On the Trapezoidal Peg Problem Among Convex Curves
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A Survey on the Square Peg Problem
A survey on the Square Peg Problem Benjamin Matschke∗ March 14, 2014 Abstract This is a short survey article on the 102 years old Square Peg Problem of Toeplitz, which is also called the Inscribed Square Problem. It asks whether every continuous simple closed curve in the plane contains the four vertices of a square. This article first appeared in a more compact form in Notices Amer. Math. Soc. 61(4), 2014. 1 Introduction In 1911 Otto Toeplitz [66] furmulated the following conjecture. Conjecture 1 (Square Peg Problem, Toeplitz 1911). Every continuous simple closed curve in the plane γ : S1 ! R2 contains four points that are the vertices of a square. Figure 1: Example for Conjecture1. A continuous simple closed curve in the plane is also called a Jordan curve, and it is the same as an injective map from the unit circle into the plane, or equivalently, a topological embedding S1 ,! R2. Figure 2: We do not require the square to lie fully inside γ, otherwise there are counter- examples. ∗Supported by Deutsche Telekom Stiftung, NSF Grant DMS-0635607, an EPDI-fellowship, and MPIM Bonn. This article is licensed under a Creative Commons BY-NC-SA 3.0 License. 1 In its full generality Toeplitz' problem is still open. So far it has been solved affirmatively for curves that are \smooth enough", by various authors for varying smoothness conditions, see the next section. All of these proofs are based on the fact that smooth curves inscribe generically an odd number of squares, which can be measured in several topological ways. -
FINITE FOURIER SERIES and OVALS in PG(2, 2H)
J. Aust. Math. Soc. 81 (2006), 21-34 FINITE FOURIER SERIES AND OVALS IN PG(2, 2h) J. CHRIS FISHERY and BERNHARD SCHMIDT (Received 2 April 2004; revised 7 May 2005) Communicated by L. Batten Abstract We propose the use of finite Fourier series as an alternative means of representing ovals in projective planes of even order. As an example to illustrate the method's potential, we show that the set [w1 + wy' + w ~y' : 0 < j < 2;'| c GF(22/1) forms an oval if w is a primitive (2h + l)sl root of unity in GF(22'') and GF(22'') is viewed as an affine plane over GF(2;'). For the verification, we only need some elementary 'trigonometric identities' and a basic irreducibility lemma that is of independent interest. Finally, we show that our example is the Payne oval when h is odd, and the Adelaide oval when h is even. 2000 Mathematics subject classification: primary 51E20, 05B25. 1. Introduction In any finite projective plane of order q, an oval is a set of q + 1 points, no three of which are collinear. In the classical plane PG(2, q) over GF(g), a nondegenerate conic is the prototypical oval. If the order of a plane is even, then the tangents to an oval all pass through a point that is called the nucleus of the oval. We call an oval together with its nucleus a hyperoval. During the 1950s Beniamino Segre proved that in PG(2, q), (1) when q is odd, then there exist no ovals other than the conies, and (2) when q — 2\ coordinates may be chosen so that the points of a hyperoval are the elements of the set {(*, f{x), 1) | JC G GF(<?)} U {(0, 1,0), (1,0, 0)}, where / is a permutation polynomial of degree at most q — 2 for which /(0) = 0, /(I) = 1, and with the additional property that for all 5 in GF(^), the function /,. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Dissertation Hyperovals, Laguerre Planes And
DISSERTATION HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY Submitted by Luke Bayens Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Spring 2013 Doctoral Committee: Advisor: Tim Penttila Jeff Achter Willem Bohm Chris Peterson ABSTRACT HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY In 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries { we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups { indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting tran- sitively on flags, with an additional hypothesis { a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order. -
A Combinatorial Approach to the Inscribed Square Problem
