Math for Mystics - From the Fibonacci Sequence to Luna’s Labyrinth to the Golden Section and Other Secrets of Sacred Geometry

Renna Shesso 

Weiser Books | 2007 | 210 páginas | rar - pdf | 2,6 Mb

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Much of math history comes to us from early astrologers who needed to be able to describe and record what they saw in the night sky. Whether you were the king's court astrologer or a farmer marking the best time for planting, timekeeping and numbers really mattered. Mistake a numerical pattern of petals and you could be poisoned. Lose the rhythm of a sacred dance or the meter of a ritually told story and the intricately woven threads that hold life together were spoiled. Ignore the celestial clock of equinoxes and solstices, and you'd risk being caught short of food for the winter.
Shesso's friendly tone and clear grasp of the information make the math "go down easy" in this marvelous book

ContentsIntroduction: “Math?! Why?” vII
Chapter 0 The Circle of Creation 1
Chapter 1 Counting 5
Chapter 2 The Moon 13
Chapter 3 Measurements 21
Chapter 4 The Days of the Week 25
Chapter 5 The Magical Squares 47
Chapter 6 The Knight’s Tour and Templar Codes? 69
Chapter 7 Shapes and Numbers Meditation 83
Chapter 8 Pythagoras 95
Chapter 9 Fibonacci, the Golden Ratio, and the Pentacle 101
Chapter 10 Venus’ Pentacle 115
Chapter 11 The Geometric Solids 123
Chapter 12 Individual Numbers 129
Chapter 13 A Tale in Which Gods Do Math 159
Chapter 14 Summing Up 161
Notes 165
Bibliography 175
Index 183

Heavenly mathematics : the forgotten art of spherical trigonometry

Glen Van Brummelen

Princeton University Press |  2012 | 215 páginas | rar - pdf | 4 Mb

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epub - 6,7 Mb - link
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Spherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. The discipline was a mainstay of mathematics education for centuries, and it was a standard subject in high schools until the 1950s. Today, however, it is rarely taught. Heavenly Mathematics traces the rich history of this forgotten art, revealing how the cultures of classical Greece, medieval Islam, and the modern West used spherical trigonometry to chart the heavens and the Earth. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation for its elegant proofs and often surprising conclusions.

Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams that have been used to teach the subject in the past. This unique compendium also features easy-to-use appendixes as well as exercises at the end of each chapter that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

Contents
Preface vii

1 Heavenly Mathematics 1
2 Exploring the Sphere 23
3 The Ancient Approach 42
4 The Medieval Approach 59
5 The Modern Approach: Right- Angled Triangles 73
6 The Modern Approach: Oblique Triangles 94
7 Areas, Angles, and Polyhedra 110
8 Stereographic Projection 129
9 Navigating by the Stars 151
Appendix A. Ptolemy’s Determination of the Sun’s Position 173
Appendix B. Textbooks 179
Appendix C. Further Reading 182
Index 189

Help Your Kids with Math: A visual problem solver for kids and parents

 

Barry Lewis

DK Publishing | 2010 | 258 páginas | rar - pdf | 9,7 Mb

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Studying math is often a source of great anxiety for children and also proves troublesome for parents helping with their homework.

Using uniquely accessible illustrated stress-free approach, Help Your Kids with Math looks at every aspect of math, from simple sums to simultaneous equations, and explains each facet in easily understandable language so that adults and kids can master the subject together.

In Help Your Kids with Math tricky concepts are explored and examined step-by-step, so that even the most math-phobic individual will be able to approach and solve complex problems with confidence.

Contents
NUMBERS
Introducing numbers ; Addition ; Subtraction ; Multiplication ; Division ; Prime numbers ; Units of measurement ; Positive and negative numbers ; Powers and roots ; Standard form ; Decimals in action; Fractions, Ratio and proportion, Percentages, Converting fractions, decimals, and percentages ; Mental math ; Rounding off ; Using a calculator ; Personal finance ; Business finance
GEOMETRY
What is geometry?; Angles; Straight lines; Symmetry; Coordinates; Vectors; Translations ; Rotations; Reflections; Enlargements; Scale drawings; Bearings; Constructions; Loci ; Triangles; Constructing triangles; Congruent triangles; Area of a triangle; Similar triangles ; Pythagorean Theorem ; Quadrilaterals; Polygons ; Circles ; Circumference and diameter ; Area of a circle ; Angles in a circle; Chords and cyclic quadrilaterals ; Tangents ; Arcs ; Sectors ; Solids ; Volumes ; Surface area 148
TRIGONOMETRY
What is trigonometry? ; Working with trigonometry ; Finding missing sides ; Finding missing angles 
ALGEBRA
What is algebra?; Sequences; Working with expressions; Expanding and factorizing expressions; Quadratic expressions; Formulas; Solving equations; Linear graphs; Simultaneous equations; Factorizing quadratic equations; The quadratic formula; Quadratic graphs ; Inequalities 
STATISTICS
What is statistics? ; Collecting and organizing data ; Bar charts ; Pie charts ; Line graphs;  Averages; Moving Averages ; Measuring spread ; Histograms ; Scatter diagrams 
PROBABILITY
What is probability? , Expectation and reality ; Multiple probability ; Dependent events ; Tree diagrams 
Reference section 232
Glossary 244
Index 252
Acknowledgments 256

Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education

Patsy Wang-Iverson, Robert J. Lang e Mark YIM

A K Peters/CRC Press | 2011 | 632 páginas | rar - pdf |11,65 Mb

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Origami⁵ continues in the excellent tradition of its four previous incarnations, documenting work presented at an extraordinary series of meetings that explored the connections between origami, mathematics, science, technology, education, and other academic fields.

The fifth such meeting, 5OSME (July 13–17, 2010, Singapore Management University) followed the precedent previous meetings to explore the interdisciplinary connections between origami and the real world. This book begins with a section on origami history, art, and design. It is followed by sections on origami in education and origami science, engineering, and technology, and culminates with a section on origami mathematics—the pairing that inspired the original meeting.

Within this one volume, you will find a broad selection of historical information, artists’ descriptions of their processes, various perspectives and approaches to the use of origami in education, mathematical tools for origami design, applications of folding in engineering and technology, as well as original and cutting-edge research on the mathematical underpinnings of origami.

Contents

Preface xvii

I Origami History, Art, and Design 1

History of Origami in the East and the West before Interfusion 3

Koshiro Hatori

Deictic Properties of Origami Technical Terms and Translatability: Cross-Linguistic Differences between English and Japanese 13

Koichi Tateishi

Betsy Ross Revisited: General Fold and One-Cut Regular and Star Polygons 29

Arnold Tubis and Crystal Elaine Mills

Reconstructing David Huffman’s Legacy in Curved-Crease Folding 39

Erik D. Demaine, Martin L. Demaine, and Duks Koschitz

Simulation of Nonzero Gaussian Curvature in Origami by Curved-Crease Couplets 53

Cheng Chit Leong

Compression and Rotational Limitations of Curved Corrugations 69

Christine E. Edison

Polygon Symmetry Systems 81

Andrew Hudson

New Collaboration on Modular Origami and LED 89

Miyuki Kawamura and Hiroyuki Moriwaki

Using the Snapology Technique to Teach Convex Polyhedra 99

Faye Goldman

A Systematic Approach to Twirl Design 111

Krystyna Burczyk and Wojciech Burczyk

Oribotics: The Future Unfolds 127

Matthew Gardiner

II Origami in Education 139

Origametria and the van Hiele Theory of Teaching Geometry 141

Miri Golan

Student Teachers Introduce Origami in Kindergarten and Primary Schools: Froebel Revisited 151

Maria Llu¨ısa Fiol, Neus Dasquens, and Montserrat Prat

Narratives of Success: Teaching Origami in Low-Income Urban Communities 165

Christine E. Edison

Origami and Spatial Thinking of College-Age Students 173

Norma Boakes

Close Observation and Reverse Engineering of Origami Models 189

James Morrow and Charlene Morrow

Origami and Learning Mathematics 205

Sue Pope and Tung Ken Lam

Hands-On Geometry with Origami 219

Michael J. Winckler, Kathrin D. Wolf, and Hans Georg Bock

My Favorite Origamics Lessons on the Volume of Solids 233

Shi-Pui Kwan

III Origami Science, Engineering, and Technology 251

Rigid-Foldable Thick Origami 253

Tomohiro Tachi

Folding a Patterned Cylinder by Rigid Origami 265

Kunfeng Wang and Yan Chen

The Origami Crash Box 277

Jiayao Ma and Zhong You

Origami Folding: A Structural Engineering Approach 291

Mark Schenk and Simon D. Guest

Designing Technical Tessellations 305

Yves Klett and Klaus Drechsler

A Simulator for Origami-Inspired Self-Reconfigurable Robots 323

Steven Gray, Nathan J. Zeichner, Mark Yim, and Vijay Kumar

A CAD System for Diagramming Origami with Prediction of Folding Processes 335

Naoya Tsuruta, Jun Mitani, Yoshihiro Kanamori, and Yukio Fukui

Development of an Intuitive Algorithm for Diagramming and 3D Animated Tutorial for Folding Crease Patterns 347

