Gems of Geometry

 

John Barnes 

Springer | 2012 - 2.ª edição | páginas | pdf | 5 Mb

link

Based on a series of lectures for adult students, this lively and entertaining book proves that, far from being a dusty, dull subject, geometry is in fact full of beauty and fascination. The author's infectious enthusiasm is put to use in explaining many of the key concepts in the field, starting with the Golden Number and taking the reader on a geometrical journey via Shapes and Solids, through the Fourth Dimension, finishing up with Einstein's Theories of Relativity.

Equally suitable as a gift for a youngster or as a nostalgic journey back into the world of mathematics for older readers, John Barnes' book is the perfect antidote for anyone whose maths lessons at school are a source of painful memories. Where once geometry was a source of confusion and frustration, Barnes brings enlightenment and entertainment.

In this second edition, stimulated by recent lectures at Oxford, further material and extra illustrations have been added on many topics including Coloured Cubes, Chaos and Crystals.

Contents

1 The Golden Number 1

Pieces of paper, The golden ratio, Fibonacci’s rabbits, Continued fractions, Pentagons, Phyllotaxis, Further reading, Exercises.

2 Shapes and Solids 27

Flatland, Polygons, Tiling, Vision and projection, Five classical polyhedra, Duality, Kepler and Poinsot, The Archimedean figures, Non-convex polyhedra, Pentagonal tilings, Further reading, Exercises.

3 The Fourth Dimension 63

What is the fourth dimension? Honeycombs, The 4-simplex, The hypercube and the 16-cell, Other regular convex figures, Non-convex regular figures, Honeycombs, five dimensions and more, Nets, Further reading, Exercises.

4 Projective Geometry 89

Pappus’ theorem, Desargues’ theorem, Duality, Duality in three dimensions, Infinity and parallels, Quadrilaterals and quadrangles, Conics, Coordinates, Finite geometries, Configurations, Further reading, Exercises.

5 Topology 113

Hairy dogs, Colour problems, Colouring maps on the torus, The Möbius band, The Klein bottle, The projective plane, Round up, Further reading, Exercises.

6 Bubbles 137

Surface tension, Two bubbles, Three bubbles, Four bubbles, Foam, Films on frames, Films on cylinders, That well known theorem, Further reading, Exercises.

7 Harmony of the Spheres 157

Steiner’s porism, Inversion, Coaxial circles, Proof of Steiner’s porism, Soddy’s hexlet, Further reading, Exercises.

8 Chaos and Fractals 179

Shaken foundations, Fractals, Fractional dimensions, Cantor sets, Population growth, Double, double, boil and trouble, Chaos and peace, And so to dust, Newton’s method, Julia and Mandelbrot sets, Natural chaos, Further reading, Exercises.

9 Relativity 205

The special theory, Time changes, The Lorentz–Fitzgerald contraction, Distortion of bodies, Lorentz transformation, Time and relativity, Mass and energy, Coordinates, Curvature, Einstein’s equations, The Schwarzchild solution, Consequences of general relativity, Black holes, Properties of black holes, Further reading, Exercise.

10 Finale 231

Squares on a quadrilateral, The Argand plane, The quadrilateral revisited, Other complex problems, Trisection, Bends, Pedal triangles, Coordinates of points and lines, Further reading.

A The Bull and the Man 247

The problem, The proof.

B Stereo Images 251

Compound figures, Desargues’ theorem.

C More on Four 261

Archimedean figures in four dimensions, Prisms and hyperprisms.

D Schlegel Images 271

Schlegel diagrams, The hypercube, The 16-cell, The 24-cell, The 120-cell, 600-cell and tetroctahedric.

E Crystals 287

Packing of spheres, Crystals, Diamonds and graphite, Silica structures, Gems, Further reading.

F Stability 305

Stability of fixed points, The fixed points, The two cycle, Three cycles.

G Fanoland 319

Seven girls and seven boys, Plus six girls and six boys, Explanation.

Bibliography 321

Index 323

1.ª edição

Gems of Geometry por John Barnes Idioma: Inglês   Editora: Berlin [u.a.] Springer 2009

Symmetry: A Very Short Introduction

Ian Stewart

Oxford University Press | 2013 | 152 páginas | rar - epub | 5,9 Mb

link (password: matav)

Symmetry is an immensely important concept in mathematics and throughout the sciences. In this Very Short Introduction, Ian Stewart demonstrates symmetry's deep implications, showing how it even plays a major role in the current search to unify relativity and quantum theory. Stewart, a respected mathematician as well as a widely known popular-science and science-fiction writer, brings to this volume his deep knowledge of the subject and his gift for conveying science to general readers with clarity and humor. He describes how symmetry's applications range across the entire field of mathematics and how symmetry governs the structure of crystals, innumerable types of pattern formation, and how systems change their state as parameters vary. Symmetry is also highly visual, with applications that include animal markings, locomotion, evolutionary biology, elastic buckling, waves, the shape of the Earth, and the form of galaxies. Fundamental physics is governed by symmetries in the laws of nature--Einstein's point that the laws should be the same at all locations and all times. 

