The distinctive approach of Henderson and Taimina's volume stimulates readers to develop a broader, deeper, understanding of mathematics through active experience—including discovery, discussion, writing fundamental ideas and learning about the history of those ideas. A series of interesting, challenging problems encourage readers to gather and discuss their reasonings and understanding. The volume provides an understanding of the possible shapes of the physical universe. The authors provide extensive information on historical strands of geometry, straightness on cylinders and cones and hyperbolic planes, triangles and congruencies, area and holonomy, parallel transport, SSS, ASS, SAA, and AAA, parallel postulates, isometries and patterns, dissection theory, square roots, pythagoras and similar triangles, projections of a sphere onto a plane, inversions in circles, projections (models) of hyperbolic planes, trigonometry and duality, 3-spheres and hyperbolic 3-spaces and polyhedra. For mathematics educators and other who need to understand the meaning of geometry.
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Editorial Reviews
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"I like the authors' writing style and I think the explanations are very clear .... I know that my students read this book, which is certainly saying something. Sometimes they think it's hard to read, but it is at least possible-and they could think it's hard to read because it might be one of the few times they have actually had to read the text .... I think the historical illustrations will be useful for creating interest with the students. This is a nice addition to this edition of the text." — Barbara Edwards, Oregon State University
"I found this text to be a wonderful, fresh, and innovative treatment of geometry. The Moore method of study by doing and involving the student works very well indeed for this subject. The style of the text is very friendly and encouraging and gets the student involved quickly with a give-and-take approach. The student develops insights and skills probably not obtainable in more traditional courses. This is a very fine text that I would strongly recommend for a beginning course in Euclidean and non-Euclidean geometry." — Norman Johnson, University of Iowa
"This book remains a treasure, an essential reference. I simply know of no other book that attempts the broad vision of geometry that this book does, that does it by-and-large so successfully, and that pays so much attention to geometric intuition, student cognitive development, and rigorous mathematics." — Judy Roitman, University of Kansas
Excerpt. © Reprinted by permission. All rights reserved.
We believe that mathematics is a natural and deep part of human experience and that experiences of meaning in mathematics are accessible to everyone. Much of mathematics is not accessible through formal approaches except to those with specialized learning. However, through the use of nonformal experience and geometric imagery, many levels of meaning in mathematics can be opened up in a way that most humans can experience and find intellectually challenging and stimulating.
Formalism contains the power of the meaning but not the meaning. It is necessary to bring the power back to the meaning.
A formal proof as we normally conceive of it is not the goal of mathematics—it is a tool—a means to an end. The goal is understanding. Without understanding we will never be satisfied—with understanding we want to expand that understanding and to communicate it to others. This book is based on a view of proof as a convincing communication that answers—Why?
Many formal aspects of mathematics have now been mechanized and this mechanization is widely available on personal computers or even handheld calculators, but the experience of meaning in mathematics is still a human enterprise that is necessary for creative work.
In this book we invite the reader to explore the basic ideas of geometry from a more mature standpoint. We will suggest some of the deeper meanings, larger contexts, and interrelations of the ideas. We are interested in conveying a different approach to mathematics, stimulating the reader to take a broader and deeper view of mathematics and to experience for herself/himself a sense of mathematizing. Through an active participation with these ideas, including exploring and writing about them, people can gain a broader context and experience. This active participation is vital for anyone who wishes to understand mathematics at a deeper level, or anyone wishing to understand something in their experience through the vehicle of mathematics.
This is particularly true for teachers or prospective teachers who are approaching related topics in the school curriculum. All too often we convey to students that mathematics is a closed system, with a single answer or approach to every problem, and often without a larger context. We believe that even where there are strict curricular constraints, there is room to change the meaning and the experience of mathematics in the classroom.
This book is based on a junior/senior-level course that David started teaching in 1974 at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication.
The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems and are encouraged to write and speak their reasonings and understandings.
Most of the problems are placed in an appropriate history perspective and approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). We find that by exploring the geometry of a sphere and a hyperbolic plane, our students gain a deeper understanding of the geometry of the (Euclidean) plane.
We introduce the modern notion of "parallel transport along a geodesic," which is a notion of parallelism that makes sense on the plane but also on a sphere or hyperbolic plane (in fact, on any surface). While exploring parallel transport on a sphere, students are able to appreciate more fully that the similarities and differences between the Euclidean geometry of the plane and the nonEuclidean geometries of a sphere or hyperbolic plane are not adequately described by the usual Parallel Postulate. We find that the early interplay between the plane and spheres and hyperbolic planes enriches all the later topics whether on the plane or on spheres and hyperbolic planes. All of these benefits will also exist by only studying the plane and spheres for those instructors that choose to do so.