Hugo Akitaya, Matheus Ribeiro, Carla Koike, and Jose Ralha

Hands-Free Microscale Origami 371

Noy Bassik, George M. Stern, Alla Brafman, Nana Y. Atuobi, and David H. Gracias

Foldable Parylene Origami Sheets Covered with Cells: Toward Applications in Bio-Implantable Devices 385

Kaori Kuribayashi-Shigetomi and Shoji Takeuchi

IV Mathematics of Origami 393

Introduction to the Study of Tape Knots 395

Jun Maekawa

Universal Hinge Patterns for Folding Orthogonal Shapes 405

Nadia M. Benbernou, Erik D. Demaine, Martin L. Demaine, and Aviv Ovadya

A General Method of Drawing Biplanar Crease Patterns 421

Herng Yi Cheng

A Design Method for Axisymmetric Curved Origami with Triangular Prism Protrusions 437

Jun Mitani

Folding Any Orthogonal Maze 449

Erik D. Demaine, Martin L. Demaine, and Jason S. Ku

Every Spider Web Has a Simple Flat Twist Tessellation 455

Robert J. Lang and Alex Bateman

Flat-Unfoldability and Woven Origami Tessellations 475

Robert J. Lang

Degenerative Coordinates in 22.5◦ Grid System 489

Tomohiro Tachi and Erik D. Demaine

Two Folding Constructions 499

Robert Orndorff

Variations on a Theorem of Haga 507

Emma Frigerio

Precise Division of Rectangular Paper into an Odd Number of Equal Parts without Tools: An Origamics Exercise 519

Kazuo Haga

The Speed of Origami Constructions Versus Other Construction Tools 531

Eul`alia Tramuns

A Note on Operations of Spherical Origami Construction 543

Toshikazu Kawasaki

Origami Alignments and Constructions in the Hyperbolic Plane 553

Roger C. Alperin

A Combinatorial Definition of 1D Flat-Folding 575

Hidefumi Kawasaki

Stamp Foldings with a Given Mountain-Valley Assignment 585

Ryuhei Uehara

Flat Vertex Fold Sequences 599

Thomas C. Hull and Eric Chang

Circle Packing for Origami Design Is Hard 609

Erik D. Demaine, S´andor P. Fekete, and Robert J. Lang

Contributors 627

Outros livros das atas das conferências:

Origami 4 : Fourth International Meeting of Origami Science, Mathematics, and Education por Robert J Lang; Origami USA.; Idioma: Inglês   Editora: Natick, Mass. : A K Peters, ©2009.

Origami³ : Third International Meeting of Origami Science, Math, and Education por Thomas Hull; Idioma: Inglês   Editora: Natick, MA : A K Peters, ©2002.

Cool Flexagon Art: Creative Activities That Make Math & Science Fun for Kids!

 

Anders Hanson e Elissa Mann

Checkerboard Library | 2013 | 34 páginas | rar - pdf | 2,45 Mb

link (password : matav)

ContentsCOOL FLEXAGON ART
HIDDEN IN THE FOLDS
POLYGON BASICS
SIDES, VERTICES, AND ANGLES
IN THE FOLD
TYPES OF FLEXAGONS
FOLDING TIPS
MAKE IT EASIER!
PROJECT 1
MAKE A HEXA-TETRA FLEXAGON
PROJECT 2
MAKE A HEXA-HEXA FLEXAGON
3-D GEOMETRY
FROM SHAPES TO SOLIDS
PROJECT 3
MAKE A 3-D FLEXAGON
PROJECT 4
TRANSFORMER STAR
MATH TERMS
GLOSSARY
WEB SITES
INDEX

Charming Proofs A Journey into Elegant Mathematics

Claudi Alsina e Roger B. Nelsen

Dolciani mathematical expositions, nº 42

The Mathematical Association of America | 2011 | 320 páginas | rar - pdf | 2,35 Mb

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Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy'. Charming Proofs presents a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, and to develop the ability to create proofs themselves. The authors consider proofs from topics such as geometry, number theory, inequalities, plane tilings, origami and polyhedra. Secondary school and university teachers can use this book to introduce their students to mathematical elegance. More than 130 exercises for the reader (with solutions) are also included.

Contents
A garden of integers
Distinguished numbers
Points in the plane
The polygonal playground
A treasury of triangle theorems
The enchantment of the equilateral triangle
The quadrilaterals' corner
Squares everywhere
Curves ahead
Adventures in tiling and coloring
Geometry in three dimensions
Additional theorems, problems, and proofs.

