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A Study of Dandelin Spheres: a Second Year Investigation
CALIFORNIA STATE SCIENCE FAIR 2009 PROJECT SUMMARY Name(s) Project Number Sundeep Bekal S1602 Project Title A Study of Dandelin Spheres: A Second Year Investigation Abstract Objectives/Goals The purpose of this investigation is to see if there is a relationship that forms between the radius of the smaller dandelin sphere to the tangent of the angle RLM and distances that form inside the ellipse. Methods/Materials I investigated a conic section of an ellipse that contained two Dandelin Spheres to determine if there was such a relationship. I used the same program as last year, Geometer's Sketchpad, to generate measurements of last year's 2D model of the conic section so that I could better investigate the relationship. After slicing the spheres along the central pole, I looked to see if there were any patterns or relationships. To do this I set the radius of the smaller circle (sphere in 3-D) to 1.09 cm. I changed the radius by .05 cm every time while also noting how the other segments changed or did not change. Materials -Geometer's Sketchpad -Calculator -Computer -Paper -Pencil Results After looking through the data tables I noticed that the radius had the same exact values as (a+c)(tan(1/2) angleRLM), where (a+c) is the distance located in the ellipse. I also noticed that the ratio of the (radius)/(a+c) had the same values as tan((1/2 angleRLM)). I figured that this would be true because the tan (theta)=(opposite)/(adjacent) which in this case, would be tan ((1/2 angleRLM)) = (radius)/(a+c). -
Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi
HEXAFLEXAGONS, PROBABILITY PARADOXES, AND THE TOWER OF HANOI For 25 of his 90 years, Martin Gard- ner wrote “Mathematical Games and Recreations,” a monthly column for Scientific American magazine. These columns have inspired hundreds of thousands of readers to delve more deeply into the large world of math- ematics. He has also made signifi- cant contributions to magic, philos- ophy, debunking pseudoscience, and children’s literature. He has produced more than 60 books, including many best sellers, most of which are still in print. His Annotated Alice has sold more than a million copies. He continues to write a regular column for the Skeptical Inquirer magazine. (The photograph is of the author at the time of the first edition.) THE NEW MARTIN GARDNER MATHEMATICAL LIBRARY Editorial Board Donald J. Albers, Menlo College Gerald L. Alexanderson, Santa Clara University John H. Conway, F.R. S., Princeton University Richard K. Guy, University of Calgary Harold R. Jacobs Donald E. Knuth, Stanford University Peter L. Renz From 1957 through 1986 Martin Gardner wrote the “Mathematical Games” columns for Scientific American that are the basis for these books. Scientific American editor Dennis Flanagan noted that this column contributed substantially to the success of the magazine. The exchanges between Martin Gardner and his readers gave life to these columns and books. These exchanges have continued and the impact of the columns and books has grown. These new editions give Martin Gardner the chance to bring readers up to date on newer twists on old puzzles and games, on new explanations and proofs, and on links to recent developments and discoveries. -
A Human Introduction to Geometry Spring 2017 UM Da Vinci Program Drew Armstrong
A Human Introduction to Geometry Spring 2017 UM da Vinci Program Drew Armstrong Contents 1 The Pythagorean Tradition? 2 1.1 Pythagoras and 2.................................. 2 1.2 Plato and Regular Polyhedra . 9 1.3 Kepler and Conic Sections . 18 2 Euclidean and Non-Euclidean Geometry 29 2.1 The Deductive Method . 29 2.2 Euclid's Elements ................................... 32 2.3 Selections from Book I . 40 2.4 Triangles and Curvature . 48 3 The Problem of Measurement 60 3.1 Pure and Applied Mathematics . 60 3.2 Eudoxus' Theory of Proportion . 63 3.3 Archimedes and the Existence of π ......................... 