Mark Bollman reviews New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian as part of MAA Reviews.
Anyone who has read Underwood Dudley’s Mathematical Cranks could be forgiven a bit of skepticism in the face of a book that promises to offer elementary geometric methods for solving classical calculus problems. New Horizons in Geometry is about as far from crank mathematics as possible. The book begins with “Mamikon’s sweeping tangent theorem”, a result first conceived in 1959, and proceeds to derive numerous formulas for arclength, area, and volume that might ordinarily be deduced with calculus. Along the way, readers will be introduced or reintroduced to such figures as the cyclogon, the autogon, and the bifocal disk. By the end, recursive formulas for volume in n-dimensional space are easily handled, with no integrals required.Read the full review here.
This volume serves as the “collected works” of a fascinating mathematical collaboration between the two authors. This is mathematics of the highest caliber; but what makes this book even more impressive is the attention paid to high-quality full-color graphics that ably illustrate the problems under consideration.
Purchase a copy today at the MAA Store.
Mark Bollman (firstname.lastname@example.org) is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.
Friday, May 24, 2013
Friday, May 17, 2013
Last week, Charles Hadlock presented a talk entitled, “Sustainability or Collapse? An Exploration of Key Dynamics That May Determine Our Future,” as part of the MAA Distinguished Lecture Series. While at the MAA, we had a chance to interview Hadlock on his latest book, Six Sources of Collapse.
Check out the interview below.
Check out the interview below.
Interested? Purchase the book at the MAA Store
or the ebook at the MAA eBooks Store.
Friday, May 10, 2013
|Martin Gardner in the Twenty-First Century edited by Michael Henle and Brian Hopkins as an MAA review.
Martin Gardner is probably as close as any author covered by MAA Reviews could come to needing no introduction. So rather than spend my own energy trying to write one, let me quote from a few other reviews of his works that have been written on this site over the years:To read the whole review, click here.
“Martin Gardner is a national treasure, someone whose
contribution to mathematics has been immense.”
“Ask almost any mathematician who grew up during the
1960s or 1970s, and they’ll tell you about the enormous influence Martin Gardner’s Mathematical Games column had on them.”
“[Gardner] has enlightened, educated and delighted readers around the world during many decades as a prolific contributor to many publications”
“It is hard to exaggerate the importance and influence of these books. You must have them! Buy one for yourself, and buy many to give away.”
Darren Glass is an Associate Professor of Mathematics at Gettysburg College, and one of the best mathematical experiences he has had was speaking at a Gathering For Gardner conference a few years ago. He would be happy to ramble about how great it was or any number of other topics, and can be reached at email@example.com.
Friday, May 3, 2013
The MAA has two new Classroom Resource Materials books!
Paradoxes and Sophisms in Calculus
By Sergiy Klymchuk and Susan Staples
Paradoxes and Sophisms in Calculus offers a delightful supplementary resource to enhance the study of single variable calculus. By the word paradox the authors mean a surprising, unexpected, counter-intuitive statement that looks invalid, but in fact is true. The word sophism describes intentionally invalid reasoning that looks formally correct, but in fact contains a subtle mistake or flaw. In other words, a sophism is a false proof of an incorrect statement. A collection of over fifty paradoxes and sophisms showcases the subtleties of this subject and leads students to contemplate the underlying concepts. A number of the examples treat historically significant issues that arose in the development of calculus, while others more naturally challenge readers to understand common misconceptions. Sophisms and paradoxes from the areas of functions, limits, derivatives, integrals, sequences, and series are explored. The book could be useful for high school teachers and university faculty as a teaching resource; high school and college students as a learning resource; and a professional development resource for calculus instructors.
Exploring Advanced Euclidean Geometry
By Gerard Venema
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.
Both books are currently available in the MAA eBooks Store.
Print copies will be available in the MAA Store shortly.
Friday, April 26, 2013
by Arthur Benjamin and Jennifer Quinn
is now available in the MAA eBooks Store!
Purchase your copy today for only $22.50.
Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments.
The book explores more than 200 identities throughout the text and exercises, frequently emphasizing numbers not often thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians.
Read the MAA Review written by Darren Glass, an associate professor at Gettysburg College, below.
"Several years ago I attended a conference at which Arthur Benjamin, one of the authors of the book under review, gave a talk about Fibonacci Numbers. In particular, he gave the following interpretation. Let fn count the number of ways to tile an n-by-1 board with 1-by-1 square tiles and 2-by-1 domino tiles. One can show that fn = Fn+1, where Fn is the standard nth Fibonacci number defined by F0 = 0, F1 = 1, and the recursion relation Fn = Fn-1 + Fn-2. He proceeded to show how this definition could be used to give a combinatorial proof of many of the Fibonacci Number identities that we are familiar with, such as Fm+n=Fm+1Fn + FmFn-1..."
Continue reading here.