DOCSLIB.ORG

Volume One: Geometry Without Multiplication

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Volume One: Geometry Without Multiplication Geometry–Do Volume One: Geometry without Multiplication by Victor Aguilar, author of Axiomatic Theory of Economics Chapters Postulates White Belt Foundations 1. Two points fully define the segment between them. Yellow Belt Congruence 2. By extending it, a segment fully defines a line. Orange Belt Parallelograms 3. Three noncollinear points fully define a triangle. Green Belt Triangle Construction 4. The center and the radius fully define a circle. Red Belt Triangles, Advanced 5. All right angles are equal; equivalently, all straight Blue Belt Quadrature Theory angles are equal. Cho–Dan Harmonic Division 6. A line and a point not on it fully define the parallel Yi–Dan Circle Inversion through that point. This book was NOT funded by the Bill and Melinda Gates Foundation. A journey of discovery? A death march? It’s the same thing! Copyright ©2020 This PDF file, which consists only of the foundational pages and the index, is the ONLY authorized free PDF file for Geometry–Do. Do not accept PDF files that do not have the copyright notice. www.researchgate.net/publication/291333791_Volume_One_Geometry_without_Multiplication Volume One, white through red belt, is available as a bound paperback, as is an abridgement consisting of white, yellow and early orange belt. Free review copies are available to mathematicians. Teachers can purchase individual copies from book sellers or contact me directly for quantity discounts, but I do not need the reviews of people with education degrees, so no free review copies are available to them. A math degree is a degree granted by a university mathematics department; people with an education degree (granted by a university education department) should not call it a math degree; doing so will not get them a free review copy. The order page and the contact page – for questions or if you wish to endorse the book – are at my website, shown below. Research Gate members may private message me or leave comments. www.axiomaticeconomics.com Volume Two, blue and black belt, is not written yet and may not be completed in 2020. Until it is written, I recommend Altshiller-Court’s College Geometry or Johnson’s Advanced Euclidean Geometry for students who have completed my Volume One. The latter is more advanced. Prerequisite for Volume One is Algebra I. No knowledge of non-linear functions, systems of linear equations or set theory is expected. But the chains of reasoning are longer and more difficult than some Algebra I students are ready for. I recommend Hall and Stevens’ A School Geometry for students who need an introduction to geometric proofs before they tackle Geometry–Do. Table of Contents The way that you wander, is the way that you chose, The day that you tarry, is the day that you lose, Sunshine or thunder, a man will always wonder, Where the fair wind blows, where the fair wind blows. If you are coming to this table of contents now for the first time, it may seem as though a black belt in Geometry–Do is a mountain too high to climb. But I tell you, the next three years are going to pass anyway, so why not go for the gold? It is an accomplishment you can boast about for the rest of your life. What else can you do as a teenager that you will be proud of as an old man? Postulates 1 The geometric postulates and the axioms of abstract algebra are explained. Notation 14 White Belt Instruction: Foundations 15 SAS, Isosceles Triangle Theorem and SSS are proven, in that order, to lay the foundation for geometry. Basic constructions (bisecting angles and segments, raising and dropping perpendiculars, replicating angles, etc.) are described. Construction workers are taught how to square foundations and walls working only inside the figure, and how to build gantries, wide gates, bridges, etc. Yellow Belt Instruction: Congruence 37 All the remaining congruence theorems are proven, and every theorem that can be proven without use of the parallel postulate. This organization is for students preparing to take a class in non-Euclidean geometry. The proofs are rigorous; we do not take similarity as an axiom, say “just set dilation to unity” and call it a proof. Architects learn of Gothic arches; they take this on faith while scholars come back to arches after completing orange belt, when they are proven to exist. Orange Belt Instruction: Parallelograms 85 Initially, the parallel postulate is needed because we speak of the intersection of lines and we can only be sure they do in Euclidean geometry. Midway through, it will be explicitly cited for the transversal theorem. There is much problem solving, including minimizing the sum of non-collinear segments and drawing a line through circles so the chords it cuts off meet given conditions. Soldiers learn how to position three guns to triangulate fire on aircraft and to enfilade roads. We conclude with a 20-question multiple-choice green-belt entrance exam. Only 10% survive, comparable to the simsa bout to get a green belt in Tang–Soo–Do. Geometry without Multiplication Victor Aguilar Green Belt Instruction: Triangle Construction 167 This begins second-year geometry, which is an honors-level class. The average students dropped out or failed orange belt and the construction workers quit after yellow belt, but it is hoped that aspiring