70 3.4 Trigonometry is Hard . 83 3.5 Rigorous and Intuitive Mathematics . 105 3.6 Impossible Problems . 109 4 Coordinate Geometry and Transformations 110 5 Projective Geometry 110 Introduction Geometry is the most human of mathematical pursuits; so much so that it was regarded as insufficiently rigorous for twentieth century tastes and was largely banished from the under- graduate curriculum. This is a shame because drawing and looking at pictures are excellent ways to engage students. Visual intuition is also the primary way that we can hack our pri- mate architecture, to allow us to make progress in more abstract kinds of mathematics that would otherwise be impossible to comprehend. In this class we will embrace the human side of mathematics by following the story of geometry|pure and applied|through the ages. On the applied side we will follow geom- etry from its earliest use in land measurement, through the discovery of perspective drawing in the Renaissance, to its use today in computer graphics. -
The Miura-Ori Opened out Like a Fan INTERNATIONAL SOCIETY for the INTERDISCIPLINARY STUDY of SYMMETRY (ISIS-SYMMETRY)
The Quarterly of the Editors: International Society for the GyiSrgy Darvas and D~nes Nag¥ interdisciplinary Study of Symmetry (ISIS-Symmetry) Volume 5, Number 2, 1994 The Miura-ori opened out like a fan INTERNATIONAL SOCIETY FOR THE INTERDISCIPLINARY STUDY OF SYMMETRY (ISIS-SYMMETRY) President ASIA D~nes Nagy, lnslltute of Apphed Physics, University of China. t~R. Da-Fu Ding, Shangha~ Institute of Biochemistry. Tsukuba, Tsukuba Soence C~ty 305, Japan Academia Stoma, 320 Yue-Yang Road, (on leave from Eotvos Lot’find Umve~ty, Budapest, Hungary) Shanghai 200031, PR China IGeometry and Crystallography, H~story of Science and [Theoreucal B~ology] Tecbnology, Lmgmsucs] Le~Xiao Yu, Department of Fine Arts. Nanjmg Normal Umvers~ty, Nanjmg 210024, P.R China Honorary Presidents }Free Art, Folk Art, Calhgraphy] Konstantin V. Frolov (Moscow) and lndta. Kirti Trivedi, Industrial Design Cenlre, lndmn Maval Ne’eman (TeI-Avw) Institute of Technology, Powa~, Bombay 400076, India lDes~gn, lndmn Art] Vice-President Israel. Hanan Bruen, School of Education, Arthur L. Loeb, Carpenter Center for the V~sual Arts, Umvers~ty of Hallo, Mount Carmel, Haffa 31999, Israel Harvard Umverslty. Cambridge, MA 02138, [Educanon] U S A. [Crystallography, Chemical Physics, Visual Art~, Jim Rosen, School of Physics and Astronomy, Choreography, Music} TeI-Av~v Umvers~ty, Ramat-Avtv, Tel-Av~v 69978. Israel and [Theoretical Physms] Sergei V Petukhov, Instnut mashmovedemya RAN (Mechamcal Engineering Research Institute, Russian, Japan. Yasushi Kajfl~awa, Synergel~cs Institute. Academy of Scmnces 101830 Moskva, ul Griboedova 4, Russia (also Head of the Russian Branch Office of the Society) 206 Nakammurahara, Odawara 256, Japan }Design, Geometry] }B~omechanlcs, B~ontcs, Informauon Mechamcs] Koichtro Mat~uno, Department of BioEngineering. -
Ellipse, Hyperbola and Their Conjunction Arxiv:1805.02111V2
Ellipse, Hyperbola and Their Conjunction Arkadiusz Kobiera Warsaw University of Technology Abstract This article presents a simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertices of the cones is a hyperbola. The hyperbola has foci which coincidence with the ellipse vertices. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the foci of the ellipse are located in the hyperbola's vertices. These two relationships create a kind of conjunction between the ellipse and the hyperbola which originate from the cones used for generation of these curves. The presented conjunction of the ellipse and hyperbola is a perfect example of mathematical beauty which may be shown by the use of very simple geometry. As in the past the conic curves appear to be arXiv:1805.02111v2 [math.HO] 26 Jan 2019 very interesting and fruitful mathematical beings. 1 Introduction The conical curves are mathematical entities which have been known for thousands years since the first Menaechmus' research around 250 B.C. [2]. Anybody who has attempted undergraduate course of geometry knows that ellipse, hyperbola and parabola are obtained by section of a cone by a plane. Every book dealing with the this subject has a sketch where the cone is sec- tioned by planes at various angles, which produces different kinds of conics. Usually authors start with the cone to produce the conic curve by section. -