A Combinatorial Approach to the Inscribed Square Problem Elizabeth Kelley Francis Edward Su, Advisor Rob Thompson, Reader Department of Mathematics May, 2015 Copyright c 2015 Elizabeth Kelley. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract The inscribed square conjecture, also known as Toeplitz’ conjecture or the square peg problem, asserts that every Jordan curve in the Euclidean plane admits an inscribed square. Although there exists no proof of the general conjecture, there are affirmative proofs of the conjecture subject to addi- tional local smoothness conditions. The weakest of these local smooth- ness conditions include the special trapezoid criterion, due to Matschke, and local monotonicity, due to Stromquist. We develop several combinato- rial approaches to the inscribed square problem, explicitly locate inscribed squares for conic sections, and explore the existence of inscribed squares in the Koch snowflake, which we prove to be not locally monotone. Contents Abstract iii 1 Historical Development of the Inscribed Square Problem 1 1.1 Statement of the Problem . .1 1.2 Past Work . .2 1.3 Related Problems . .6 2 Conic Sections 9 3 Koch Snowflake 17 3.1 Inscribed Squares in the Koch Snowflake . 20 4 Discretizing the Parameter Space 35 4.1 Pivot vertex and side length . 36 4.2 Two pivot vertices . -
Isoptics of a Closed Strictly Convex Curve. - II Rendiconti Del Seminario Matematico Della Università Di Padova, Tome 96 (1996), P
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA W. CIESLAK´ A. MIERNOWSKI W. MOZGAWA Isoptics of a closed strictly convex curve. - II Rendiconti del Seminario Matematico della Università di Padova, tome 96 (1996), p. 37-49 <http://www.numdam.org/item?id=RSMUP_1996__96__37_0> © Rendiconti del Seminario Matematico della Università di Padova, 1996, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Isoptics of a Closed Strictly Convex Curve. - II. W. CIE015BLAK (*) - A. MIERNOWSKI (**) - W. MOZGAWA(**) 1. - Introduction. ’ This article is concerned with some geometric properties of isoptics which complete and deepen the results obtained in our earlier paper [3]. We therefore begin by recalling the basic notions and necessary results concerning isoptics. An a-isoptic Ca of a plane, closed, convex curve C consists of those points in the plane from which the curve is seen under the fixed angle .7r - a. We shall denote by C the set of all plane, closed, strictly convex curves. Choose an element C and a coordinate system with the ori- gin 0 in the interior of C. Let p(t), t E [ 0, 2~c], denote the support func- tion of the curve C. -
Ellipse and Oval in Baroque Sacral Architecture in Slovakia
Vol. 13, Issue 1/2017, 30-41, DOI: 10.1515/cee-2017-0004 ELLIPSE AND OVAL IN BAROQUE SACRAL ARCHITECTURE IN SLOVAKIA Zuzana GRÚ ŇOVÁ 1* , Michaela HOLEŠOVÁ 2 1 Department of Building Engineering and Urban Planning, Faculty of Civil Engineering, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia. 2 Department of Structural Mechanics and Applied Mathematics, Faculty of Civil Engineering, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia. * corresponding author: [email protected]. Abstract Keywords: Oval, circular and elliptic forms appear in the architecture from the Elliptic and oval plans; very beginning. The basic problem of the geometric analysis of the Slovak baroque sacral spaces with an elliptic or oval ground plan is a great sensitivity of the architecture; outcome calculations to the plan's precision, mainly to distinguish Geometrical analysis of Serlio's between oval and ellipse. Sebastiano Serlio and Guarino Guarini and Guarini's ovals. belong to those architects, theoreticians, who analysed the potential of circular or oval forms and some of their ideas are analysed in the paper. Elliptical or oval plans were used also in Slovak baroque architecture or interior elements and the paper introduce some of the most known examples as a connection to the world architecture ideas. 1. Introduction Oval and elliptic forms belonged to architectural dictionary of forms from the very beginning. The basic shape, easy to construct was of course a circle. Circular or many forms of circular-like, not precisely shaped forms could be found at many Neolithic sites like Skara Brae (profane and supposed sacral spaces), temples of Malta and many others. -
Types of Coordinate Systems What Are Map Projections?