military officers remain, for there is much to learn about machine gun emplacement. Mariners learn to navigate using lighthouses and structural engineers learn about skew bridges, but most of the applications are for infantry officers, which many boys find interesting even if they do not plan to enlist. For aspiring mathematicians, there is much use of loci and we work through many of the triangle construction problems that are a mainstay of Russian geometry but are rarely considered in the West. Military officers will be pleased that they are learning to fight the Russian way, which is more scientific and makes more use of automatic cannons than NATO does. The positioning of Shilkas is not random; they put a lot of thought into triangulating fire, enfilading roads, and covering possible sniper positions. Red Belt Instruction: Triangles, Advanced 215 By advanced I mean theorems difficult enough that they went unsolved for decades and are now named after famous mathematicians. Volume Two will assume an audience of Olympians; here, while we are still being helpful to engineers and military officers, we are transitioning into helping students compete in the International Mathematical Olympiad. We consider the work of Miquel, Wallace, Torricelli, Napoleon, Fagnano, Euler, et. al. Torricelli’s problem of minimizing the sum of the distances to vertices is solved; and in reverse, to recover the triangle. The Euler segment theorem is proven and the Euler (nine- point) circle is discussed. Homothecy is introduced. Air Force officers learn how to bomb the defense that Army officers designed as orange belts. There are results of interest to military officers that go beyond just laying ambushes. Index of Postulates, Theorems and Constructions 299 Principal Results of Geometry–Do in Alphabetical Order 347 Index of Names 351 References 359 Glossary 363 Glosario Español 373 Das englisch-deutsche Glossar 385 Англо-русский глоссарий 399 Euclid’s Postulates Plus One More Segment Two points fully define the segment between them. Line By extending it, a segment fully defines a line. Triangle Three noncollinear points fully define a triangle. Circle The center and the radius fully define a circle. Right Angle All right angles are equal; equivalently, all straight angles are equal. Parallel A line and a point not on it fully define the parallel through that point. Segments are denoted with a bar, 퐸퐹; rays with an arrow, 퐸퐹⃗⃗⃗⃗⃗ ; lines with a double arrow, 퐸퐹⃡⃗⃗⃗⃗ ; and angles as ∠퐸퐹퐺. The postulates are in terms of fully defined, which means that a figure with the given characteristics is unique, if it exists. Under defined means figures with the given characteristics are legion; more information is needed. John Playfair stated the parallel postulate roughly as I and David Hilbert do, which can be proven to be equivalent to Euclid’s Fifth Postulate. If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. While Hilbert and I both found Euclid’s postulate to be convoluted and chose Playfair’s version, and we both reject real numbers as unsupported by our postulates, we otherwise took separate paths. Geometry–Do is like Hilbert’s geometry, but it is unique and has its own postulates. Euclid also had five “common notions,” which vaguely describe what modern mathematicians call equivalence relations, total orderings, and additive groups. Equivalence Relations and Total Orderings A relation is an operator, 푅, that returns either a “true” or a “false” when applied to an ordered pair of elements from a given set. For instance, if the set is integers and the relation is equality, then 5 = 5 is true, but 5 = 4 is false. Relations must be applied to objects from the same set. For instance, 퐸퐹 = ∠퐺 is neither true nor false; it is incoherent. There are four ways that relations may be characterized. It is never true that a relation has all four, but some have three. Reflexive 푎 푅 푎 Symmetric 푎 푅 푏 implies 푏 푅 푎 Anti-Symmetric 푎 푅 푏 and 푏 푅 푎 implies 푎 = 푏 Transitive 푎 푅 푏 and 푏 푅 푐 implies 푎 푅 푐 Geometry without Multiplication Victor Aguilar A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
Load more

Recommended publications
  • Angle Chasing
    Angle Chasing Ray Li June 12, 2017 1 Facts you should know 1. Let ABC be a triangle and extend BC past C to D: Show that \ACD = \BAC + \ABC: 2. Let ABC be a triangle with \C = 90: Show that the circumcenter is the midpoint of AB: 3. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove that H is the incenter of 4DEF . 4. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove (i) that A; E; F; H lie on a circle diameter AH and (ii) that B; E; F; C lie on a circle with diameter BC. 5. Let ABC be a triangle with circumcenter O and orthocenter H: Show that \BAH = \CAO: 6. Let ABC be a triangle with circumcenter O and orthocenter H and let AH and AO meet the circumcircle at D and E, respectively. Show (i) that H and D are symmetric with respect to BC; and (ii) that H and E are symmetric with respect to the midpoint BC: 7. Let ABC be a triangle with altitudes AD; BE; and CF: Let M be the midpoint of side BC. Show that ME and MF are tangent to the circumcircle of AEF: 8. Let ABC be a triangle with incenter I, A-excenter Ia, and D the midpoint of arc BC not containing A on the circumcircle. Show that DI = DIa = DB = DC: 9. Let ABC be a triangle with incenter I and D the midpoint of arc BC not containing A on the circumcircle.