Contents
List of illustrations
Introduction
1 What is symmetry?
2 Origins of symmetry
3 Types of symmetry
4 Structure of groups
5 Groups and games
6 Nature’s patterns
7 Nature’s laws
8 Atoms of symmetry
Further reading

Index

The king of infinite space : Euclid and his Elements

David Berlinski 

Basic Books |  2013 | 187 páginas | pdf | 1,3  Mb

link
link1

epub - 5,9 Mb - link

Geometry defines the world around us, helping us make sense of everything from architecture to military science to fashion. And for over two thousand years, geometry has been equated with Euclid’s Elements, arguably the most influential book in the history of mathematics In The King of Infinite Space, renowned mathematics writer David Berlinski provides a concise homage to this elusive mathematician and his staggering achievements. Berlinski shows that, for centuries, scientists and thinkers from Copernicus to Newton to Einstein have relied on Euclid’s axiomatic system, a method of proof still taught in classrooms around the world. Euclid’s use of elemental logic—and the mathematical statements he and others built from it—have dramatically expanded the frontiers of human knowledge.
The King of Infinite Space presents a rich, accessible treatment of Euclid and his beautifully simple geometric system, which continues to shape the way we see the world.

Contents
Preface xi
I Signs of Men 1
II An Abstraction from the Gabble 11
III Common Beliefs 19
IV Darker by Definition 33
V The Axioms 45
VI The Greater Euclid 57
VII Visible and Invisible Proof 77
VIII The Devil’s Offer 91
IX The Euclidean Joint Stock Company 117
X Euclid the Great 147
Teacher’s Note 155
A Note on Sources 157
Appendix: Euclid’s Definitions 159
Index 163


Outros livros do mesmos autor:

One, two, three : absolutely elementary mathematics por David Berlinski Idioma: Inglês   Editora: New York : Pantheon Books, ©2011.

Infinite ascent : a short history of mathematics por David Berlinski Idioma: Inglês   Editora: New York : Modern Library, 2005.

Geometric Puzzle Design

Stewart Coffin

A K Peters/CRC Press | 2007 - 2ª edição | pdf | 5,4 Mb

link

This book discusses how to design "good" geometric puzzles: two-dimensional dissection puzzles, polyhedral dissections, and burrs. It outlines major categories of geometric puzzles and provides examples, sometimes going into the history and philosophy of those examples. The author presents challenges and thoughtful questions, as well as practical design and woodworking tips to encourage the reader to build his own puzzles and experiment with his own designs. Aesthetics, phychology, and mathematical considerations all factor into the definition of the quality of a puzzle.

Contents
Preface vii
Introduction ix
1 Two-Dimensional Dissections 1
2 Two-Dimensional Combinatorial Puzzles 17
3 Misdirection-Type Puzzles 37
4 Variations on Sliding Block Puzzles 41
5 Cubic Block Puzzles 45
6 Interlocking Block Puzzles 55
7 The Six-Piece Burr 59
8 Larger (and Smaller) Burrs 69
9 The Diagonal Burr 81
10 The Rhombic Dodecahedron and Its Stellations 87
11 Polyhedral Puzzles with Dissimilar Pieces 99
12 Intersecting Prisms 107
13 Puzzles that Make Different Shapes 113
14 Coordinate-Motion Puzzles 117
15 Puzzles Using Hexagonal or Rhombic Sticks 121
16 Split Triangular Sticks 129
17 Dissected Rhombic Dodecahedra 133
18 Miscellaneous Confusing Puzzles 139
19 Triacontahedral Designs 143
20 Puzzles Made of Polyhedral Blocks 153
21 Intermezzo 161
22 Theme and Variations 167
23 Blocks and Pins 175
24 Woodworking Techniques 187
Finale 197
Bibliography 199

Index 201

Geometry of Complex Numbers

Hans Schwerdtfeger

Dover Publications | 1979 | 200 Páginas | rar - epub | 11,7 Mb

link (password: matav)
(novo fciheiro)

DjVu | 2,1 Mb

link direto
link
link1

depositfiles.com 

rapidshare.com

rapidgator.net

Since its initial publication in 1962, Professor Schwerdtfeger's illuminating book has been widely praised for generating a deeper understanding of the geometrical theory of analytic functions as well as of the connections between different branches of geometry. Its focus lies in the intersection of geometry, analysis, and algebra, with the exposition generally taking place on a moderately advanced level. Much emphasis, however, has been given to the careful exposition of details and to the development of an adequate algebraic technique.
In three broad chapters, the author clearly and elegantly approaches his subject. The first chapter, Analytic Geometry of Circles, treats such topics as representation of circles by Hermitian matrices, inversion, stereographic projection, and the cross ratio. The second chapter considers in depth the Moebius transformation: its elementary properties, real one-dimensional projectivities, similarity and classification of various kinds, anti-homographies, iteration, and geometrical characterization. The final chapter, Two-Dimensional Non-Euclidean Geometries, discusses subgroups of Moebius transformations, the geometry of a transformation group, hyperbolic geometry, and spherical and elliptic geometry. For this Dover edition, Professor Schwerdtfeger has added four new appendices and a supplementary bibliography.Advanced undergraduates who possess a working knowledge of the algebra of complex numbers and of the elements of analytical geometry and linear algebra will greatly profit from reading this book. It will also prove a stimulating and thought-provoking book to mathematics professors and teachers.