The Geometry of Numbers

 C. D. Olds, Anneli Lax and Giuliana P. Davidoff

The Mathematical Association of America | 2000 | 193 páginas | rar - pdf | 786 kb

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djvu -3,5 Mb link

This is a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice points on lines, circles and inside simple polygons in the plane. A minimum of mathematical expertise is required beyond an acquaintance with elementary geometry. The authors gradually lead up to the theorems of Minkowski and others who succeeded him. On the way the reader will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres.

Contents
Lattice Points and Number Theory
An Introduction to the Geometry of Numbers
Gaussian Integers, by Peter D. Lax
The Closest Packing of Convex Bodies
Brief Biographies
Solutions and Hints.

The Glorious Golden Ratio

Alfred S. Posamentier e Ingmar Lehmann

Prometheus Books | 2011 | 363 páginas | rar - epub | 20 Mb

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What exactly is the Golden Ratio? How was it discovered? Where is it found? These questions and more are thoroughly explained in this engaging tour of one of mathematics' most interesting phenomena. The authors trace the appearance of the Golden Ratio throughout history, demonstrate a variety of ingenious techniques used to construct it, and illustrate the many surprising geometric figures in which the Golden Ratio is embedded.
Requiring no more than an elementary knowledge of geometry and algebra, the authors give readers a new appreciation of the indispensable qualities and inherent beauty of mathematics.

Contents
Acknowledgments
Introduction
Chapter 1: Defining and Constructing the Golden Ratio
Chapter 2: The Golden Ratio in History
Chapter 3: The Numerical Value of the Golden Ratio and Its Properties
Chapter 4: Golden Geometric Figures
Chapter 5: Unexpected Appearances of the Golden Ratio
Chapter 6: The Golden Ratio in the Plant Kingdom
Chapter 7: The Golden Ratio and Fractals
Concluding Thoughts
Appendix: Proofs and Justifications of Selected Relationships
Notes
Index

Non-Euclidean Geometry

(Mathematical Association of America Textbooks)

H. S. M. Coxeter

The Mathematical Association of America | 1998 | 355 páginas | rar - pdf | 11 Mb

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DjVu | 2,8 Mb
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Referência em: MathEduc

This is a reissue of Professor Coxeter's classic text on non-Euclidean geometry. It begins with a historical introductory chapter, and then devotes three chapters to surveying real projective geometry, and three to elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'. This is essential reading for anybody with an interest in geometry.