    [Show full text]
  • Extremal Problems for Convex Polygons ∗
    Extremal Problems for Convex Polygons ∗ Charles Audet Ecole´ Polytechnique de Montreal´ Pierre Hansen HEC Montreal´ Fred´ eric´ Messine ENSEEIHT-IRIT Abstract. Consider a convex polygon Vn with n sides, perimeter Pn, diameter Dn, area An, sum of distances between vertices Sn and width Wn. Minimizing or maximizing any of these quantities while fixing another defines ten pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization meth- ods. Numerous open problems are mentioned, as well as series of test problems for global optimization and nonlinear programming codes. Keywords: polygon, perimeter, diameter, area, sum of distances, width, isoperimeter problem, isodiametric problem. 1. Introduction Plane geometry is replete with extremal problems, many of which are de- scribed in the book of Croft, Falconer and Guy [12] on Unsolved problems in geometry. Traditionally, such problems have been solved, some since the Greeks, by geometrical reasoning. In the last four decades, this approach has been increasingly complemented by global optimization methods. This allowed solution of larger instances than could be solved by any one of these two approaches alone. Probably the best known type of such problems are circle packing ones: given a geometrical form such as a unit square, a unit-side triangle or a unit- diameter circle, find the maximum radius and configuration of n circles which can be packed in its interior (see [46] for a recent survey and the site [44] for a census of exact and approximate results with up to 300 circles).
    [Show full text]
  • Properties of Equidiagonal Quadrilaterals (2014)
    Forum Geometricorum Volume 14 (2014) 129–144. FORUM GEOM ISSN 1534-1178 Properties of Equidiagonal Quadrilaterals Martin Josefsson Abstract. We prove eight necessary and sufficient conditions for a convex quadri- lateral to have congruent diagonals, and one dual connection between equidiag- onal and orthodiagonal quadrilaterals. Quadrilaterals with both congruent and perpendicular diagonals are also discussed, including a proposal for what they may be called and how to calculate their area in several ways. Finally we derive a cubic equation for calculating the lengths of the congruent diagonals. 1. Introduction One class of quadrilaterals that have received little interest in the geometrical literature are the equidiagonal quadrilaterals. They are defined to be quadrilat- erals with congruent diagonals. Three well known special cases of them are the isosceles trapezoid, the rectangle and the square, but there are other as well. Fur- thermore, there exists many equidiagonal quadrilaterals that besides congruent di- agonals have no special properties. Take any convex quadrilateral ABCD and move the vertex D along the line BD into a position D such that AC = BD. Then ABCD is an equidiagonal quadrilateral (see Figure 1). C D D A B Figure 1. An equidiagonal quadrilateral ABCD Before we begin to study equidiagonal quadrilaterals, let us define our notations. In a convex quadrilateral ABCD, the sides are labeled a = AB, b = BC, c = CD and d = DA, and the diagonals are p = AC and q = BD. We use θ for the angle between the diagonals. The line segments connecting the midpoints of opposite sides of a quadrilateral are called the bimedians and are denoted m and n, where m connects the midpoints of the sides a and c.