Elementary Mathematics from an Advanced Standpoint: Geometry

Felix Klein

Dover Publications | 2004 | 224 páginas | rar - epub | 6,2 Mb

link (password : matav)
(novo ficheiro)

djvu | 2 Mb
link
link1
link2

This text begins with the simplest geometric manifolds, the Grassmann determinant principle for the plane and the Grassmann principle for space; and more. Also explores affine and projective transformations; higher point transformations; transformations with change of space element; and the theory of the imaginary. Concludes with a systematic discussion of geometry and its foundations

CONTENTS
Introduction
Part One: The Simplest Geometric Manifolds
I. Line-Segment, Area, Volume, as Relative Magnitudes
Definition by means of determinants; interpretation of the sign
Simplest applications, especially the cross ratio
Area of rectilinear polygons
Curvilinear areas
Theory of Amsler’s polar planimeter
Volume of polyhedrons, the law of edges
One-sided polyhedrons
II. The Grassmann Determinant Principle for the Plane
Line-segment (vectors)
Application in statics of rigid systems
Classification of geometric magnitudes according to their behavior under transformation of rectangular coordinates
Application of the principle of classification to elementary magnitudes
III. The Grassmann Principle for Space
Line-segment and plane-segment
Application to statics of rigid bodies
Relation to Mobius’ null-system
Geometric interpretation of the null-system
Connection with the theory of screws
IV. Classification of the Elementary Configurations of Space according to their Behavior under Transformation of Rectangular Coordinates
Generalities concerning transformations of rectangular space coordinates
Transformation formulas for some elementary magnitudes
Couple and free plane magnitude as equivalent manifolds
Free line-segment and free plane magnitude (“polar” and “axial” vector)
Scalars of first and second kind
Outlines of a rational vector algebra
Lack of a uniform nomenclature in vector calculus
V. Derivative Manifolds
Derivatives from points (curves, surfaces, point sets)
Difference between analytic and synthetic geometry
Projective geometry and the principle of duality
Plücker’s analytic method and the extension of the principle of duality (line coordinates)
Grassmann’s Ausdehnungslehre; n-dimensional geometry
Scalar and vector fields; rational vector analysis
Part Two: Geometric Transformations
Transformations and their analytic representation
I. Affine Transformations
Analytic definition and fundamental properties
Application to theory of ellipsoid
Parallel projection from one plane upon another
Axonometric mapping of space (affine transformation with vanishing determinant)
Fundamental theorem of Pohlke
II. Projective Transformations
Analytic definition; introduction of homogeneous coordinates
Geometric definition: Every collineation is a projective transformation
Behavior of fundamental manifolds under projective transformation
Central projection of space upon a plane (projective transformation with vanishing determinant)
Relief perspective
Application of projection in deriving properties of conics
III. Higher Point Transformations
1. The Transformation by Reciprocal Radii
Peaucellier’s method of drawing a line
Stereographic projection of the sphere
2. Some More General Map Projections
Mercator’s projection
Tissot theorems
3. The Most General Reversibly Unique Continuous Point Transformations
Genus and connectivity of surfaces
Euler’s theorem on polyhedra
IV. Transformations with Change of Space Element
1. Dualistic Transformations
2. Contact Transformations
3. Some Examples
Forms of algebraic order and class curves
Application of contact transformations to theory of cog wheels
V. Theory of the Imaginary
Imaginary circle-points and imaginary sphere-circle
Imaginary transformation
Von Staudt’s interpretation of self-conjugate imaginary manifolds by means of real polar systems
Von Staudt’s complete interpretation of single imaginary elements
Space relations of imaginary points and lines
Part Three: Systematic Discussion of Geometry and Its Foundations
I. The Systematic Discussion
1. Survey of the Structure of Geometry
Theory of groups as a geometric principle of classification
Cayley’s fundamental principle: Projective geometry is all geometry
2. Digression on the Invariant Theory of Linear Substitutions
Systematic discussion of invariant theory
Simple examples
3. Application of Invariant Theory to Geometry
Interpretation of invariant theory of n variables in affine geometry of Rn with fixed origin
Interpretation in projective geometry of Rn−1
4. The Systematization of Affine and Metric Geometry Based on Cayley’s Principle
Fitting the fundamental notions of affine geometry into the projective system
Fitting the Grassmann determinant principle into the invariant-theoretic conception of geometry. Concerning tensors
Fitting the fundamental notions of metric geometry into the projective system
Projective treatment of the geometry of the triangle
II. Foundations of Geometry
General statement of the question: Attitude to analytic geometry
Development of pure projective geometry with subsequent addition of metric geometry
1. Development of Plane Geometry with Emphasis upon Motions
Development of affine geometry from translation
Addition of rotation to obtain metric geometry
Final deduction of expressions for distance and angle
Classification of the general notions surface-area and curve-length
2. Another Development of Metric Geometry—the Role of the Parallel Axiom
Distance, angle, congruence, as fundamental notions
Parallel axiom and theory of parallels (non-euclidean geometry
Significance of non-euclidean geometry from standpoint of philosophy
Fitting non-euclidean geometry into the projective system
Modern geometric theory of axioms
3. Euclid’s Elements
Historical place and scientific worth of the Elements
Contents of thirteen books of Euclid
Foundations
Beginning of the first book
Lack of axiom of betweenness in Euclid; possibility of the sophisms
Axiom of Archimedes in Euclid; horn-shaped angles as example of a system of magnitudes excluded by this axiom