Mathematical Theory of Big Game Hunting
DOCUMEWT RESUME ED 121 522 SE 020 770 AUTHOR Schaaf, Willie' L. TITLE A Bibliography of Recreational Mathematics, Volume 1. Fourth, Edition. INSTITUTION National Council of Teachers of Mathematics, Inc., Reston, Va. PUB DATE 70 NOTE 160p.; For related documents see Ed 040 874 and ED 087 631 AVAILABLE FROMNational Council of Teachers of Mathematics, Inc., 1906 Association Drive, Reston, Virginia 22091 EDRS PRICE MP-$0.83 Plus Postage. HC Not Available from EDRS. DESCRIPTORS *Annotated Bibliographies; *Games; Geometric Concepts; History; *Literature Guides; *Mathematical Enrichment; Mathematics; *Mathematics Education; Number Concepts; Reference Books ABSTRACT This book is a partially annotated bibliography of books, articles, and periodicals concerned with mathematical games, puzzles, and amusements. It is a reprinting of Volume 1 of a three-volume series. This volume, originally published in 1955, treats problems and recreations which have been important in the history of mathematics as well as some of more modern invention. The book is intended for use by both professional and amateur mathematicians. Works on recreational mathematics are listed in eight broad categories: general works, arithmetic and algebraic recreations, geometric recreations, assorted recreations, magic squares, the Pythagorean relationship, famous problems of antiquity, and aathematical 'miscellanies. (SD) *********************************************************************** Documents acquired by ERIC include many informal unpublished * materials not available from other sources. ERIC sakes every effort * * to obtain the best copy available. Nevertheless, items of marginal * * reproducibility are often encountered and this affects the quality* * of the microfiche and hardcopy reproductions ERIC aakes available * * via the ERIC Document Reproduction Service (EDRS). EDRS is not * responsible for the quality of the original document. -
Exploring Mathematics
ii A Teacher’s Guide to Exploring Mathematics Ames Bryant Kit Pavlekovsky Emily Turner Tessa Whalen-Wagner Adviser: Deanna Haunsperger ii Contents Contents I Introduction V I Number Sense 1 Counting Basics 3 Grocery Shopping 4 Units 5 Modular Arithmetic 6 Logarithms and Exponents 8 Changing Bases 11 II Geometry 15 Construction I 17 Construction II 21 Möbius Strips 22 Pythagorean Theorem 23 How Far Can a Robot Reach? 25 Spherical Geometry 27 I II CONTENTS III Sets 29 De Morgan’s Law 31 Building Sets 32 Subsets 34 Investigating Infinity 36 IV Probability 39 Probability vs. Reality 41 Game Theory 43 Continuous Probability 45 Bayes’ Theorem 45 V Patterns 51 Math in Nature 53 Fibonacci in Nature 53 Counting Patterns 55 Tiling 57 VI Data 61 Bad Graphs 63 Graphs 64 Introduction to Statistics 66 CONTENTS III VII Logic 71 Pigeonhole Principle 73 Formal Logic 74 Logical Paradoxes 76 Logic Puzzles 77 VIII Miscellaneous 85 Map Coloring 87 Shortest Path Problem 88 Flexagons 90 Math in Literature 92 IV CONTENTS Introduction V VI Part I Number Sense 1 Number Sense Counting Basics Goals Learn basic skills for counting possible choices, estimating number of possible outcomes Supplies N/A Prior Knowledge N/A 9. How many ways are there for the word "light?" How many ways can you rearrange the word "happy" and end up with "happy"? 10. Think about using the shepherd’s principle. The sheep are the number of circular arrange- ments, and their legs seem to be linear arrangements of the same number of people. How many legs on each sheep? 11. -
Cylinder-Hyperboloid-Cone 15