    [Show full text]
  • Ethnomathematics and Education in Africa
    Copyright ©2014 by Paulus Gerdes www.lulu.com http://www.lulu.com/spotlight/pgerdes 2 Paulus Gerdes Second edition: ISTEG Belo Horizonte Boane Mozambique 2014 3 First Edition (January 1995): Institutionen för Internationell Pedagogik (Institute of International Education) Stockholms Universitet (University of Stockholm) Report 97 Second Edition (January 2014): Instituto Superior de Tecnologias e Gestão (ISTEG) (Higher Institute for Technology and Management) Av. de Namaacha 188, Belo Horizonte, Boane, Mozambique Distributed by: www.lulu.com http://www.lulu.com/spotlight/pgerdes Author: Paulus Gerdes African Academy of Sciences & ISTEG, Mozambique C.P. 915, Maputo, Mozambique ([email protected]) Photograph on the front cover: Detail of a Tonga basket acquired, in January 2014, by the author in Inhambane, Mozambique 4 CONTENTS page Preface (2014) 11 Chapter 1: Introduction 13 Chapter 2: Ethnomathematical research: preparing a 19 response to a major challenge to mathematics education in Africa Societal and educational background 19 A major challenge to mathematics education 21 Ethnomathematics Research Project in Mozambique 23 Chapter 3: On the concept of ethnomathematics 29 Ethnographers on ethnoscience 29 Genesis of the concept of ethnomathematics among 31 mathematicians and mathematics teachers Concept, accent or movement? 34 Bibliography 39 Chapter 4: How to recognize hidden geometrical thinking: 45 a contribution to the development of an anthropology of mathematics Confrontation 45 Introduction 46 First example 47 Second example
    [Show full text]
  • 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]
  • Allowing Coverage Holes of Bounded Diameter in Wireless Sensor
    1 Trap Coverage: Allowing Coverage Holes of Bounded Diameter in Wireless Sensor Networks Paul Balister x Zizhan Zhengy Santosh Kumarx Prasun Sinhay x University of Memphis y The Ohio State University fpbalistr,[email protected] fzhengz,[email protected] Abstract—Tracking of movements such as that of people, or blanket coverage [19]), as has been the case in prototype animals, vehicles, or of phenomena such as fire, can be achieved deployments [14], [17], [22]. The requirement of full coverage by deploying a wireless sensor network. So far only prototype will soon become a bottleneck as we begin to see real-life systems have been deployed and hence the issue of scale has not become critical. Real-life deployments, however, will be at large deployments. scale and achieving this scale will become prohibitively expensive In this paper, we therefore propose a new model of coverage, if we require every point in the region to be covered (i.e., full called Trap Coverage, that scales well with large deployment coverage), as has been the case in prototype deployments. regions. We define a Coverage Hole in a target region of In this paper we therefore propose a new model of coverage, deployment A to be a connected component1 of the set of called Trap Coverage, that scales well with large deployment regions. A sensor network providing Trap Coverage guarantees uncovered points of A. A sensor network is said to provide that any moving object or phenomena can move at most a Trap Coverage with diameter d to A if the diameter of any (known) displacement before it is guaranteed to be detected by Coverage Hole in A is at most d.
    [Show full text]
  • Art Problem Solving
    the ART of PROBLEM SOLVING Volume 1: the BASICS Sandor Lehoczky Richard Rusczyk Copyright © 1993,1995, 2003,2004, 2006 Sandor Lehoczky and Richard Rusczyk. All Rights Reserved. Reproduction of any part of this book without expressed permission of the authors is strictly forbid­ den. For use outside the classroom of the problems contained herein, permission must be acquired from the cited sources. ISBN-10: 0-9773045-6-6 ISBN-13: 978-0-9773045-6-1 Published by: AoPS Incorporated P.O. Box 2185 Alpine, CA 91903-2185 (619) 659-1612 [email protected] Visit the Art of Problem Solving website at http: //www. artofproblemsolving. com Printed in the United States of America. Seventh Edition; printed in 2006. Editor: David Patrick Cover image designed by Vanessa Rusczyk using KaleidoTile software. Cover Image: "Grand Canyon from South Rim" by Ansel Adams. No permissions required; National Archive photo 79-AAF-8. This book was produced using the KTgX document processing system. Diagrams created by Maria Monks using METRPOST. To Ameyalli, for laughter as clear as your lake, for spirit strong and serene as your skyline, for all that you have taught and learned in four winters. And to Mrs. Wendt, who is still by far the best teacher I ever had. —SL For my desert flower Vanessa. Told you we'd make it there eventually. —RR Special thanks to the following people who helped make this possible: William and Claire Devlin, Sandor L. and Julieanne G. Lehoczky, Steve and Ann Rubio, Richard and Claire Rusczyk, Stanley Rusczyk. Thanks A large number of individuals and organizations have helped make the ART of PROBLEM SOLVING possible.