Revolutions of Geometry

 (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

Michael O'Leary

Wiley | 2010 | 607 páginas | pdf | 7 Mb

link
link1

Guides readers through the development of geometry and basic proof writing using a historical approach to the topic
In an effort to fully appreciate the logic and structure of geometric proofs, Revolutions of Geometry places proofs into the context of geometry's history, helping readers to understand that proof writing is crucial to the job of a mathematician. Written for students and educators of mathematics alike, the book guides readers through the rich history and influential works, from ancient times to the present, behind the development of geometry. As a result, readers are successfully equipped with the necessary logic to develop a full understanding of geometric theorems.
Following a presentation of the geometry of ancient Egypt, Babylon, and China, the author addresses mathematical philosophy and logic within the context of works by Thales, Plato, and Aristotle. Next, the mathematics of the classical Greeks is discussed, incorporating the teachings of Pythagoras and his followers along with an overview of lower-level geometry using Euclid's Elements. Subsequent chapters explore the work of Archimedes, Viete's revolutionary contributions to algebra, Descartes' merging of algebra and geometry to solve the Pappus problem, and Desargues' development of projective geometry. The author also supplies an excursion into non-Euclidean geometry, including the three hypotheses of Saccheri and Lambert and the near simultaneous discoveries of Lobachevski and Bolyai. Finally, modern geometry is addressed within the study of manifolds and elliptic geometry inspired by Riemann's work, Poncelet's return to projective geometry, and Klein's use of group theory to characterize different geometries

Table of Contents
Preface.
Acknowledgments.
PART I FOUNDATIONS.
1 The First Geometers.
1.1 Egypt.
1.2 Babylon.
1.3 China.
2 Thales.
2.1 The Axiomatic System.
2.2 Deductive Logic.
2.3 Proof Writing.
3 Plato and Aristotle.
3.1 Form.
3.2 Categorical Propositions..
3.3 Categorical Syllogisms.
3.4 Figures.
PART II THE GOLDEN AGE.
4 Pythagoras.
4.1 Number Theory.
4.2 The Pythagorean Theorem.
4.3 Archytas.
4.4 The Golden Ratio.
5 Euclid.
5.1 The Elements.
5.2 Constructions.
5.3 Triangles.
5.4 Parallel Lines.
5.5 Circles.
5.6 The Pythagorean Theorem Revisited.
6 Archimedes.
6.1 The Archimedean Library.
6.2 The Method of Exhaustion.
6.3 The Method.
6.4 Preliminaries to the Proof.
6.5 The Volume of a Sphere.
PART III ENLIGHTENMENT.
7 François Viète.
7.1 The Analytic Art.
7.2 Three Problems.
7.3 Conic Sections.
7.4 The Analytic Art in Two Variables.
8 René Descartes.
8.1 Compasses.
8.2 Method.
8.3 Analytic Geometry.
9 Gérard Desargues.
9.1 Projections.
9.2 Points at Infinity.
9.3 Theorems of Desargues and Menelaus.
9.4 Involutions.
PART IV A STRANGE NEW WORLD.
10 Giovanni Saccheri.
10.1 The Question of Parallels.
10.2 The Three Hypotheses.
10.3 Conclusions for Two Hypotheses.
10.4 Properties of Parallel Lines.
10.5 Parallelism Redefined.
11 Johann Lambert.
11.1 The Three Hypotheses Revisited.
11.2 Polygons.
11.3 Omega Triangles.
11.4 Pure Reason.
12 Nicolai Lobachevski and János Bolyai.
12.1 Parallel Fundamentals.
12.2 Horocycles.
12.3 The Surface of a Sphere.
12.4 Horospheres.
12.5 Evaluating the Pi Function.
PART V NEW DIRECTIONS.
13 Bernhard Riemann.
13.1 Metric Spaces.
13.2 Topological Spaces.
13.3 Stereographic Projection.
13.4 Consistency of Non-Euclidean Geometry.
14 Jean-Victor Poncelet.
14.1 The Projective Plane.
14.2 Duality.
14.3 Perspectivity.
14.4 Homogeneous Coordinates.
15 Felix Klein.
15.1 Group Theory.
15.2 Transformation Groups.
15.3 The Principal Group.
15.4 Isometries of the Plane.
15.5 Consistency of Euclidean Geometry.
References.
Index.