CONTENTS
I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY
1.1 Euclid 1
1.2 Saccheri and Lambert 5
1.3 Gauss, Wächter, Schweikart, Taurinus 7
1.4 Lobatschewsky 8
1.5 Bolyai 10
1.6 Riemann 11
1.7 Klein 13
II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS
2.1 Definitions and axioms 16
2.2 Models 23
2.3 The principle of duality 26
2.4 Harmonic sets 28
2.5 Sense 31
2.6 Triangular and tetrahedral regions 34
2.7 Ordered correspondences 35
2.8 One-dimensional projectivities 40
2.9 Involutions 44
III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS
3.1 Two-dimensional projectivities 48
3.2 Polarities in the plane 52
3.3 Conies 55
3.4 Projectivities on a conic 59
3.5 The fixed points of a collineation 61
3.6 Cones and reguli 62
3.7 Three-dimensional projectivities 63
3.8 Polarities in space 65
IV. HOMOGENEOUS COORDINATES
4.1 The von Staudt-Hessenberg calculus of points 7
4.2 One-dimensional projectivities 74
4.3 Coordinates in one and two dimensions 76
4.4 Collineations and coordinate transformations 8
4.5 Polarities 85
4.6 Coordinates in three dimensions 87
4.7 Three-dimensional projectivities 90
4.8 Line coordinates for the generators of a quadric 9
4.9 Complex projective geometry 94
V. ELLIPTIC GEOMETRY IN ONE DIMENSION
5.1 Elliptic geometry in general 95
5.2 Models 96
5.3 Reflections and translations 97
5.4 Congruence 100
5.5 Continuous translation 101
5.6 The length of a segment 103
5.7 Distance in terms of cross ratio 104
5.8 Alternative treatment using the complex line 10
VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS
6.1 Spherical and elliptic geometry 109
6.2 Reflection 110
6.3 Rotations and angles Ill
6.4 Congruence 113
6.5 Circles 115
6.6 Composition of rotations 118
6.7 Formulae for distance and angle 120
6.8 Rotations and quaternions 122
6.9 Alternative treatment using the complex plane 126
VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS
7.1 Congruent transformations 128
7.2 Clifford parallels 133
7.3 The Stephanos-Cartan representation of rotations by points 136
7.4 Right translations and left translations 138
7.5 Right parallels and left parallels 141
7.6 Study's representation of lines by pairs of points 146
7.7 Clifford translations and quaternions 148
7.8 Study's coordinates for a line 151
7.9 Complex space 153
VIII. DESCRIPTTE GEOMETRY
8.1 Klein's projective model for hyperbolic geometry 157
8.2 Geometry in a convex region 159
8.3 Veblen's axioms of order 161
8.4 Order in a pencil 162
8.5 The geometry of lines and planes through a fixed point . . 164
8.6 Generalized bundles and pencils 165
8.7 Ideal points and lines 171
8.8 Verifying the projective axioms 172
8.9 Parallelism 174
IX EUCLIDEAN AND HYPERBOLIC GEOMETRY
9.1 The introduction of congruence 179
9.2 Perpendicular lines and planes 181
9.3 Improper bundles and pencils 184
9.4 The absolute polarity 185
9.5 The Euclidean case 186
9.6 The hyperbolic case 187
9.7 The Absolute 192
9.8 The geometry of a bundle 197
X. HYPERBOLIC GEOMETRY IN TWO DIMENSIONS
10.1 Ideal elements 199
10.2 Angle-bisectors 200
10.3 Congruent transformations 201
10.4 Some famous constructions 204
10.5 An alternative expression for distance 206
10.6 The angle of parallelism 207
10.7 Distance and angle in terms of poles and polars 208
10.8 Canonical coordinates 209
10.9 Euclidean geometry as a limiting case 211
XI CIRCLES AND TRIANGLES
11.1 Various definitions for a circle 213
11.2 The circle as a special conic 215
11.3 Spheres 218
11.4 The in- and ex-circles of a triangle 220
11.5 The circum-circles and centroids 221
11.6 The polar triangle and the orthocentre 223
XII. THE USE OF A GENERAL TRIANGLE OF REFERENCE
12.1 Formulae for distance and angle 224
12.2 The general circle 226
12.3 Tangential equations 228
12.4 Circum-circles and centroids 229
12.5 In- and ex-circles 231
12.6 The orthocentre 231
12.7 Elliptic trigonometry 232
12.8 The radii 235
12.9 Hyperbolic trigonometry 237
XIII. AREA
13.1 Equivalent regions 241
13.2 The choice of a unit 241
13.3 The area of a triangle in elliptic geometry 242
13.4 Area in hyperbolic geometry 243
13.5 The extension to three dimensions 247
13.6 The differential of distance 248
13.7 Arcs and areas of circles 249
13.8 Two surfaces which can be developed on the Euclidean plane 251
XIV. EUCLIDEAN MODELS
14.1 The meaning of "elliptic" and "hyperbolic" 252
14.2 Beltrami's model 252
14.3 The differential of distance 254
14.4 Gnomonic projection 255
14.5 Development on surfaces of constant curvature 256
14.6 Klein's conformai model of the elliptic plane 258
14.7 Klein's conformai model of the hyperbolic plane 260
14.8 Poincaré's model of the hyperbolic plane 263
14.9 Conformai models of non-Euclidean space 264
XV. CONCLUDING REMARKS
15.1 HjelmsleVs mid-line 267
15.2 The Napier chain 273
15.3 The Engel chain 277
15.4 Normalized canonical coordinates 281
15.5 Curvature 283
15.6 Quadratic forms 284
15.7 The volume of a tetrahedron 285
15.8 A brief historical survey of construction problems . . . . 289
15.9 Inversive distance and the angle of parallelism 292
APPENDIX: ANGLES AND ARCS IN THE HYPERBOLIC PLANE 299
BIBLIOGRAPHY 317
INDEX 327

Making Mathematics with Needlework: Ten Papers and Ten Projects

 Sarah-Marie Belcastro e Carolyn Yackel 

A K Peters/CRC Press | 2007 | 202 páginas | rar - pdf | 25,3 Mb

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Mathematical craftwork has become extremely popular, and mathematicians and crafters alike are fascinated by the relationship between their crafts. The focus of this book, written for mathematicians, needleworkers, and teachers of mathematics, is on the relationship between mathematics and the fiber arts (including knitting, crocheting, cross-stitch, and quilting). Each chapter starts with an overview of the mathematics and the needlework at a level understandable to both mathematicians and needleworkers, followed by more technical sections discussing the mathematics, how to introduce the mathematics in the classroom through needlework, and how to make the needlework project, including patterns and instructions.

