The Changing Shape of Geometry, the Mathematical Association
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Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). -
Art Concepts for Kids: SHAPE & FORM!
Art Concepts for Kids: SHAPE & FORM! Our online art class explores “shape” and “form” . Shapes are spaces that are created when a line reconnects with itself. Forms are three dimensional and they have length, width and depth. This sweet intro song teaches kids all about shape! Scratch Garden Value Song (all ages) https://www.youtube.com/watch?v=coZfbTIzS5I Another great shape jam! https://www.youtube.com/watch?v=6hFTUk8XqEc Cookie Monster eats his shapes! https://www.youtube.com/watch?v=gfNalVIrdOw&list=PLWrCWNvzT_lpGOvVQCdt7CXL ggL9-xpZj&index=10&t=0s Now for the activities!! Click on the links in the descriptions below for the step by step details. Ages 2-4 Shape Match https://busytoddler.com/2017/01/giant-shape-match-activity/ Shapes are an easy art concept for the littles. Lots of their environment gives play and art opportunities to learn about shape. This Matching activity involves tracing the outside shape of blocks and having littles match the block to the outline. It can be enhanced by letting your little ones color the shapes emphasizing inside and outside the shape lines. Block Print Shapes https://thepinterestedparent.com/2017/08/paul-klee-inspired-block-printing/ Printing is a great technique for young kids to learn. The motion is like stamping so you teach little ones to press and pull rather than rub or paint. This project requires washable paint and paper and block that are not natural wood (which would stain) you want blocks that are painted or sealed. Kids can look at Paul Klee art for inspiration in stacking their shapes into buildings, or they can chose their own design Ages 4-6 Lois Ehlert Shape Animals http://www.momto2poshlildivas.com/2012/09/exploring-shapes-and-colors-with-color.html This great project allows kids to play with shapes to make animals in the style of the artist Lois Ehlert. -
Second Rapport Intermédiaire
Compte-rendu intermédiaire 01/04/2016-31/03/2107 Projet ANR- AA-PPPP-000 Cirmath Circulations des mathématiques dans et par les journaux : histoire, territoires et publics Programme xxxxx 200x A IDENTIFICATION ................................................................ 3 B LIVRABLES ET JALONS ......................................................... 4 C RAPPORT D’AVANCEMENT SUR LA PERIODE CONCERNEE (01/04/16- 31/03/17) .................................................................... 6 C.1 Description des travaux effectués ............................................. 6 C.1.1 Rappel : Une présentation rapide du programme de recherche de Cirmath 6 C.1.2 Le séminaire Cirmath 7 C.1.3 Les colloques Cirmath 7 C.1.4 La base de données des journaux mathématiques - Cirmath-data 8 C.1.5 Conclusion 11 C.2 Résultats marquants (si applicable) ........................................ 11 C.3 Réunions du consortium (si applicable) ................................... 11 C.4 Commentaires libres............................................................. 12 D IMPACT DU PROJET DEPUIS LE DEBUT ...................................... 13 D.1 Indicateurs d’impact ............................................................. 13 D.2 Liste des publications et communications ................................ 14 Référence du formulaire : ANR-FORM-090601-02-01 D.2.1 Liste des publications et communications directement liées au projet Cirmath. 14 D.2.2 Liste des publications et communications des membres du projet Cirmath ayant trait à la thématique de la circulation -
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. -
Geometric Modeling in Shape Space
Geometric Modeling in Shape Space Martin Kilian Niloy J. Mitra Helmut Pottmann Vienna University of Technology Figure 1: Geodesic interpolation and extrapolation. The blue input poses of the elephant are geodesically interpolated in an as-isometric- as-possible fashion (shown in green), and the resulting path is geodesically continued (shown in purple) to naturally extend the sequence. No semantic information, segmentation, or knowledge of articulated components is used. Abstract space, line geometry interprets straight lines as points on a quadratic surface [Berger 1987], and the various types of sphere geometries We present a novel framework to treat shapes in the setting of Rie- model spheres as points in higher dimensional space [Cecil 1992]. mannian geometry. Shapes – triangular meshes or more generally Other examples concern