The foundations of geometry

Gerard Venema

Pearson | 2011 - 2ª edição | 407 páginas | pdf | 3 Mb

link direto
link
link1

Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers—and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra.

This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites.

Table of Contents
1. Prologue: Euclid’s Elements
1.1 Geometry before Euclid
1.2 The logical structure of Euclid’s Elements
1.3 The historical significance of Euclid’s Elements
1.4 A look at Book I of the Elements
1.5 A critique of Euclid’s Elements
1.6 Final observations about the Elements
2. Axiomatic Systems and Incidence Geometry
2.1 The structure of an axiomatic system
2.2 An example: Incidence geometry
2.3 The parallel postulates in incidence geometry
2.4 Axiomatic systems and the real world
2.5 Theorems, proofs, and logic
2.6 Some theorems from incidence geometry
3. Axioms for Plane Geometry
3.1 The undefined terms and two fundamental axioms
3.2 Distance and the Ruler Postulate
3.3 Plane separation
3.4 Angle measure and the Protractor Postulate
3.5 The Crossbar Theorem and the Linear Pair Theorem
3.6 The Side-Angle-Side Postulate
3.7 The parallel postulates and models
4. Neutral Geometry
4.1 The Exterior Angle Theorem and perpendiculars
4.2 Triangle congruence conditions
4.3 Three inequalities for triangles
4.4 The Alternate Interior Angles Theorem
4.5 The Saccheri-Legendre Theorem
4.6 Quadrilaterals
4.7 Statements equivalent to the Euclidean Parallel Postulate
4.8 Rectangles and defect
4.9 The Universal Hyperbolic Theorem
5. Euclidean Geometry
5.1 Basic theorems of Euclidean geometry
5.2 The Parallel Projection Theorem
5.3 Similar triangles
5.4 The Pythagorean Theorem
5.5 Trigonometry
5.6 Exploring the Euclidean geometry of the triangle
6. Hyperbolic Geometry
6.1 The discovery of hyperbolic geometry
6.2 Basic theorems of hyperbolic geometry
6.3 Common perpendiculars
6.4 Limiting parallel rays and asymptotically parallel lines
6.5 Properties of the critical function
6.6 The defect of a triangle
6.7 Is the real world hyperbolic?
7. Area
7.1 The Neutral Area Postulate
7.2 Area in Euclidean geometry
7.3 Dissection theory in neutral geometry
7.4 Dissection theory in Euclidean geometry
7.5 Area and defect in hyperbolic geometry
8. Circles
8.1 Basic definitions
8.2 Circles and lines
8.3 Circles and triangles
8.4 Circles in Euclidean geometry
8.5 Circular continuity
8.6 Circumference and area of Euclidean circles
8.7 Exploring Euclidean circles
9. Constructions
9.1 Compass and straightedge constructions
9.2 Neutral constructions
9.3 Euclidean constructions
9.4 Construction of regular polygons
9.5 Area constructions
9.6 Three impossible constructions
10. Transformations
10.1 The transformational perspective
10.2 Properties of isometries
10.3 Rotations, translations, and glide reflections
10.4 Classification of Euclidean motions
10.5 Classification of hyperbolic motions
10.6 Similarity transformations in Euclidean geometry
10.7 A transformational approach to the foundations
10.8 Euclidean inversions in circles
11. Models
11.1 The significance of models for hyperbolic geometry
11.2 The Cartesian model for Euclidean geometry
11.3 The Poincaré disk model for hyperbolic geometry
11.4 Other models for hyperbolic geometry
11.5 Models for elliptic geometry
11.6 Regular Tessellations
12. Polygonal Models and the Geometry of Space
12.1 Curved surfaces
12.2 Approximate models for the hyperbolic plane
12.3 Geometric surfaces
12.4 The geometry of the universe
12.5 Conclusion
12.6 Further study
12.7 Templates
APPENDICES
A. Euclid’s Book I
A.1 Definitions
A.2 Postulates
A.3 Common Notions
A.4 Propositions
B. Systems of Axioms for Geometry
B.1 Filling in Euclid’s gaps
B.2 Hilbert’s axioms
B.3 Birkhoff’s axioms
B.4 MacLane’s axioms
B.5 SMSG axioms
B.6 UCSMP axioms
C. The Postulates Used in this Book
C.1 The undefined terms
C.2 Neutral postulates
C.3 Parallel postulates
C.4 Area postulates
C.5 The reflection postulate
C.6 Logical relationships
D. Set Notation and the Real Numbers
D.1 Some elementary set theory
D.2 Properties of the real numbers
D.3 Functions
E. The van Hiele Model
F. Hints for Selected Exercises

Bibliography

The Fifty-Nine Icosahedra

(Lecture Notes in Statistics)

 H. S. M. Coxeter, P. DuVal, H. T. Flather e J. F. Petrie

Springer | 2013 -  reprint of the original 1st ed. 1982 edition | 52 páginas | rar - pdf | 1,7 Mb

link (password: matav)

Edição anterior - 1938 (link)

This is a completely new edition of the classic book which has been out of print for many years. The plans and illustrations of all 59 of the stellations of the icosahedron have been redrawn by Kate and David Krennell and there is a new introduction by Professor Coxeter. For a thorough understanding of the process of stellation and for splendid examples of polyhedra, this book will be a valuable addition to any mathematics library.

