Turtle Geometry

The Computer as a Medium for Exploring Mathematics
Author: Harold Abelson,Andrea A. DiSessa
Publisher: MIT Press
ISBN: 9780262510370
Category: Computers
Page: 477
View: 8257

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Turtle Geometry presents an innovative program of mathematical discovery that demonstrates how the effective use of personal computers can profoundly change the nature of a student's contact with mathematics. Using this book and a few simple computer programs, students can explore the properties of space by following an imaginary turtle across the screen.The concept of turtle geometry grew out of the Logo Group at MIT. Directed by Seymour Papert, author of Mindstorms, this group has done extensive work with preschool children, high school students and university undergraduates. Harold Abelson is an associate professor in the Department of Electrical Engineering and Computer Science at MIT. Andrea diSessa is an associate professor in the Graduate School of Education, University of California, Berkeley.

Exploring Geometry, Second Edition


Author: Michael Hvidsten
Publisher: CRC Press
ISBN: 1498760988
Category: Mathematics
Page: 558
View: 3688

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This text promotes student engagement with the beautiful ideas of geometry. Every major concept is introduced in its historical context and connects the idea with real-life. A system of experimentation followed by rigorous explanation and proof is central. Exploratory projects play an integral role in this text. Students develop a better sense of how to prove a result and visualize connections between statements, making these connections real. They develop the intuition needed to conjecture a theorem and devise a proof of what they have observed.

Geometry Turned On

Dynamic Software in Learning, Teaching, and Research
Author: James King,Doris Schattschneider
Publisher: Cambridge University Press
ISBN: 9780883850992
Category: Mathematics
Page: 206
View: 7771

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This volume is a collection of articles about dynamic geometry: active, exploratory geometry carried out with interactive computer software. This software has had a profound effect on classroom teaching wherever it has been introduced. Unconstrained parts of the configurations are moveable - they can literally be grabbed with a cursor (using a mouse) and be dragged or stretched - and as they move, all other objects in the configuration automatically self-adjust, preserving all dependent relationships and constraints. The software has also become an indispensable research tool for mathematicians and scientists. This book gives many examples of the ways in which it can be used, and some of the effects it can have. It raises various questions for teaching and research. Some articles address the basic question, 'What is dynamic geometry good for?' as they discuss: accuracy of construction, visualization, exploration and discovery, motivating proof, transformations, tracing loci, simulation, and creating microworlds.

An Integrated Introduction to Computer Graphics and Geometric Modeling


Author: Ronald Goldman
Publisher: CRC Press
ISBN: 1439803358
Category: Computers
Page: 574
View: 3946

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Taking a novel, more appealing approach than current texts, An Integrated Introduction to Computer Graphics and Geometric Modeling focuses on graphics, modeling, and mathematical methods, including ray tracing, polygon shading, radiosity, fractals, freeform curves and surfaces, vector methods, and transformation techniques. The author begins with fractals, rather than the typical line-drawing algorithms found in many standard texts. He also brings the turtle back from obscurity to introduce several major concepts in computer graphics. Supplying the mathematical foundations, the book covers linear algebra topics, such as vector geometry and algebra, affine and projective spaces, affine maps, projective transformations, matrices, and quaternions. The main graphics areas explored include reflection and refraction, recursive ray tracing, radiosity, illumination models, polygon shading, and hidden surface procedures. The book also discusses geometric modeling, including planes, polygons, spheres, quadrics, algebraic and parametric curves and surfaces, constructive solid geometry, boundary files, octrees, interpolation, approximation, Bezier and B-spline methods, fractal algorithms, and subdivision techniques. Making the material accessible and relevant for years to come, the text avoids descriptions of current graphics hardware and special programming languages. Instead, it presents graphics algorithms based on well-established physical models of light and cogent mathematical methods.

Turtles, Termites, and Traffic Jams

Explorations in Massively Parallel Microworlds
Author: Mitchel Resnick
Publisher: MIT Press
ISBN: 9780262680936
Category: Computers
Page: 163
View: 6565

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How does a bird flock keep its movements so graceful and synchronized? Most people assume that the bird in front leads and the others follow. In fact, bird flocks don't have leaders: they are organized without an organizer, coordinated without a coordinator. And a surprising number of other systems, from termite colonies to traffic jams to economic systems, work the same decentralized way. Turtles, Termites, and Traffic Jams describes innovative new computational tools that can qhelp people (even young children) explore the workings of such systems--and help them move beyond the centralized mindset.

Paris--a Musical Gazetteer


Author: Nigel Simeone
Publisher: Yale University Press
ISBN: 9780300080544
Category: History
Page: 299
View: 8717

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This essential guidebook is designed for all travelers interested in exploring the historic musical sites of Paris - in person or from an armchair. Paris is a uniquely rich music capital, its streets echoing with centuries of great music that has been created and performed there. Virtually every neighborhood boasts a concert hall, church, museum, or home that has played a significant role in the extraordinary musical tradition of the city. This gazetteer will guide you to the important musical landmarks in Paris and explain why each is noteworthy.

The Algorithmic Beauty of Plants


Author: Przemyslaw Prusinkiewicz,Aristid Lindenmayer
Publisher: Springer Science & Business Media
ISBN: 1461384761
Category: Computers
Page: 228
View: 6441

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Now available in an affordable softcover edition, this classic in Springer's acclaimed Virtual Laboratory series is the first comprehensive account of the computer simulation of plant development. 150 illustrations, one third of them in colour, vividly demonstrate the spectacular results of the algorithms used to model plant shapes and developmental processes. The latest in computer-generated images allow us to look at plants growing, self-replicating, responding to external factors and even mutating, without becoming entangled in the underlying mathematical formulae involved. The authors place particular emphasis on Lindenmayer systems - a notion conceived by one of the authors, Aristid Lindenmayer, and internationally recognised for its exceptional elegance in modelling biological phenomena. Nonetheless, the two authors take great care to present a survey of alternative methods for plant modelling.

