Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists.

Author: Michael Henle

Publisher: Pearson College Division

ISBN: UOM:39076002874936

Category: Mathematics

Page: 389

View: 228

Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.

C H A P T E R Although the term "modern geometries" traditionally refers to post-
Euclid geometries, namely the non-Euclidean and projective geometries
presented in Chapters 2 and 4, it seems ironic to describe topics formalized
hundreds of ...

Author: Judith N. Cederberg

Publisher: Springer Science & Business Media

ISBN: 9781475734904

Category: Mathematics

Page: 441

View: 297

Designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. The first chapter presents several finite geometries in an axiomatic framework, while Chapter 2 continues the synthetic approach in introducing both Euclids and ideas of non-Euclidean geometry. There follows a new introduction to symmetry and hands-on explorations of isometries that precedes an extensive analytic treatment of similarities and affinities. Chapter 4 presents plane projective geometry both synthetically and analytically, and the new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Throughout, each chapter includes a list of suggested resources for applications or related topics in areas such as art and history, plus this second edition points to Web locations of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions are available for "Cabri Geometry" and "Geometers Sketchpad".

Discuss early applications of geometry found in history of mathematics texts from
African cultures , cultures of the Far East , and Central and South American
cultures . 1 . 2 DEVELOPMENT OF MODERN GEOMETRIES The title of this book
...

Author: James R. Smart

Publisher: Brooks/Cole Publishing Company

ISBN: UOM:39015055843331

Category: Mathematics

Page: 451

View: 742

This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. This edition reflects the recommendations of the COMAP proceedings on Geometry's Future, the NCTM standards, and the Professional Standards for Teaching Mathematics. References to a new companion text, Active Geometry by David A. Thomas encourage students to explore the geometry of motion through the use of computer software. Using Active Geometry at the beginning of various sections allows professors to give students a somewhat more intuitive introduction using current technology before moving on to more abstract concepts and theorems.

This book emphasizes the beauty of geometry using a modern approach.

Author: Arthur Baragar

Publisher: Pearson College Division

ISBN: UCSC:32106017935435

Category: Mathematics

Page: 370

View: 314

This book emphasizes the beauty of geometry using a modern approach. Models & computer exercises help readers to cultivate geometric intuition. Topics include Euclidean Geometry, Hand Constructions, Geometer's Sketch Pad, Hyperbolic Geometry, Tilings & Lattices, Spherical Geometry, Projective Geometry, Finite Geometry, and Modern Geometry Research. Ideal for geometry at an intermediate level.

This text updates the teaching of college geometry based upon three fundamental ideas: geometries only approximate reality; the best presentation of a geometry is by transformation groups; and points and other geometric objects should be co ...

Author: Michael Henle

Publisher: Pearson College Division

ISBN: 013193418X

Category: Mathematics

Page: 372

View: 672

This text updates the teaching of college geometry based upon three fundamental ideas: geometries only approximate reality; the best presentation of a geometry is by transformation groups; and points and other geometric objects should be co-ordinated. The work is designed to be engaging and accessible, and it describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists.

MODERN GEOMETRY was written to provide undergraduate and graduate level mathematics education students with an introduction to both Euclidean and non-Euclidean geometries, appropriate to their needs as future junior and senior high school ...

Author: David A. Thomas

Publisher: Brooks/Cole Publishing Company

ISBN: 0534365507

Category: Mathematics

Page: 348

View: 976

MODERN GEOMETRY was written to provide undergraduate and graduate level mathematics education students with an introduction to both Euclidean and non-Euclidean geometries, appropriate to their needs as future junior and senior high school mathematics teachers. MODERN GEOMETRY provides a systematic survey of Euclidean, hyperbolic, transformation, fractal, and projective geometries. This approach is consistent with the recommendations of the National Council of Teachers of Mathematics (NCTM), the International Society for Technology in Education (ISTE), and other professional organizations active in the preparation and continuing professional development of K-12 mathematics teachers.

School Mathematics Study Group. (1965). Geometry:Student's Text, rev. ed.
Pasadena, CA: A.C. Vroman. Seidenberg, A. (1962). Lectures in Projective
Geometry. New York: Van Nostrand Reinhold. Smart, J.R. (1978). Modern Geometries, 2nd ...

Author: Judith Cederberg

Publisher: Springer

ISBN: UOM:39076000911177

Category: Mathematics

Page: 232

View: 850

A Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 introduces Euclid's geometry and the basic ideas of non-Euclidean geometry. The synthetic approach of Chapters 1 - 2 is followed by the analytic treatment of transformations of the Euclidean plane in Chapter 3. Chapter 4 presents plane projective geometry both synthetically and analytically. The extensive use of matrix representations of groups of transformations in Chapters 3 - 4 reinforces ideas from linear algebra and serves as excellent preparation for a course in abstract algebra. Each chapter includes a list of suggested sources for applications and/or related topics.

Theorem 1.8.1 depends strongly on the parallel postulate-it is not true in non-
Euclidean geometries where “space” is “curved”. The difference between the sum
of the angles of a triangle and 180° is a measure of the curvature of space; if the ...

Author: George A. Jennings

Publisher: Springer Science & Business Media

ISBN: 9781461208556

Category: Mathematics

Page: 204

View: 170

This introduction to modern geometry differs from other books in the field due to its emphasis on applications and its discussion of special relativity as a major example of a non-Euclidean geometry. Additionally, it covers the two important areas of non-Euclidean geometry, spherical geometry and projective geometry, as well as emphasising transformations, and conics and planetary orbits. Much emphasis is placed on applications throughout the book, which motivate the topics, and many additional applications are given in the exercises. It makes an excellent introduction for those who need to know how geometry is used in addition to its formal theory.