Paraboloid Parabola Hyperbola Line 15 Hyperboloid Ellipse Hyperbolic Cylinder-Hyperboloid-Cone paraboloid Sphere www.magicmathworks.org/geomlab15 Line pair Find the model with strings joining two disks. Cone Begin with the strings vertical so that the resulting solid Sine is a cylinder. curve Cylinder Rotate the top progressively to obtain in turn a Tractrix hyperboloid ('of one sheet') and a double cone. Exponential Circle The cylinder and cone are limiting cases of the curve hyperboloid, from which all the sections 14.1 to 14.4 can be obtained. Try experiment 7 again with the model Line Catenary family where the hyperboloid is fixed in position and contains two spheres. Catenoid Loxodrome Since both the cylinder and cone have elliptical sections, it must be possible to orientate the two so that their intersection has such a boundary. Study the models of success and failure. Helix Equiangular Proof in Geomlab file. spiral Helicoid Archimedean Plane spiral Tiling Polygon Polyhedron To see why an oblique closed section of a hyperboloid of one sheet is an ellipse, we begin with the circular section of a cylinder and transform it in two stages. First we transform it into an oblique section of a cylinder. Then we take that and transform it into the section we want. We shear the cylinder parallel to the black plane. The effect is to stretch the circle in the direction AB. The result is an ellipse. The projection perpendicular to the vertical plane through AB looks like this: B B A A We therefore know that a curve which projects as a circle in plan and a straight line in elevation is an ellipse. -
Conic Sections
Conic sections Conics have recurred throughout this book in both geometric and algebraic settings. Hence, I have decided to dedicate the final chapter to them. As the only conics appearing on IMO geometry problems are invariably circles, the results proved in this chapter are largely irrelevant. Nevertheless, the material is sufficiently interesting to be worthy of inclusion. Sections of cones With the obvious exception of the circle, the conics were first discovered by the Greek mathematician Menaech- mus who contemplated slicing a right circular cone C with a flat plane . Indeed, the term ‘conic’ is an abbrevia- tion of conic section . It is more natural to consider C as the double cone with equation x2 y2 z2. If cuts both cones, then the conic section is a hyperbola . If it cuts only one cone in a closed curve, it is an ellipse . The intermediate case, where the plane is inclined at exactly the same slope as the cone, results in a parabola . Observe that x2 y2 z2 is the equation of a projective circle; this explains why all conic sections are equivalent under projective transformations. We define a Dandelin sphere 5 to be a sphere tangent to both C (at a circle *) and (at a point F, namely the focus ). The plane containing * intersects at a line ", known as the directrix . 1. Prove that the directrix is the polar of the focus. For an arbitrary point P on the conic, we let P R meet * at Q. 2. Prove that P Q P F . 3. Let A be the foot of the perpendicular from P to the plane containing *. -
Contents More Information
Cambridge University Press 0521819709 - Flexagons Inside Out Les Pook Table of Contents More information Contents Preface page ix List of notation xi Introduction 1 1 Making and flexing flexagons 5 1.1 The trihexaflexagon 5 1.2 The pinch flex 6 1.3 A simple square flexagon 7 1.4 General assembly instructions 8 1.5 Decoration of faces 13 2 Early history of flexagons 14 2.1 The discovery of flexagons 14 2.2 The Flexagon Committee 15 2.3 Revival of interest 16 3 Geometry of flexagons 18 3.1 Flexagon characteristics 19 3.2 Appearance of main positions 20 3.3 Flexagons as linkages 21 3.4 Mapping flexagons 24 3.5 Flexagon figures and symbols 29 4 Hexaflexagons 31 4.1 Degrees of freedom of hexaflexagons 32 4.2 The trihexaflexagon revisited 33 4.3 Multicycle hexaflexagons 36 4.4 Hexaflexagon design 40 4.5 Hexaflexagon face arrangements 42 5 Hexaflexagon variations 47 5.1 Triangle flexagons 47 5.2 The V-flex 49 v © Cambridge University Press www.cambridge.org Cambridge University Press 0521819709 - Flexagons Inside Out Les Pook Table of Contents More information vi Contents 5.3 Origami with hexaflexagons 52 6 Square flexagons 53 6.1 Single cycle square flexagons 54 6.2 Reconstruction from a flexagon figure 58 6.3 Square flexagons with main position links 59 6.4 Distinct types of square flexagon with main position links 69 6.5 Square flexagons with box position links 70 6.6 Square flexagon face arrangements 76 6.7 The interleaf flex 78 7 Introduction to convex polygon flexagons 82 7.1 The convex polygon family 83 7.2 Degrees of freedom of intermediate positions 84 7.3 -