kinematic spaces and Lie groups which straight line graphs in Euclidean space – are treated as points in are convenient for handling congruent shapes and motion design. a shape space. We introduce useful Riemannian metrics in this These examples show that it is often beneficial and insightful to space to aid the user in design and modeling tasks, especially to endow the set of objects under consideration with additional struc- explore the space of (approximately) isometric deformations of a ture and to work in more abstract spaces. We will show that many given shape. Much of the work relies on an efficient algorithm to geometry processing tasks can be solved by endowing the set of compute geodesics in shape spaces; to this end, we present a multi- closed orientable surfaces – called shapes henceforth – with a Rie- resolution framework to solve the interpolation problem – which mannian structure. -
Shape Synthesis Notes
SHAPE SYNTHESIS OF HIGH-PERFORMANCE MACHINE PARTS AND JOINTS By John M. Starkey Based on notes from Walter L. Starkey Written 1997 Updated Summer 2010 2 SHAPE SYNTHESIS OF HIGH-PERFORMANCE MACHINE PARTS AND JOINTS Much of the activity that takes place during the design process focuses on analysis of existing parts and existing machinery. There is very little attention focused on the synthesis of parts and joints and certainly there is very little information available in the literature addressing the shape synthesis of parts and joints. The purpose of this document is to provide guidelines for the shape synthesis of high-performance machine parts and of joints. Although these rules represent good design practice for all machinery, they especially apply to high performance machines, which require high strength-to-weight ratios, and machines for which manufacturing cost is not an overriding consideration. Examples will be given throughout this document to illustrate this. Two terms which will be used are part and joint. Part refers to individual components manufactured from a single block of raw material or a single molding. The main body of the part transfers loads between two or more joint areas on the part. A joint is a location on a machine at which two or more parts are fastened together. 1.0 General Synthesis Goals Two primary principles which govern the shape synthesis of a part assert that (1) the size and shape should be chosen to induce a uniform stress or load distribution pattern over as much of the body as possible, and (2) the weight or volume of material used should be a minimum, consistent with cost, manufacturing processes, and other constraints. -
Geometry and Art LACMA | | April 5, 2011 Evenings for Educators
Geometry and Art LACMA | Evenings for Educators | April 5, 2011 ALEXANDER CALDER (United States, 1898–1976) Hello Girls, 1964 Painted metal, mobile, overall: 275 x 288 in., Art Museum Council Fund (M.65.10) © Alexander Calder Estate/Artists Rights Society (ARS), New York/ADAGP, Paris EOMETRY IS EVERYWHERE. WE CAN TRAIN OURSELVES TO FIND THE GEOMETRY in everyday objects and in works of art. Look carefully at the image above and identify the different, lines, shapes, and forms of both GAlexander Calder’s sculpture and the architecture of LACMA’s built environ- ment. What is the proportion of the artwork to the buildings? What types of balance do you see? Following are images of artworks from LACMA’s collection. As you explore these works, look for the lines, seek the shapes, find the patterns, and exercise your problem-solving skills. Use or adapt the discussion questions to your students’ learning styles and abilities. 1 Language of the Visual Arts and Geometry __________________________________________________________________________________________________ LINE, SHAPE, FORM, PATTERN, SYMMETRY, SCALE, AND PROPORTION ARE THE BUILDING blocks of both art and math. Geometry offers the most obvious connection between the two disciplines. Both art and math involve drawing and the use of shapes and forms, as well as an understanding of spatial concepts, two and three dimensions, measurement, estimation, and pattern. Many of these concepts are evident in an artwork’s composition, how the artist uses the elements of art and applies the principles of design. Problem-solving skills such as visualization and spatial reasoning are also important for artists and professionals in math, science, and technology. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: [email protected]