What are map projections? Page 1 of 155 What are map projections? ArcGIS 10 Within ArcGIS, every dataset has a coordinate system, which is used to integrate it with other geographic data layers within a common coordinate framework such as a map. Coordinate systems enable you to integrate datasets within maps as well as to perform various integrated analytical operations such as overlaying data layers from disparate sources and coordinate systems. What is a coordinate system? Coordinate systems enable geographic datasets to use common locations for integration. A coordinate system is a reference system used to represent the locations of geographic features, imagery, and observations such as GPS locations within a common geographic framework. Each coordinate system is defined by: Its measurement framework which is either geographic (in which spherical coordinates are measured from the earth's center) or planimetric (in which the earth's coordinates are projected onto a two-dimensional planar surface). Unit of measurement (typically feet or meters for projected coordinate systems or decimal degrees for latitude–longitude). The definition of the map projection for projected coordinate systems. Other measurement system properties such as a spheroid of reference, a datum, and projection parameters like one or more standard parallels, a central meridian, and possible shifts in the x- and y-directions. Types of coordinate systems There are two common types of coordinate systems used in GIS: A global or spherical coordinate system such as latitude–longitude. These are often referred to file://C:\Documents and Settings\lisac\Local Settings\Temp\~hhB2DA.htm 10/4/2010 What are map projections? Page 2 of 155 as geographic coordinate systems. -
The Shape and History of the Ellipse in Washington, D.C
The Shape and History of the Ellipse in Washington, D.C. Clark Kimberling Department of Mathematics University of Evansville 1800 Lincoln Avenue, Evansville, IN 47722 email: [email protected] 1 Introduction When conic sections are introduced to mathematics classes, certain real-world examples are often cited. Favorites include lamp-shade shadows for hyperbolas, paths of baseballs for parabolas, and planetary orbits for ellipses. There is, however, another outstanding example of an ellipse. Known simply as the Ellipse, it is a gathering place for thousands of Americans every year, and it is probably the world’s largest noncircular ellipse. Situated just south of the White House in President’sPark, the Ellipse has an interesting shape and an interesting history. Figure 1. Looking north from the top of Washington Monument: the Ellipse and the White House [19] When Charles L’Enfant submitted his Plan for the American capital city to President George Washington, he included many squares, circles, and triangles. Today, well known shapes in or near the capital include the Federal Triangle, McPherson Square, the Pentagon, the Octagon House Museum, Washington Circle, and, of course, the Ellipse. Regarding the Ellipse in its present size and shape, a map dated September 29, 1877 (Figure 7) was probably used for the layout. The common gardener’smethod would not have been practical for so large an ellipse –nearly 17 acres –so that the question, "How was the Ellipse laid out?" is of considerable interest. (The gardener’s method uses three stakes and a rope. Drive two stakes into the ground, and let 2c be the distance between them. -
Oval Domes: History, Geometry and Mechanics
Santiago Huerta Research E. T.S. de Arquitectura Oval Domes: Universidad Politécnica de Madrid Avda. Juan de Herrera, 4 History, Geometry and Mechanics 28040 Madrid SPAIN Abstract. An oval dome may be defined as a dome whose [email protected] plan or profile (or both) has an oval form. The word “oval” Keywords: oval domes, history of comes from the Latin ovum, egg. The present paper contains engineering, history of an outline of the origin and application of the oval in construction, structural design historical architecture; a discussion of the spatial geometry of oval domes, that is, the different methods employed to lay them out; a brief exposition of the mechanics of oval arches and domes; and a final discussion of the role of geometry in oval arch and dome design. Introduction An oval dome may be defined as a dome whose plan or profile (or both) has an oval form. The word