Kepler's Laws and Conic Sections
Arnold Math J. (2016) 2:139–148 DOI 10.1007/s40598-015-0030-6 RESEARCH EXPOSITION Kepler’s Laws and Conic Sections Alexander Givental1,2 To the memory of Valery A. Senderov Received: 5 July 2015 / Revised: 7 September 2015 / Accepted: 24 October 2015 / Published online: 23 December 2015 © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2015 Abstract The geometry of Kepler’s problem is elucidated by lifting the motion from the (x, y)-plane to the cone r 2 = x2 + y2. Keywords Kepler’s laws · Conic section Introduction. In the senior year at the Moscow high school no. 2, my astronomy term paper1 was about Kepler’s laws of planetary motion. The intention was to derive the three famous empirical laws from the first principles of Newtonian gravitation. Assuming the 1st law, which stipulates that each orbit is a conic section with the center of attraction at its focus, I had no difficulty deriving the 2nd and 3rd laws (about conservation of sectorial velocity, and the relation between the periods of revolutions with the orbits’ sizes respectively). However, the derivation of the 1st law per se resisted my effort, apparently, for the lack of skill in solving ordinary differential equations. Physics and math were taught in our school at an advanced level, and so I had no problems with differentiation of functions, and even with setting Kepler’s problem in the ODE form: r r¨ =−k . |r|3 1 Astronomy teacher: N. V Mart’yanova, geometry and algebra teacher: Z. M. Fotieva, calculus teacher: V. A. Senderov. -
Paper-Fold Physics: Newton’S Three Laws of Motion Lisa (Yuk Kuen) Yau Revised Date: Monday, July 30, 2018
Unit Title: Paper-fold Physics: Newton’s Three Laws of Motion Lisa (Yuk Kuen) Yau Revised Date: Monday, July 30, 2018 1. Abstract Can you teach physics effectively using only paper and the simple act of folding? This interdisciplinary curriculum unit is designed for a 5th grade class focusing on how to unpack Newton’s Three Laws of Motion using paper folding (i.e. origami) as a teaching tool to promote inquiry and project-based learning. Students will learn scientific concepts by making paper models, testing them to form hypotheses, analyzing the collected data with graphs, researching supporting theories of Newton, making real-world connections, writing scientific conclusions, and applying their new understandings to solve challenges such as how to design a paper container to protect a fragile egg from cracking under a stressful landing. Can paper folding be a teaching strategy to promote an equitable and competitive learning environment for all students from the hyperactive to the hypersensitive, from the “underachievers” to the “overachievers”, and all other biased labels in between? Folding in solitude is a calming activity that builds concentration and focus (i.e., mindfulness), allows each student multiple opportunities to fail and succeed. In the repeating actions of folding in order to gain mastery, student can retain the learned knowledge better and longer. From my own experience, I have witnessed how powerful origami can help students to verbalize what they are learning to each other. Students gain a better sense of self and community as they make personal connections to real problems and situations with their hands “thinking out loud.” Of course, origami is not the magic pill but if uses properly, it can open many doors of positive learning for both students and teachers from all walks of life.