CHANGES IN THIS EDITION
This book is an expansion and revision of the book Experiencing Geometry on Plane and Sphere (1996) and the book Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces (2001). There are several important changes: First, there are now coauthors—Daina was a "contributor" to the second edition. She brings considerable experience with and knowledge of the history of mathematics. We start in Chapter 0 with an introduction to four strands in the history of geometry and use the framework of these strands to infuse history into (almost) every chapter in the book in ways to enhance the students understanding and to clear up many misconceptions. There are two new chapters—the old Chapter 14 (Circles in the Plane) has been split into two new chapters: Chapter 15 (on circles with added results on spheres and hyperbolic plane and about trisecting angles and other constructions) and Chapter 16 (on inversions with added material on applications). There is also a new Chapter 21, on the geometry of mechanisms that includes historical machines and results in modern mathematics.
We have included discussions of four new geometric results announced' in 2003-2004: In Chapter 12 we describe the discovery and solution of Archimedes' Stomacion Problem. Problem 15.2 is based on the 2003 generalization of the notion of power of a point to spheres. In Chapter 16 we talk about applications of a problem of Apollonius to modern pharmacology. In Chapter 18 we discuss the newly announced solution of the Poincare Conjecture. In Chapter 22 we bring in a new result about unfolding linkages. In Chapter 24 we discuss the latest updates on the shape of space, including the possibility that the shape of the universe is based on a dodecahedron. In addition, we have rearranged and clarified other chapters from the earlier editions.
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Most Helpful Customer Reviews
5 of 6 people found the following review helpful:
4.0 out of 5 stars
Some good surface geometry, then poor,
This review is from: Experiencing Geometry (3rd Edition) (Paperback)
The first five chapters or so are an interesting and intuitive introduction to surface geometry. We "experience" the geometry of a sphere by stretching rubber bands on tennis balls or rolling them across freshly painted floors etc., and make observations like "The water strider is very sensitive to motion and vibration on the water's surface, but it can be approached from above or below without its knowledge. Hungry birds and fish take advantage of this fact." Then we study cylinders and cones. Since they have zero curvature their straight lines can be be understood as straight lines on their covering surfaces (straightness is preserved when a cylinder unfolds like a roll of toilet paper, etc.). Then hyperbolic geometry. The plane of hyperbolic geometry can be built by pasting together thin, semi-circular annuli. It can also be crocheted by the same principle. The pseudosphere (a hyperbolic cylinder) is easier to build: just pile up increasingly pointy cones. It can also be successfully crocheted. The annular plane is essentially a pseudosphere cut lengthwise. Thus it is a hyperbolic piece of paper rather than a hyperbolic cylinder, which should be a good thing. But unfortunately the annular plane cannot be extended very far without scrunching up. This defeats its purpose. It would be very difficult to develop any intuition for Euclidean geometry by working on a piece of paper scrunched up to a ball; a good, clean cylinder would be much more useful, provided of course that its imperfections are kept in mind. So I think we might as well stick to the pseudosphere. Over the next few chapters the "experiencing" approach deteriorates slightly, and necessarily so I think, as we study triangle congruencies and parallelism. The fruits of our efforts are limited. We make a start at the problem of determining the area of hyperbolic triangles, but only later will we be able to complete it using double integrals. And the relation between hyperbolic geometry and the parallel postulate is not done justice: it appears to be incidental rather than a major reason for studying hyperbolic geometry in the first place, and besides it is not very clear since lines to us are stitches in the scrunchy waltz-dress model (the standard plane models for hyperbolic geometry will appear briefly only later). The second half of the book is quite useless. There are isolated chapters on all sorts of completely unconnected topics: isometries and patters, Euclidean circle geometry, 2-manifolds, geometric solution of quadratic and cubic equations, polyhedra, etc. There is not much "experiencing" going on, although we do experience that life is easy for our authors and hard for us: they introduce something new, draw a few pictures, and as soon as there is any work to be done they claim it as a problem for the reader and move on to something else.
6 of 8 people found the following review helpful:
3.0 out of 5 stars
interesting approach,
This review is from: Experiencing Geometry (3rd Edition) (Paperback)
This book takes an interesting approach to exploring geometry with a lot of drawings and instructions on how to create physical models. however a lot of the knowledge is placed on the user to explore the subject and think. a lot of theorems and lemmas are not given or proven in the book. i think this book is great along side a course in undergraduate geometry, but lacks examples, excersises, and solutions, too much so to be used for self study. so overall a great textbook but not a great book to read during your free time. for that reason i only give it 3 stars.
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