CONTENTS
1. Introduction... 3
2. Complete Enumeration of Stellated Icosahedra, by Considering the Possible Faces. . 8
3. An Alternative Enumeration, by Considering Solid Cells.... 15
4. Notes on the Plates. . 18

The Secrets of Triangles: A Mathematical Journey

Alfred S. Posamentier e Ingmar Lehmann 

Prometheus Books | 2012 | 387 páginas | rar - epub |9,9 Mb

link (password : matav)

Everyone knows what a triangle is, yet very few people appreciate that the common three-sided figure holds many intriguing "secrets." For example, if a circle is inscribed in any random triangle and then three lines are drawn from the three points of tangency to the opposite vertices of the triangle, these lines will always meet at a common point - no matter what the shape of the triangle. This and many more interesting geometrical properties are revealed in this entertaining and illuminating book about geometry. Flying in the face of the common impression that mathematics is usually dry and intimidating, this book proves that this sometimes-daunting, abstract discipline can be both fun and intellectually stimulating. 
The authors, two veteran math educators, explore the multitude of surprising relationships connected with triangles and show some clever approaches to constructing triangles using a straightedge and a compass. Readers will learn how they can improve their problem-solving skills by performing these triangle constructions. The lines, points, and circles related to triangles harbor countless surprising relationships that are presented here in a very engaging fashion.
Requiring no more than a knowledge of high school mathematics and written in clear and accessible language, this book will give all readers a new insight into some of the most enjoyable and fascinating aspects of geometry. 

Contents
Acknowledgments
Preface
1. Introduction to the Triangle
2. Concurrencies of a Triangle
3. Noteworthy Points in a Triangle
4. Concurrent Circles of a Triangle
5. Special Lines of a Triangle
6. Useful Triangle Theorems
7. Areas of and within Triangles
8. Triangle Constructions
9. Inequalities in a Triangle
10. Triangles and Fractals
Appendix

Geometry from Euclid to Knots

(Dover Books on Mathematics)

Saul Stahl

Dover Publications | 2010 | 480 páginas | rar - epub | 22,7 Mb

link
password: matav

Designed to inform readers about the formal development of Euclidean geometry and to prepare prospective high school mathematics instructors to teach Euclidean geometry, this text closely follows Euclid's classic, Elements. The text augments Euclid's statements with appropriate historical commentary and many exercises — more than 1,000 practice exercises provide readers with hands-on experience in solving geometrical problems.

In addition to providing a historical perspective on plane geometry, this text covers non-Euclidean geometries, allowing students to cultivate an appreciation of axiomatic systems. Additional topics include circles and regular polygons, projective geometry, symmetries, inversions, knots and links, graphs, surfaces, and informal topology. This republication of a popular text is substantially less expensive than prior editions and offers a new Preface by the author.

Contents
Preface to the Dover Edition
Preface
1 Other Geometries: A Computational Introduction
1.1 Spherical Geometry
1.2 Hyperbolic Geometry
1.3 Other Geometries
2 The Neutral Geometry of the Triangle
2.1 Introduction
2.2 Preliminaries
2.3 Propositions 1 through 28
2.4 Postulate 5 Revisited
3 Nonneutral Euclidean Geometry
3.1 Parallelism
3.2 Area
3.3 The Theorem of Pythagoras
3.4 Consequences of the Theorem of Pythagoras
3.5 Proportion and Similarity
4. Circles and Regular Polygons
4.1 The Neutral Geometry of the Circle
4.2 The Nonneutral Euclidean Geometry of the Circle
4.3 Regular Polygons
4.4 Circle Circumference and Area
4.5 Impossible Constructions
5 Toward Projective Geometry
5.1 Division of Line Segments
5.2 Collinearity and Concurrence
5.3 The Projective Plane
6 Planar Symmetries
6.1 Translations, Rotations, and Fixed Points
6.2 Reflections
6.3 Glide Reflections
6.4 The Main Theorems
6.5 Symmetries of Polygons
6.6 Frieze Patterns
6.7 Wallpaper Designs
7 Inversions
7.1 Inversions as Transformations
7.2 Inversions to the Rescue
7.3 Inversions as Hyperbolic Motions
8 Symmetry in Space
8.1 Regular and Semiregular Polyhedra
8.2 Rotational Symmetries of Regular Polyhedra
8.3 Monstrous Moonshine
9. Informal Topology
10 Graphs
10.1 Nodes and Arcs
10.2 Traversability
10.3 Colorings
10.4 Planarity
10.5 Graph Homeomorphisms
11 Surfaces
11.1 Polygonal Presentations
11.2 Closed Surfaces
11.3 Operations on Surfaces
11.4 Bordered Surfaces
12 Knots and Links
12.1 Equivalence of Knots and Links
12.2 Labelings
12.3 The Jones Polynomial
Appendix A: A Brief Introduction to The Geometer's Sketchpad®
Appendix B: Summary of Propositions
Appendix C: George D. Birkhoff's Axiomatization of Euclidean Geometry
Appendix D: The University of Chicago School Mathematics Project's Geometrical Axioms
Appendix E: David Hilbert's Axiomatization of Euclidean Geometry

