*New Horizons in Geometry* represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors—Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order.

The hundreds of full color illustrations by Mamikon are visually enticing and provide great motivation to read further and savor the wonderful results. Lengths, areas, and volumes of curves, surfaces, and solids are explored from a visually captivating perspective. It is an understatement to say that Apostol and Mamikon have breathed new life into geometry.

Preface

Introduction

Foreword

1. Mamikon’s Sweeping Tangent Theorem

2. Cycloids and Trochoids

3. Cyclogons and Trochogons

4. Circumgons and Circumsolids

5. The Method of Punctured Containers

6. Unwrapping Curves from Cylinders and Cones

7. New Descriptions of Conics via Twisted Cylinders, Focal Disks, and Directors

8. Ellipse to Hyperbola: “With This String I Thee Wed”

9. Trammels

10. Isoperimetric and Isoparametric Problems

11. Arclength and Tanvolutes

12. Centroids

13. New Balancing Principles with Applications

14. Sums of Squares

15. Appendix

Bibliography

Index

About the Authors

For the average lay person the word roulette means a gambling game, or perhaps a small toothed wheel that makes equally spaced perforations like those on sheets of postage stamps. In geometry a *roulette* is the locus of a point attached to a plane curve that rolls along a fixed base curve without slipping. A surprising number of classical curves can be generated as roulettes – the cycloid, cardioid, tractrix, catenary, parabola, and ellipse, to name just a few. If the rolling curve is a circle, the roulette is called a* trochoid* (from the Greek work* **τροχός* for wheel). Both the cycloid and cardioid are examples of trochoids.

**Tom M. Apostol** joined the Caltech faculty in 1950 and is Professor of Mathematics, Emeritus. He is internationally known for his books on Calculus, Analysis, and Analytic Number Theory, (translated into 7 languages), and for creating *Project MATHEMATICS!*, a video series that brings mathematics to life with computer animation, live action, music, and special effects. The videos have won first-place honors at a dozen international festivals, and were translated into Hebrew, Portuguese, French, and Spanish. Apostol has published 102 research papers, has written two chapters for the Digital Library of Mathematical Functions (2010), and is coauthor of three texts for the physics telecourse: *The Mechanical Universe ... and Beyond.* He has received several awards for research and teaching. In 1978 he was a visiting professor at the University of Patras, Greece, and in 2001 was elected a Corresponding Member of the Academy of Athens, where he delivered his inaugural lecture in Greek. In 2012 he was selected to be a Fellow of the American Mathematical Society.

**Mamikon A. Mnatsakanian** was Professor of Astrophysics at Yerevan State University and Director of the Mathematical Modeling Center of Physical Processes, Armenian Academy of Sciences. As an undergraduate he invented ‘Visual Calculus’, more fully described in this book. As an astrophysicist he developed a generalized theory of relativity with variable gravitational constant that resolves observational controversies in Cosmology. He also developed new methods that he applied to radiation transfer theory and to stellar statistics and dynamics.

After the 1988 devastating earthquake in Armenia he began seismic safety investigations that brought him to California. After the Soviet Union collapsed, he stayed in the USA where he created assessment problems for the California State Department of Education and UC Davis, and participated in various educational programs, in the process of which he created hundreds of mathematics educational games and puzzles. This work eventually led him to Caltech and *Project MATHEMATICS!*, where he began fruitful collaboration with Tom Apostol. He is the author of 100 scientific papers, 30 of them on mathematics co-authored with Apostol. Website: www.mamikon.com

Review Pending