Contents
Quilted mobius band
Picking up stitches and Diophantine equations
The Sierpinski variations: self-similar crochet
Ony two knit stitches can create a torus
Symmetry patterns in cross-stitch
Socks with Algebraic structure
Fortunatus's purse
(K)not cables, braids
The graph theory of blackwork embroidery
Stop those pants!

Livro relacionado: 

Crocheting adventures with hyperbolic planes por Daina Taimin̦a Idioma: Inglês   Editora: Wellesley, MA : A.K. Peters, ©2009.

Measurement

Paul Lockhart 

Belknap Press of Harvard University Press | 2012 | 416 páginas | rar - pdf | 1,44 Mb

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For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.

In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.

Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.

CONTENTS

Reality and Imagination…1

On Problems…5

Part One: Size and Shape…21

In which we begin our investigation of abstract geometrical figures.

Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves.

Part Two: Time and Space…199

Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader.

Acknowledgments…399

Index…401

Outro livro do mesmo autor:

A mathematician's lament por Paul Lockhart   Idioma: Inglês   Editora: New York, NY : Bellevue Literary Press, 2009.

New Foundations for Physical Geometry: The Theory of Linear Structures

 

Tim Maudlin

Oxford University Press | 2014 | 374 páginas | rar - pdf | 1,94 Mb

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Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. This book proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time. This is the initial volume in a two-volume set, the first of which develops the mathematical structure and the second of which applies it to classical and Relativistic physics.
The book begins with a brief historical review of the development of mathematics as it relates to geometry, and an overview of standard topology. The new theory, the Theory of Linear Structures, is presented and compared to standard topology. The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line. Axioms for the Theory of Linear Structures are laid down, and definitions of other geometrical notions developed in those terms. Various novel geometrical properties, such as a space being intrinsically directed, are defined using these resources. Applications of the theory to discrete spaces (where the standard theory of open sets gets little purchase) are particularly noted. The mathematics is developed up through homotopy theory and compactness, along with ways to represent both affine (straight line) and metrical structure.
ContentsAcknowledgments x
Introduction 1
Metaphorical and Geometrical Spaces 6
A Light Dance on the Dust of the Ages 9
The Proliferation of Numbers 12
Descartes and Coordinate Geometry 14
John Wallis and the Number Line 16
Dedekind and the Construction of Irrational Numbers 20
Overview and Terminological Conventions 25
1. Topology and Its Shortcomings 28
Standard Topology 31
Closed Sets, Neighborhoods, Boundary Points, and Connected Spaces 33
The Hausdorff Property 36
Why Discrete Spaces Matter 45
The Relational Nature of Open Sets 47
The Bill of Indictment (So Far) 49
2. Linear Structures, Neighborhoods, Open Sets 54
Methodological Morals 54
The Essence of the Line 57
The (First) Theory of Linear Structures 59
Proto-Linear Structures 69
Discrete Spaces, Mr Bush’s Wild Line, the Woven Plane, and the Affine Plane 74
A Taxonomy of Linear Structures 79
Neighborhoods in a Linear Structure 81
Open Sets 85
Finite-Point Spaces 86
Return to Intuition 89
Directed Linear Structures 92
Linear Structures and Directed Linear Structures 96
Neighborhoods, Open Sets, and Topologies Again 97
Finite-Point Spaces and Geometrical Interpretability 99
A Geometrically Uninterpretable Topological Space 103
Segment-Spliced Linear Structures 104
Looking Ahead 107
Exercises 107
Appendix: Neighborhoods and Linear Structures 108
3. Closed Sets, Open Sets (Again), Connected Spaces 113
Closed Sets: Preliminary Observations 113
Open and Closed Intervals 114
IP-closed and IP-open Sets 115
IP-open Sets and Open Sets, IP-closed Sets and Closed Sets 117
Zeno’s Combs 120
Closed Sets, Open Sets, and Complements 123
Interiors, Boundary Points, and Boundaries 127
Formal Properties of Boundary Points 136
Connected Spaces 140
Chains and Connectedness 143
Directedness and Connectedness 148
Exercises 150
4. Separation Properties, Convergence, and Extensions 152
Separation Properties 152
Convergence and Unpleasantness 155
Sequences and Convergence 160
Extensions 163
The Topologist’s Sine Curve 165
Physical Interlude: Thomson’s Lamp 168
Exercises 172
5. Properties of Functions 174
Continuity: an Overview 174
The Intuitive Explication of Continuity and Its Shortcomings 175
The Standard Definition and Its Shortcomings 178
What the Standard Definition of “Continuity” Defines 183
The Essence of Continuity 186
Continuity at a Point and in a Direction 190
An Historical Interlude 192
Remarks on the Architecture of Definitions; Lineal Functions 194
Lines and Continuity in Standard Topology 199
Exercises 201
6. Subspaces and Substructures; Straightness and Differentiability 203
The Geometrical Structure of a Subspace: Desiderata 203
Subspaces in Standard Topology 205
Subspaces in the Theory of Linear Structures 206
Substructures 211
One Way Forward 218
Euclid’s Postulates and the Nature of Straightness 220
Convex Affine Spaces 227
Example: Some Conical Spaces 233
Tangents 235
Upper and Lower Tangents, Differentiability 244
Summation 253
Exercises 254
7. Metrical Structure 256
Approaches to Metrical Structure 256
Ratios Between What? 258
The Additive Properties of Straight Lines 260
Congruence and Comparability 262
Eudoxan and Anthyphairetic Ratios 274
The Compass 280
Metric Linear Structures and Metric Functions 285
Open Lines, Curved Lines, and Rectification 287
Continuity of the Metric 291
Exercises 294
Appendix: A Remark about Minimal Regular Metric Spaces 294
8. Product Spaces and Fiber Bundles 297
New Spaces from Old 297
Constructing Product Linear Structures 300
Examples of Product Linear Structures 303
Neighborhoods and Open Sets in Product Linear Structures 307
Fiber Bundles 309
Sections 313
Additional Structure 315
Exercises 318
9. Beyond Continua 320
How Can Continua and Non-Continua Approximate Each Other? 320
Continuous Functions 321
Homotopy 334
Compactness 339
Summary of Mathematical Results and Some Open Questions 345
Exercises 346
Axioms and Definitions 347
Bibliography 358
Index 361