Famous geometrical theorems and problems, with their history

William Whitehead Rupert

Boston, D.C. Heath & Co. | 1900

online: 
archive.org
hathitrust.org
forgottenbooks.org

The author, having derived much pleasure and inspiration from the brief historical notes in some of the mathematical text-books that he studied when a student in college, has thought that, by giving the history of a few of the most celebrated geometrical theorems and problems, he might place a light in the window which may throw a cheerful ray adown the long and sometimes dusty pathway that leads to geometrical truth. In the preparation of this little book most valuable assistance has been derived from Florian Cajori sHistory of Mathematics, James Gows History of Greek Mathematics, and G, J. Allmans Greek Geometry from Thales to Euclid, It is, however, toW. W.Rourse Balls reniarkably interesting Short History of Mathematics that Famous Geometrical Theorems and Problems owes the largest debt. To Professor A, D. Eisenhower, Principal of the Norristown High School, George Q.Sheppard, Professor of Mathematics, Hill School, Pottstown, Pa., Dr. George M.Philips, Principal West Chester State Normal School, and Daniel Carhart, Ce., Dean and Professor of Civil Engineering, Western University of Pennsylvania, who have read this book in manuscript, the author is indebted for valuable, suggestions and many kind words of encouragement.

Second Handbook of Research on Mathematics Teaching and Learning

Frank K. Jr. Lester

Information Age Publishing | 2007 | 1381 páginas | rar - pdf | 11,4 Mb

link
password: matav

The audience remains much the same as for the 1992 Handbook, namely, mathematics education researchers and other scholars conducting work in mathematics education. This group includes college and university faculty, graduate students, investigators in research and development centers, and staff members at federal, state, and local agencies that conduct and use research within the discipline of mathematics.
The intent of the authors of this volume is to provide useful perspectives as well as pertinent information for conducting investigations that are informed by previous work. The Handbook should also be a useful textbook for graduate research seminars. In addition to the audience mentioned above, the present Handbook contains chapters that should be relevant to four other groups: teacher educators, curriculum developers, state and national policy makers, and test developers and others involved with assessment.

Taken as a whole, the chapters reflects the mathematics education research community's willingness to accept the challenge of helping the public understand what mathematics education research is all about and what the relevance of their research fi ndings might be for those outside their immediate community.
The intent of the authors of this volume is to provide useful perspectives as well as pertinent information for conducting investigations that are informed by previous work. The Handbook should also be a useful textbook for graduate research seminars. In addition to the audience mentioned above, the present Handbook contains chapters that should be relevant to four other groups: teacher educators, curriculum developers, state and national policy makers, and test developers and others involved with assessment.
Taken as a whole, the chapters reflects the mathematics education research community's willingness to accept the challenge of helping the public understand what mathematics education research is all about and what the relevance of their research fi ndings might be for those outside their immediate community.
Taken as a whole, the chapters reflects the mathematics education research community's willingness to accept the challenge of helping the public understand what mathematics education research is all about and what the relevance of their research fi ndings might be for those outside their immediate community. 

