tangent segments DB and DC , what relationship exists? [use distance tool] f. Grab and move point B around the circle. What happens to the segment measures of DBand DC ? What conjecture can you make about tangent segments from the same exterior point? Fig. 2 Project AMP Dr. Antonio R. Quesada – Director, Project AMP Circle Conjectures Activity Three: Central and Inscribed angles Team members’ names: __________________________________________________ File name: _____________________________________________________________ Goal: Investigate the relationship between inscribed and central angles. a. Construct a circle A. b. Construct an inscribed angle Ð BCD. c. Construct a central angle Ð BAD (see Fig. 3). d. Measure both angles and compare. e. Construct diameter DE f. Drag point B around the arc DBE. What happens to the angles measures? g. What conjecture can you say exists between central and inscribed angles that share the same intercepted arc? h. Does your conjecture remain if you drag point B pass point E? Explain why. Fig. 3 Project AMP Dr. Antonio R. Quesada – Director, Project AMP Circle Conjectures Activity Four: Congruent Inscribed Angles Team members’ names: __________________________________________________ File name: _____________________________________________________________ Goal: Investigate the relationship between congruent inscribed angles. a. Construct a circle A. b. Construct two inscribed angles, Ð BCD and Ð BED, having them share the intercepted arc BD (See Fig. 4). c. Measure the angles and compare. d. Drag point B around the arc CDE, what happens to the angle measurements? e. What conjecture can you say exists between inscribed angles that share an intercepted arc? f. What happens when you drag point B pass C to the arc CE? Explain Fig. 4 Project AMP Dr. Antonio R. Quesada – Director, Project AMP Circle Conjectures ActivityFive: Angles inscribed in a Semi-circle Team members’ names: __________________________________________________ File name: _____________________________________________________________ Goal: Investigate angles inscribed in a semi-circle. a. Draw a circle A with chord BC running through the center of the circle. b. Construct an inscribed angle Ð BDC that intercepts the arc BC (see fig. 5). c. Measure the angle Ð BDC. d. Drag point D around the circle, what happens to the angle measure? e. What conjecture can you say exists for an inscribed angle which intercepts a half circle? Fig. 5 Project AMP Dr. Antonio R. Quesada – Director, Project AMP Circle Conjectures Activity Six: Congruent Chords Team members’ names: __________________________________________________ File name: _____________________________________________________________ Goal: Investigate the relationship between congruent chords. a. Construct a circle A with a chord BC. b. Construct a new chord DE, which is a reflection of chord BC with respect to the center of the circle, point A (use the symmetry tool). Chords DE and BC should be congruent. Measure their lengths to make sure. c. Construct central angles Ð BAC and Ð DAE. Measure each of these angles and compare their values, what relationship exists? d. Through experimenting, what other conjecture(s) can you find that exist between congruent chords in a circle? (Hint: examine arcs and distance from center).
Load more

Recommended publications
  • Proofs with Perpendicular Lines
    3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • History of Mathematics
    Georgia Department of Education History of Mathematics K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by using manipulatives and a variety of representations, working independently and cooperatively to solve problems, estimating and computing efficiently, and conducting investigations and recording findings. There is a shift towards applying mathematical concepts and skills in the context of authentic problems and for the student to understand concepts rather than merely follow a sequence of procedures. In mathematics classrooms, students will learn to think critically in a mathematical way with an understanding that there are many different ways to a solution and sometimes more than one right answer in applied mathematics. Mathematics is the economy of information. The central idea of all mathematics is to discover how knowing some things well, via reasoning, permit students to know much else—without having to commit the information to memory as a separate fact. It is the reasoned, logical connections that make mathematics coherent. The implementation of the Georgia Standards of Excellence in Mathematics places a greater emphasis on sense making, problem solving, reasoning, representation, connections, and communication. History of Mathematics History of Mathematics is a one-semester elective course option for students who have completed AP Calculus or are taking AP Calculus concurrently. It traces the development of major branches of mathematics throughout history, specifically algebra, geometry, number theory, and methods of proofs, how that development was influenced by the needs of various cultures, and how the mathematics in turn influenced culture. The course extends the numbers and counting, algebra, geometry, and data analysis and probability strands from previous courses, and includes a new history strand.
    [Show full text]
  • 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.
    [Show full text]
  • Squaring the Circle in Elliptic Geometry
    Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 2 Article 1 Squaring the Circle in Elliptic Geometry Noah Davis Aquinas College Kyle Jansens Aquinas College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Davis, Noah and Jansens, Kyle (2017) "Squaring the Circle in Elliptic Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 2 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss2/1 Rose- Hulman Undergraduate Mathematics Journal squaring the circle in elliptic geometry Noah Davis a Kyle Jansensb Volume 18, No. 2, Fall 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a [email protected] Aquinas College b scholar.rose-hulman.edu/rhumj Aquinas College Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 2, Fall 2017 squaring the circle in elliptic geometry Noah Davis Kyle Jansens Abstract. Constructing a regular quadrilateral (square) and circle of equal area was proved impossible in Euclidean geometry in 1882. Hyperbolic geometry, however, allows this construction. In this article, we complete the story, providing and proving a construction for squaring the circle in elliptic geometry. We also find the same additional requirements as the hyperbolic case: only certain angle sizes work for the squares and only certain radius sizes work for the circles; and the square and circle constructions do not rely on each other. Acknowledgements: We thank the Mohler-Thompson Program for supporting our work in summer 2014. Page 2 RHIT Undergrad. Math. J., Vol. 18, No. 2 1 Introduction In the Rose-Hulman Undergraduate Math Journal, 15 1 2014, Noah Davis demonstrated the construction of a hyperbolic circle and hyperbolic square in the Poincar´edisk [1].
    [Show full text]
  • Chapter 1 – Symmetry of Molecules – P. 1
    Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • SOME GEOMETRY in HIGH-DIMENSIONAL SPACES 11 Containing Cn(S) Tends to ∞ with N
    SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES MATH 527A 1. Introduction Our geometric intuition is derived from three-dimensional space. Three coordinates suffice. Many objects of interest in analysis, however, require far more coordinates for a complete description. For example, a function f with domain [−1; 1] is defined by infinitely many \coordi- nates" f(t), one for each t 2 [−1; 1]. Or, we could consider f as being P1 n determined by its Taylor series n=0 ant (when such a representation exists). In that case, the numbers a0; a1; a2;::: could be thought of as coordinates. Perhaps the association of Fourier coefficients (there are countably many of them) to a periodic function is familiar; those are again coordinates of a sort. Strange Things can happen in infinite dimensions. One usually meets these, gradually (reluctantly?), in a course on Real Analysis or Func- tional Analysis. But infinite dimensional spaces need not always be completely mysterious; sometimes one lucks out and can watch a \coun- terintuitive" phenomenon developing in Rn for large n. This might be of use in one of several ways: perhaps the behavior for large but finite n is already useful, or one can deduce an interesting statement about limn!1 of something, or a peculiarity of infinite-dimensional spaces is illuminated. I will describe some curious features of cubes and balls in Rn, as n ! 1. These illustrate a phenomenon called concentration of measure. It will turn out that the important law of large numbers from probability theory is just one manifestation of high-dimensional geometry.
    [Show full text]
  • 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.
    [Show full text]