Aoval@ comes from the Latin ovum, egg. Thus, an oval dome is egg-shaped. The first buildings with oval plans were built without a predetermined form, just trying to enclose a space in the most economical form. Eventually, the geometry was defined by using circular arcs with common tangents at the points of change of curvature. Later the oval acquired a more regular form with two axes of symmetry. Therefore, an “oval” may be defined as an egg-shaped form, doubly symmetric, constructed with circular arcs; an oval needs a minimum of four centres, but it is possible also to build ovals with multiple centres. The preceding definition corresponds with the origin and the use of oval forms in building and may be applied without problem up to, say, the eighteenth century. -
3D Squircle, Quadrics, Sextic Surface, Mapping a Sphere to a Cube, Algebraic Geometry
Squircular Calculations Chamberlain Fong [email protected] Joint Mathematics Meetings 2018, SIGMAA-ARTS Abstract – The Fernandez-Guasti squircle is a plane algebraic curve that is an intermediate shape between the circle and the square. It has qualitative features that are similar to the more famous Lamé curve. However, unlike the Lamé curve which has unbounded polynomial exponents, the Fernandez-Guasti squircle is a low degree quartic curve. This makes it more amenable to algebraic manipulation and simplification. In this paper, we will analyze the squircle and derive formulas for its area, arc length, and polar form. We will also provide several parametric equations of the squircle. Finally, we extend the squircle to three dimensions by coming up with an analogous surface that is an intermediate shape between the sphere and the cube. Keywords – Squircle, Lamé curve, Circle and Square Homeomorphism, Algebraic surface, 3D squircle, Quadrics, Sextic Surface, Mapping a Sphere to a Cube, Algebraic geometry Figure 1: The FG-squircle at varying values of s. 1 Introduction In 1992, Manuel Fernandez-Guasti discovered an intermediate shape between the circle and the square [Fernandez-Guasti 1992]. This shape is a quartic plane curve. The equation for this shape is provided in Figure 1. There are two parameters for this equation: s and r. The squareness parameter s allows the shape to interpolate between the circle and the square. When s = 0, the equation produces a circle with radius r. When s = 1, the equation produces a square with a side length of 2r. In between, the equation produces a smooth planar curve that resembles both the circle and the square. -
Planar Circle Geometries an Introduction to Moebius–, Laguerre– and Minkowski–Planes
Planar Circle Geometries an Introduction to Moebius–, Laguerre– and Minkowski–planes Erich Hartmann Department of Mathematics Darmstadt University of Technology 2 Contents 0 INTRODUCTION 7 1 RESULTS ON AFFINE AND PROJECTIVE GEOMETRY 11 1.1 Affineplanes.................................. 11 1.1.1 Theaxiomsofanaffineplane . 11 1.1.2 Collineations of an affine plane . 12 1.1.3 Desarguesianaffineplanes . 13 1.1.4 Pappianaffineplanes......................... 14 1.2 Projectiveplanes ............................... 15 1.2.1 Theaxiomsofaprojectiveplane . 15 1.2.2 Collineations of a projective plane . 15 1.2.3 Desarguesianprojectiveplanes. 16 1.2.4 Homogeneous coordinates of a desarguesian plane . 17 1.2.5 Collineations of a desarguesian projective plane . 17 1.2.6 Pappianprojectiveplanes . 17 1.2.7 The groups P ΓL(2,K),PGL(2,K) and P SL(2,K) ........ 18 1.2.8 Basic 1–dimensional projective configurations . 19 2 OVALS AND CONICS 23 2.1 Oval, parabolic and hyperbolic curve . 23 2.2 The ovals c1 and c2 .............................. 24 2.3 Properties of the ovals c1 and c2 ...................... 26 2.4 Ovalconicsandtheirproperties . 27 2.5 Ovalconicsinaffineplanes ......................... 39 2.6 Finiteovals .................................. 40 2.7 Furtherexamplesofovals .......................... 43 2.7.1 Translation–ovals ........................... 43 2.7.2 Moufang–ovals ............................ 44 2.7.3 Ovals in Moulton–planes . 45 2.7.4 Ovals in planes over nearfields . 45 2.7.5 Ovals in finite planes over quasifields . 45 2.7.6 Realovalsonconvexfunctions. 46 3 3 MOEBIUS–PLANES 47 3.1 The classical real Moebius–plane . 47 3.2 TheaxiomsofaMoebius–plane . 48 3.3 Miquelian Moebius–planes . 50 3.3.1 The incidence structure M(K, q) .................