However, the latter does not often occur except in high school Euclidean
geometry. In modern geometries such as algebraic geometry, the associated
number fields are usually the complex field and arbitrary fields of characteristic 0,
so no ...

Author: Wen-tsün Wu

Publisher: Springer Science & Business Media

ISBN: 3211825061

Category: Computers

Page: 288

View: 767

This book is a translation of Professor Wu's seminal Chinese book of 1984 on Automated Geometric Theorem Proving. The translation was done by his former student Dongming Wang jointly with Xiaofan Jin so that authenticity is guaranteed. Meanwhile, automated geometric theorem proving based on Wu's method of characteristic sets has become one of the fundamental, practically successful, methods in this area that has drastically enhanced the scope of what is computationally tractable in automated theorem proving. This book is a source book for students and researchers who want to study both the intuitive first ideas behind the method and the formal details together with many examples.

In order to avoid uninteresting cases from the point of view of geometry, we will
assume throughout the whole book that there exist elements a, b in X which are
linearly independent. But, on the other hand, we do not ask for the existence of a
...

Author: Walter Benz

Publisher: Springer Science & Business Media

ISBN: 3764374322

Category: Mathematics

Page: 244

View: 871

Presents the real inner product spaces of arbitrary (finite or infinite) dimension greater than or equal to 2. This book studies the sphere geometries of Mobius and Lie for these spaces, besides euclidean and hyperbolic geometry, as well as geometries where Lorentz transformations play the key role.

Boston, Mass.: Prindle, Weber & Schmidt, 1966. CHOQUET, GUSTAve, Geometry
in a Modern Setting. Boston, Mass.: Houghton Mifflin Company, 1969. CoxETER,
H. S. M., Introduction to Geometry. New York: John Wiley and Sons, 1961.

Author: Howard Whitley Eves

Publisher: Jones & Bartlett Learning

ISBN: 0867204753

Category: Mathematics

Page: 370

View: 435

College Geometry is divided into two parts. Part I is a sequel to basic high school geometry and introduces the reader to some of the important modern extensions of elementary geometry- extension that have largely entered into the mainstream of mathematics. Part II treats notions of geometric structure that arose with the non-Euclidean revolution in the first half of the nineteenth century.

We can now see why Klein introduced the terminology hyperbolic for
Lobatchevsky's geometry , elliptic for the case of ... Moreover , the non -
Euclidean and Euclidean geometries were also seen to be special cases or
subgeometries of ...

Author: Morris Kline

Publisher: OUP USA

ISBN: 0195061373

Category: Mathematics

Page: 448

View: 168

Traces the development of mathematics from its beginnings in Babylonia and ancient Egypt to the work of Riemann and Godel in modern times

This old theorem , used by Euclid as a first step in the comparison of areas , and
leading up to his proof of the theorem of Pythagoras , has been generally
dropped from the modern geometries . It can be made useful occasionally , as for
...

This old theorem , used by Euclid as a first step in the comparison of areas , and
leading up to his proof of the theorem of Pythagoras , has been generally
dropped from the modern geometries . It can be made useful occasionally , as for
...

Author: Roger Arthur Johnson

Publisher:

ISBN: PSU:000016990890

Category: Geometry

Page: 319

View: 275

The author deals with the geometry of the triangle and the circle, concentrating on extensions of Euclidean theory, and exploring in unusual detail the many theorems that have been developed by relatively recent geometers. Several hundred of these theorems and proofs are available only in widely scattered journals, this book also has value as a time-saver and reference work to geometers and teachers of geometry. In addition the author has included a number of theorems left unproved, to be used by the student as exercises.

This little book attempts to bring modern geometry nearer to the imagination of
the reader . We must apologise to the late Rev. Charles Dodgson ( alias Lewis
Carroll ) and Mrs. Hargrave ( née Alice Liddell ) for sublimating their historical ...

Cf . also Convegno reticoli e geometrie proiettive , Palermo 1957 . Rome ,
Cremonese , 1958 ... Tallini , G . [ 8 ] : Le geometrie di Galois e le loro
applicazioni alla statistica e alla teoria ... WAERDEN ( Van der ) , B . L . [ 3 ] : Modern Algebra .

Incidence geometry has a long tradition starting with Euclid [25] and his
predecessors and is at the same time a central part of modern mathematics. The
main progress in incidence geometry until about 1900 are the axioms of affine
and ...

Author: Johannes Ueberberg

Publisher: Springer Science & Business Media

ISBN: 9783642209727

Category: Mathematics

Page: 248

View: 686

Incidence geometry is a central part of modern mathematics that has an impressive tradition. The main topics of incidence geometry are projective and affine geometry and, in more recent times, the theory of buildings and polar spaces. Embedded into the modern view of diagram geometry, projective and affine geometry including the fundamental theorems, polar geometry including the Theorem of Buekenhout-Shult and the classification of quadratic sets are presented in this volume. Incidence geometry is developed along the lines of the fascinating work of Jacques Tits and Francis Buekenhout. The book is a clear and comprehensible introduction into a wonderful piece of mathematics. More than 200 figures make even complicated proofs accessible to the reader.

con diret Introduction to Modern Geometry by Lathematics J . L . Vanderslice . 1 -
15 - 52 . 111 JT When a student who has been raised on the plane and solid
geometry of high schools first becomes aware of the existence of many other
types ...

With Chapters on Mensuration and Modern Geometry Isaac Sharpless.
PREFACE . This little work on elementary Geometry is offered to those who
desire to make themselves familiar with the general principles of the science in
the limited time ...

LINES, PLANES, ANGLES, h chapter AND SETS The mathematics taught in the
years preceding high school almost always includes Some geometry presented
in an informal way. As a rule, students acquire a very good basic knowledge of ...