Leveled Texts for Mathematics: Geometry

Lori Barker

Shell Education | 2011 | 147 páginas | rar - pdf | 4,25 Mb

link (password: matav)

Highlighting geometry, this resource provides the know-how to use leveled texts to differentiate instruction in mathematics. A total of 15 different topics are featured in and the high-interest text is written at four different reading levels with matching visuals. Practice problems are provided to reinforce what is taught in the passage.

Table of Contents
What Is Differentiation?...4
How to Differentiate Using This Product...5
General Information About Student Populations...6
Below-Grade-Level Students...6
English Language Learners...6
On-Grade-Level Students....7
Above-Grade-Level Students...7
Strategies for Using the Leveled Texts.....8
Below-Grade-Level Students...8
English Language Learners.... 11
Above-Grade-Level Students.... 14
How to Use This Product.... 16
Readability Chart.... 16
Components of the Product.... 16
Tips for Managing the Product.... 18
Correlations to Mathematics Standards... 19
Leveled Texts..... 21
Angles All Around.... 21
Understanding Triangles.... 29
To Cross or Not to Cross... 37
Quadrilaterals... 45
Classifying 2-D Shapes.... 53
Irregular Shapes... 61
Congruent and Similar Figures..... 69
Understanding 3-D Shapes.... 77
Understanding Prisms... 85
The Coordinate Plane.... 93
Circles... 101
Symmetry... 109
Reflections..... 117
Rotations... 125
Translations... 133
Appendices..... 141
References Cited... 141
Contents of Teacher Resource CD... 142

New Horizons in Geometry

(Dolciani Mathematical Expositions)

Tom Apostol e Mamikon Mnatsakania

The Mathematical Association of America | 2013 | 528 páginas | rar - pdf | 5 Mb

link (password: matav)


New Horizons in Geometry represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order. 
The hundreds of full color illustrations by Mamikon are visually enticing and provide great motivation to read further and savor the wonderful results. Lengths, areas, and volumes of curves, surfaces, and solids are explored from a visually captivating perspective. It is an understatement to say that Apostol and Mamikon have breathed new life into geometry.

CONTENTS
Preface . ix
Introduction . . xi
Foreword . . xiii
Chapter 1. Mamikon’s Sweeping Tangent Theorem  . . 1
Chapter 2. Cycloids and Trochoids . . 31
Chapter 3. Cyclogons and Trochogons  . . 65
Chapter 4. Circumgons and Circumsolids . . 101
Chapter 5. The Method of Punctured Containers. . 135
Chapter 6. Unwrapping Curves from Cylinders and Cones . . 169
Chapter 7. New Descriptions of Conics via Twisted Cylinders, Focal Disks, and Directors . 213
Chapter 8. Ellipse to Hyperbola: “With This String I Thee Wed” . . 243
Chapter 9. Trammels . . 267
Chapter 10. Isoperimetric and Isoparametric Problems . . 295
Chapter 11. Arclength and Tanvolutes  331
Chapter 12. Centroids . . 375
Chapter 13. New Balancing Principles with Applications . . 401
Chapter 14. Sums of Squares. . 443
Chapter 15. Appendix . . 473

Exploring Advanced Euclidean Geometry with GeoGebra

(Classroom Resource Materials)

Gerard A. Venema

Mathematical Association of America | 2013 | 146 páginas | rar - pdf | 820 kb

link (password: matav)

This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry.