    [Show full text]
  • Cyclic Quadrilaterals — the Big Picture Yufei Zhao [email protected]
    Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao [email protected] An important skill of an olympiad geometer is being able to recognize known configurations. Indeed, many geometry problems are built on a few common themes. In this lecture, we will explore one such configuration. 1 What Do These Problems Have in Common? 1. (IMO 1985) A circle with center O passes through the vertices A and C of triangle ABC and intersects segments AB and BC again at distinct points K and N, respectively. The circumcircles of triangles ABC and KBN intersects at exactly two distinct points B and M. ◦ Prove that \OMB = 90 . B M N K O A C 2. (Russia 1995; Romanian TST 1996; Iran 1997) Consider a circle with diameter AB and center O, and let C and D be two points on this circle. The line CD meets the line AB at a point M satisfying MB < MA and MD < MC. Let K be the point of intersection (different from ◦ O) of the circumcircles of triangles AOC and DOB. Show that \MKO = 90 . C D K M A O B 3. (USA TST 2007) Triangle ABC is inscribed in circle !. The tangent lines to ! at B and C meet at T . Point S lies on ray BC such that AS ? AT . Points B1 and C1 lies on ray ST (with C1 in between B1 and S) such that B1T = BT = C1T . Prove that triangles ABC and AB1C1 are similar to each other. 1 Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao A B S C C1 B1 T Although these geometric configurations may seem very different at first sight, they are actually very related.
    [Show full text]
  • A Condition That a Tangential Quadrilateral Is Also Achordalone
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE Mathematical Communications 12(2007), 33-52 33 A condition that a tangential quadrilateral is also achordalone Mirko Radic´,∗ Zoran Kaliman† and Vladimir Kadum‡ Abstract. In this article we present a condition that a tangential quadrilateral is also a chordal one. The main result is given by Theo- rem 1 and Theorem 2. Key words: tangential quadrilateral, bicentric quadrilateral AMS subject classifications: 51E12 Received September 1, 2005 Accepted March 9, 2007 1. Introduction A polygon which is both tangential and chordal will be called a bicentric polygon. The following notation will be used. If A1A2A3A4 is a considered bicentric quadrilateral, then its incircle is denoted by C1, circumcircle by C2,radiusofC1 by r,radiusofC2 by R, center of C1 by I, center of C2 by O, distance between I and O by d. A2 C2 C1 r A d 1 O I R A3 A4 Figure 1.1 ∗Faculty of Philosophy, University of Rijeka, Omladinska 14, HR-51 000 Rijeka, Croatia, e-mail: [email protected] †Faculty of Philosophy, University of Rijeka, Omladinska 14, HR-51 000 Rijeka, Croatia, e-mail: [email protected] ‡University “Juraj Dobrila” of Pula, Preradovi´ceva 1, HR-52 100 Pula, Croatia, e-mail: [email protected] 34 M. Radic,´ Z. Kaliman and V. Kadum The first one who was concerned with bicentric quadrilaterals was a German mathematicianNicolaus Fuss (1755-1826), see [2]. He foundthat C1 is the incircle and C2 the circumcircle of a bicentric quadrilateral A1A2A3A4 iff (R2 − d2)2 =2r2(R2 + d2).