Mindstorms

Children, Computers, And Powerful Ideas
Author: Seymour A. Papert
Publisher: Basic Books
ISBN: 9780465046744
Category: Education
Page: 252
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Mindstorms has two central themes: that children can learn to use computers in a masterful way and that learning to use computers can change the way they learn everything else. Even outside the classroom, Papert had a vision that the computer could be used just as casually and as personally for a diversity of purposes throughout a person's entire life. Seymour Papert makes the point that in classrooms saturated with technology there is actually more socialization and that the technology often contributes to greater interaction among students and among students and instructors.

Windows on Mathematical Meanings

Learning Cultures and Computers
Author: Richard Noss,Celia Hoyles
Publisher: Springer Science & Business Media
ISBN: 9400916965
Category: Education
Page: 278
View: 1966

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This book challenges some of the conventional wisdoms on the learning of mathematics. The authors use the computer as a window onto mathematical meaning-making. The pivot of their theory is the idea of webbing, which explains how someone struggling with a new mathematical idea can draw on supportive knowledge, and reconciles the individual's role in mathematical learning with the part played by epistemological, social and cultural forces.

Advanced Logo

A Language for Learning
Author: Michael Friendly
Publisher: Psychology Press
ISBN: 1317766784
Category: Psychology
Page: 676
View: 7017

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Advanced Logo shows how LOGO can be used as a vehicle to promote problem solving skills among secondary students, college students, and instructors. The book demonstrates the wide range of educational domains that can be explored through LOGO including generative grammars, physical laws of motion and mechanics, artificial intelligence, robotics, and calculus.

Perspectives on the Teaching of Geometry for the 21st Century

An ICMI Study
Author: Carmelo Mammana,V. Villani
Publisher: Springer Science & Business Media
ISBN: 9401152268
Category: Education
Page: 353
View: 394

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In recent years geometry seems to have lost large parts of its former central position in mathematics teaching in most countries. However, new trends have begun to counteract this tendency. There is an increasing awareness that geometry plays a key role in mathematics and learning mathematics. Although geometry has been eclipsed in the mathematics curriculum, research in geometry has blossomed as new ideas have arisen from inside mathematics and other disciplines, including computer science. Due to reassessment of the role of geometry, mathematics educators and mathematicians face new challenges. In the present ICMI study, the whole spectrum of teaching and learning of geometry is analysed. Experts from all over the world took part in this study, which was conducted on the basis of recent international research, case studies, and reports on actual school practice. This book will be of particular interest to mathematics educators and mathematicians who are involved in the teaching of geometry at all educational levels, as well as to researchers in mathematics education.

Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics


Author: Robert B. Banks
Publisher: Princeton University Press
ISBN: 1400843030
Category: Mathematics
Page: 304
View: 9255

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Have you ever daydreamed about digging a hole to the other side of the world? Robert Banks not only entertains such ideas but, better yet, he supplies the mathematical know-how to turn fantasies into problem-solving adventures. In this sequel to the popular Towing Icebergs, Falling Dominoes (Princeton, 1998), Banks presents another collection of puzzles for readers interested in sharpening their thinking and mathematical skills. The problems range from the wondrous to the eminently practical. In one chapter, the author helps us determine the total number of people who have lived on earth; in another, he shows how an understanding of mathematical curves can help a thrifty lover, armed with construction paper and scissors, keep expenses down on Valentine's Day. In twenty-six chapters, Banks chooses topics that are fairly easy to analyze using relatively simple mathematics. The phenomena he describes are ones that we encounter in our daily lives or can visualize without much trouble. For example, how do you get the most pizza slices with the least number of cuts? To go from point A to point B in a downpour of rain, should you walk slowly, jog moderately, or run as fast as possible to get least wet? What is the length of the seam on a baseball? If all the ice in the world melted, what would happen to Florida, the Mississippi River, and Niagara Falls? Why do snowflakes have six sides? Covering a broad range of fields, from geography and environmental studies to map- and flag-making, Banks uses basic algebra and geometry to solve problems. If famous scientists have also pondered these questions, the author shares the historical details with the reader. Designed to entertain and to stimulate thinking, this book can be read for sheer personal enjoyment.

Geometric Integration Theory


Author: Hassler Whitney
Publisher: Courier Corporation
ISBN: 048615470X
Category: Mathematics
Page: 400
View: 3884

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Geared toward upper-level undergraduates and graduate students, this treatment of geometric integration theory consists of an introduction to classical theory, a postulational approach to general theory, and a section on Lebesgue theory. 1957 edition.

Functional Differential Geometry


Author: Gerald Jay Sussman,Jack Wisdom,Will Farr
Publisher: MIT Press
ISBN: 0262019345
Category: Mathematics
Page: 228
View: 1374

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Physics is naturally expressed in mathematical language. Students new to the subject must simultaneously learn an idiomatic mathematical language and the content that is expressed in that language. It is as if they were asked to read Les Misérables while struggling with French grammar. This book offers an innovative way to learn the differential geometry needed as a foundation for a deep understanding of general relativity or quantum field theory as taught at the college level. The approach taken by the authors (and used in their classes at MIT for many years) differs from the conventional one in several ways, including an emphasis on the development of the covariant derivative and an avoidance of the use of traditional index notation for tensors in favor of a semantically richer language of vector fields and differential forms. But the biggest single difference is the authors' integration of computer programming into their explanations. By programming a computer to interpret a formula, the student soon learns whether or not a formula is correct. Students are led to improve their program, and as a result improve their understanding.