The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.

Contents

Preface vii

0 A Quick Review of Elementary Euclidean Geometry 1

0.1 Measurement and congruence. . 1

0.2 Angle addition  . . 2

0.3 Triangles and triangle congruence conditions . . 3

0.4 Separation and continuity. . 4

0.5 The exterior angle theorem . . 5

0.6 Perpendicular lines and parallel lines . . 5

0.7 The Pythagorean theorem. . 7

0.8 Similar triangles . . . 8

0.9 Quadrilaterals  . . 9

0.10 Circles and inscribed angles . . 10

0.11 Area . . 11

1 The Elements of GeoGebra 13

1.1 Getting started: the GeoGebra toolbar. . 13

1.2 Simple constructions and the drag test  . . 16

1.3 Measurement and calculation . . 18

1.4 Enhancing the sketch .. . 20

2 The Classical Triangle Centers 23

2.1 Concurrent lines . . 23

2.2 Medians and the centroid . . 24

2.3 Altitudes and the orthocenter .. 25

2.4 Perpendicular bisectors and the circumcenter . . 26

2.5 The Euler line . . 27

3 Advanced Techniques in GeoGebra 31

3.1 User-defined tools . . 31

3.2 Check boxes . . 33

3.3 The Pythagorean theorem revisited  . . 34

4 Circumscribed, Inscribed, and Escribed Circles 39

4.1 The circumscribed circle and the circumcenter  . . 39

4.2 The inscribed circle and the incenter . . 41

4.3 The escribed circles and the excenters . . 42

4.4 The Gergonne point and the Nagel point . . 43

4.5 Heron’s formula . . 44

5 The Medial and Orthic Triangles 47

5.1 The medial triangle  . . 47

5.2 The orthic triangle . . 48

5.3 Cevian triangles  . . 50

5.4 Pedal triangles . . 51

6 Quadrilaterals 53

6.1 Basic definitions . . 53

6.2 Convex and crossed quadrilaterals. . 54

6.3 Cyclic quadrilaterals . . 55

6.4 Diagonals . . 56

7 The Nine-Point Circle 57

7.1 The nine-point circle . . . 57

7.2 The nine-point center . . 59

7.3 Feuerbach’s theorem  . . 60

8 Ceva’s Theorem 63

8.1 Exploring Ceva’s theorem . . 63

8.2 Sensed ratios and ideal points  . . 65

8.3 The standard form of Ceva’s theorem  . . 68

8.4 The trigonometric form of Ceva’s theorem  . . 71

8.5 The concurrence theorems  . . 72

8.6 Isotomic and isogonal conjugates and the symmedian point  . . 73

9 The Theorem of Menelaus 77

9.1 Duality  . . 77

9.2 The theorem of Menelaus . . 78

10 Circles and Lines 81

10.1 The power of a point  . . 81

10.2 The radical axis . . 83

10.3 The radical center  . . 84

11 Applications of the Theorem of Menelaus 85

11.1 Tangent lines and angle bisectors . . . 85

11.2 Desargues’ theorem  . . 86

11.3 Pascal’s mystic hexagram . . 88

11.4 Brianchon’s theorem  . . 90

11.5 Pappus’s theorem  . . 91

11.6 Simson’s theorem. . 93

11.7 Ptolemy’s theorem . . 96

11.8 The butterfly theorem  . . 97

12 Additional Topics in Triangle Geometry 99

12.1 Napoleon’s theorem and the Napoleon point  . . 99

12.2 The Torricelli point . . 100

12.3 van Aubel’s theorem. . 100

12.4 Miquel’s theorem and Miquel points . . 101

12.5 The Fermat point  . . 101

12.6 Morley’s theorem . . 102

13 Inversions in Circles 105

13.1 Inverting points .  . 105

13.2 Inverting circles and lines  . . 107

13.3 Othogonality  . . 108

13.4 Angles and distances. . 110

14 The Poincar´e Disk 111

14.1 The Poincar´e disk model for hyperbolic geometry  . . 111

14.2 The hyperbolic straightedge . . 113

14.3 Common perpendiculars . . 114

14.4 The hyperbolic compass. . 116

14.5 Other hyperbolic tools . . 117

14.6 Triangle centers in hyperbolic geometry . . 118

References 121

Index 123