    [Show full text]
  • Angle Bisectors in a Quadrilateral Are Concurrent
    Angle Bisectors in a Quadrilateral in the classroom A Ramachandran he bisectors of the interior angles of a quadrilateral are either all concurrent or meet pairwise at 4, 5 or 6 points, in any case forming a cyclic quadrilateral. The situation of exactly three bisectors being concurrent is not possible. See Figure 1 for a possible situation. The reader is invited to prove these as well as observations regarding some of the special cases mentioned below. Start with the last observation. Assume that three angle bisectors in a quadrilateral are concurrent. Join the point of T D E H A F G B C Figure 1. A typical configuration, showing how a cyclic quadrilateral is formed Keywords: Quadrilateral, diagonal, angular bisector, tangential quadrilateral, kite, rhombus, square, isosceles trapezium, non-isosceles trapezium, cyclic, incircle 33 At Right Angles | Vol. 4, No. 1, March 2015 Vol. 4, No. 1, March 2015 | At Right Angles 33 D A D A D D E G A A F H G I H F F G E H B C E Figure 3. If is a parallelogram, then is a B C B C rectangle B C Figure 2. A tangential quadrilateral Figure 6. The case when is a non-isosceles trapezium: the result is that is a cyclic Figure 7. The case when has but A D quadrilateral in which : the result is that is an isosceles ∘ trapezium ( and ∠ ) E ∠ ∠ ∠ ∠ concurrence to the fourth vertex. Prove that this line indeed bisects the angle at the fourth vertex. F H Tangential quadrilateral A quadrilateral in which all the four angle bisectors G meet at a pointincircle is a — one which has an circle touching all the four sides.
    [Show full text]
  • The Role of Euclidean Geometry in High School
    JOURNAL OF MATHEMATICAL BEHAVIOR 15, 221-237 (1996) The Role of Euclidean Geometry in High School HUNG-HSI Wu University of California When I was a high school student long ago, the need to study Euclidean geome- try was taken for granted. In fact, the novelty of learning to prove something was so overwhelming to me and some of my friends that, years later, we would look back at Euclidean geometry as the high point of our mathematics education. In recent years, the value of Euclid has depreciated considerably. A vigorous debate is now going on about how much, if any, of his work is still relevant. In this article, I would like to give my perspective on this subject, and in so doing, I will be taking a philosophical tour, so to speak, of a small portion of mathematics. This may not be so surprising because after all, in life as in mathematics, one does need a little philosophy for guidance from time to time. The bone of contention in the geometry curriculum is of course how many of the traditional “two-column proofs” should be retained. Let me first briefly indicate the range of available options in this regard: 1. A traditional text in use in a local high school in Berkeley not too long ago begins on page 1 with the undefined terms of the axioms, and formal proofs start on page 22. There is no motivation or explanation of the whys and hows of an axiomatic system. 2. A recently published text does experimental geometry (i.e., “hands-on ge- ometry” with no proofs) all the way until the last 130 pages of its 700 pages of exposition.
    [Show full text]
  • Largest Small N-Polygons: Numerical Results and Conjectured Optima
    Largest Small n-Polygons: Numerical Results and Conjectured Optima János D. Pintér Department of Industrial and Systems Engineering Lehigh University, Bethlehem, PA, USA [email protected] Abstract LSP(n), the largest small polygon with n vertices, is defined as the polygon of unit diameter that has maximal area A(n). Finding the configuration LSP(n) and the corresponding A(n) for even values n 6 is a long-standing challenge that leads to an interesting class of nonlinear optimization problems. We present numerical solution estimates for all even values 6 n 80, using the AMPL model development environment with the LGO nonlinear solver engine option. Our results compare favorably to the results obtained by other researchers who solved the problem using exact approaches (for 6 n 16), or general purpose numerical optimization software (for selected values from the range 6 n 100) using various local nonlinear solvers. Based on the results obtained, we also provide a regression model based estimate of the optimal area sequence {A(n)} for n 6. Key words Largest Small Polygons Mathematical Model Analytical and Numerical Solution Approaches AMPL Modeling Environment LGO Solver Suite For Nonlinear Optimization AMPL-LGO Numerical Results Comparison to Earlier Results Regression Model Based Optimum Estimates 1 Introduction The diameter of a (convex planar) polygon is defined as the maximal distance among the distances measured between all vertex pairs. In other words, the diameter of the polygon is the length of its longest diagonal. The largest small polygon with n vertices is the polygon of unit diameter that has maximal area. For any given integer n 3, we will refer to this polygon as LSP(n) with area A(n).
    [Show full text]