CONTENTS

Preface. 
Acknowledgements.
Part I: Foundations. 
Putting Philosophy to Work: Coping With Multiple Theoretical Perspectives, Paul Cobb.
Theory in Mathematics Education Scholarship, Edward A. Silver & Patricio G. Herbst
Method, Alan H. Schoenfeld. 
Part II: Teachers and Teaching. 
Assessing Teachers' Mathematical Knowledge: What Knowledge Matters and What Evidence Counts? Heather C. Hill, Laurie Sleep, Jennifer M. Lewis, & Deborah Loewenberg Ball. 
The Mathematical Education and Development of Teachers, Judith T. Sowder.
Understanding Teaching and Classroom Practice in Mathematics, Megan Loef Franke, Elham Kazemi and Daniel Battey. 
Mathematics Teachers' Beliefs and Affect, Randolph A. Philipp. 
Part III: Influences on Student Outcomes. 
How Curriculum Influences Student Learning, Mary Kay Stein, Janine Remillard and Margaret Smith. 
The Effects of Classroom Mathematics Teaching on Students' Learning, James S. Hiebert and Douglas A. Grouws. 
Culture, Race, Power, and Mathematics Education, Diversity in Mathematics Education Center for Learning and Teaching. 
The Role of Culture in Teaching and Learning Mathematics, Norma G. Presmeg. 
Part IV: Students and Learning. 
Early Childhood Mathematics Learning, Douglas H. Clements and Julie Sarama. 
Whole Number Concepts and Operations, Lieven Verschaffel, Brian Greer, and Erik DeCorte.
Rational Numbers and Proportional Reasoning: Toward a Theoretical Framework for Research, Susan J. Lamon. Early Algebra, David W. Carraher and Analucia D. Schliemann. 
Learning and Teaching of Algebra at the Middle School through College Levels: Building Meaning for Symbols and Their Manipulation, Carolyn Kieran. 
Problem Solving and Modeling, Richard Lesh and Judith Zawejewski. 
Toward Comprehensive Perspectives on the Learning and Teaching of Proof, Guershon Harel and Larry Sowder. 
The Development of Geometric and Spatial Thinking, Michael T. Battista. 
Research in Probability: Responding to Classroom Realities, Graham A. Jones, Cynthia W. Langrall and Edward S. Mooney. 
Research on Statistics Learning and Reasoning, J. Michael Shaughnessy. 
Mathematics Thinking and Learning at Post-secondary Level, Michele Artigue, Carmen Batanero and Phillip Kent. 
Part V: Assessment. 
Keeping Learning on Track: Classroom Assessment and the Regulation of Learning, Dylan Wiliam. 
High Stakes Testing in Mathematics, Linda Dager Wilson. 
Large-scale Assessment of Mathematics Education, Jan DeLange. 
Part VI: Issues and Perspectives. 
Issues in Access and Equity in Mathematics Education, Alan J. Bishop and Helen J. Forgasz. 
Research on Technology in Mathematics Education: The Perspective of Constructs, Rose Mary Zbiek, M. Kathleen Heid, Glendon Blume and Thomas P. Dick. 
Engineering Change in Mathematics Education: Research, Policy, and Practice, William F. Tate and Celia Rousseau. 
Educational Policy Research and Mathematics Education, Joan Ferrini-Mundy & Robert Floden. 
Mathematics Content Specification in the Age of Assessment, Norman L. Webb. 
Reflections on the State and Trends in Research on Mathematics Teaching and Learning: From Here to Utopia, Mogens Niss.

From Calculus to Computers Using the last 200 years of mathematics history in the classroom

(Mathematical Association of America Notes)

Amy Shell-Gellasch e Dick Jardine

The Mathematical Association of America | 2005 | 268 páginas | rar - pdf | 1,9 Mb

link (password: matav)

To date, much of the literature prepared on the topic of integrating mathematics history into undergraduate teaching contains, predominantly, ideas from the 18th century and earlier. This volume focuses on nineteenth- and twentieth-century mathematics, building on the earlier efforts but emphasizing recent history in the teaching of mathematics, computer science, and related disciplines. From Calculus to Computers is a resource for undergraduate teachers that provides ideas and materials for immediate adoption in the classroom and proven examples to motivate innovation by the reader. Contributions to this volume are from historians of mathematics and college mathematics instructors with years of experience and expertise in these subjects. Examples of topics covered are probability in undergraduate statistics courses, logic and programming for computer science, undergraduate geometry to include non-Euclidean geometries, numerical analysis, and abstract algebra.
Emphasizes mathematics history from the nineteenth and twentieth centuries
Provides ideas and material for immediate adoption in the classroom
Topics covered range from Galois theory to using the history of women and minorities in teaching

Table of Contents
Preface
Introduction
Part I. Algebra, Number Theory, Calculus, and Dynamical Systems:
1. Arthur Cayley and the first paper on group theory David J. Pengelley
2. Putting the differential back into differential calculus Robert Rogers
3. Using Galois' idea in the teaching of abstract algebra Matt D. Lunsford
4. Teaching elliptic curves using original sources Lawrence D'Antonio
5. Using the historical development of predator-prey models to teach mathematical modeling Holly P. Hirst
Part II. Geometry:
6. How to use history to clarify common confusions in geometry Daina Taimina and David W. Henderson
7. Euler on Cevians Eisso J. Atzema and Homer White
8. Modern geometry after the end of mathematics Jeff Johannes
Part III. Discrete Mathematics, Computer Science, Numerical Methods, Logic, and Statistics:
9. Using 20th century history in a combinatorics and graph theory class Linda E. MacGuire
10. Public key cryptography Shai Simonson
11. Introducing logic via Turing machines Jerry M. Lodder
12. From Hilbert's program to computer programming William Calhoun
13. From the tree method in modern logic to the beginning of automated theorem proving Francine F. Abeles
14. Numerical methods history projects Dick Jardine
15. Foundations of Statistics in American Textbooks: probability and pedagogy in historical context Patti Wilger Hunter
Part IV. History of Mathematics and Pedagogy:
16. Incorporating the mathematical achievements of women and minority mathematicians into classrooms Sarah J. Greenwald
17. Mathematical topics in an undergraduate history of science course David Lindsay Roberts
18. Building a history of mathematics course from a local perspective Amy Shell-Gellasch
19. Protractors in the classroom: an historical perspective Amy Ackerberg-Hastings
20. The metric system enters the American classroom:
1790-1890 Peggy Aldrich Kidwell
21. Some wrinkles for a history of mathematics course Peter Ross
22. Teaching history of mathematics through problems John R. Prather