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania e-mail: [email protected] NEWSLETTER No. 47 KRZYSZTOF CIESIELSKI Mathematics Institute March 2003 Jagiellonian University Reymonta 4 EMS Agenda ................................................................................................. 2 30-059 Kraków, Poland e-mail: [email protected] Editorial by Sir John Kingman .................................................................... 3 STEEN MARKVORSEN Department of Mathematics Executive Committee Meeting ....................................................................... 4 Technical University of Denmark Building 303 Introducing the Committee ............................................................................ 7 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] An Answer to the Growth of Mathematical Knowledge? ............................... 9 SPECIALIST EDITORS Interview with Vagn Lundsgaard Hansen .................................................. 15 INTERVIEWS Steen Markvorsen [address as above] Interview with D V Anosov .......................................................................... 20 SOCIETIES Krzysztof Ciesielski [address as above] Israel Mathematical Union ......................................................................... 25 EDUCATION Tony Gardiner -
HENRI BROCARD (1865) UN INGÉNIEUR SAVANT DU Xixe SIÈCLE
OCTOBRE 2015 • LA JAUNE ET LA ROUGE 36 PARCOURS TRAJECTOIRES PAULINE ROMERA-LEBRET chercheur associé au Groupe d’histoire et diffusion des sciences d’Orsay (GHDSO) HENRI BROCARD (1865) UN INGÉNIEUR SAVANT DU XIXe SIÈCLE Dans une nécrologie d’Henri Brocard 1, on peut lire que sa curiosité s’étendait « à de nombreux domaines : sciences naturelles, économie rurale, météorologie », même si « c’est surtout comme mathématicien qu’il s’est fait un nom » et s’il vouait une passion à la bibliographie. Henri Brocard 2 est un exemple d’ingénieur-savant du XIXe siècle, à savoir un polytechnicien à la charnière entre la théorie et la pratique mais aussi entre les mathématiques et la physique. DR Brocard fournit sur son temps libre une importante production mathématique, en particulier sur la nouvelle géométrie du triangle. ENRI BROCARD est né le 12 mai au 2e régiment de Montpellier, il fait 1845 dans une petite commune ensuite un bref passage dans l’armée H de la Meuse, à quarante kilo- du Rhin. Il participe à ses premières mètres de Bar-le-Duc. Il entre à campagnes pendant la guerre franco- l’École polytechnique allemande de 1870. en 1865 puis rejoint le Il est fait prison- Corps des ingénieurs « Durant trente-sept ans nier lors de la de l’armée française. défaite de Sedan. En 1869, après deux dans l’armée, il participe Il fait aussi par- années d’études à à onze campagnes » tie de la colonne l’École d’application, de Sétif lors de il est nommé lieute- la révolte des nant en second du Génie. -
A CONCISE MINI HISTORY of GEOMETRY 1. Origin And
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 5{21. A CONCISE MINI HISTORY OF GEOMETRY LEOPOLD VERSTRAELEN 1. Origin and development in Old Greece Mathematics was the crowning and lasting achievement of the ancient Greek cul- ture. To more or less extent, arithmetical and geometrical problems had been ex- plored already before, in several previous civilisations at various parts of the world, within a kind of practical mathematical scientific context. The knowledge which in particular as such first had been acquired in Mesopotamia and later on in Egypt, and the philosophical reflections on its meaning and its nature by \the Old Greeks", resulted in the sublime creation of mathematics as a characteristically abstract and deductive science. The name for this science, \mathematics", stems from the Greek language, and basically means \knowledge and understanding", and became of use in most other languages as well; realising however that, as a matter of fact, it is really an art to reach new knowledge and better understanding, the Dutch term for mathematics, \wiskunde", in translation: \the art to achieve wisdom", might be even more appropriate. For specimens of the human kind, \nature" essentially stands for their organised thoughts about sensations and perceptions of \their worlds outside and inside" and \doing mathematics" basically stands for their thoughtful living in \the universe" of their idealisations and abstractions of these sensations and perceptions. Or, as Stewart stated in the revised book \What is Mathematics?" of Courant and Robbins: \Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either". -
A Survey of the Development of Geometry up to 1870
A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics.