This book is a compendium of joint work produced by the authors during the period 1998-2012, most of it published in the American Mathematical Monthly, Math Horizons, Mathematics Magazine, and The Mathematical Gazette. The published papers have been edited, augmented, and rearranged into 15 chapters. Each chapter is preceded by a sample of problems that can be solved by the methods developed in that chapter. Each opening page contains a brief abstract of the chapter's contents.

Chapter 1, entitled “Mamikon's Sweeping-Tangent Theorem,” was the starting point of this collaboration. It describes an innovative and visual approach for solving many standard calculus problems by a geometric method that makes little or no use of formulas. The method was conceived in 1959 by my co-author (who prefers to be called Mamikon), when he was an undergraduate student at Yerevan University in Armenia. When young Mamikon showed his method to Soviet mathematicians they dismissed it out of hand and said “It can't be right. You can't solve calculus problems that easily.”

Mamikon went on to get a Ph.D. in physics, was appointed a professor of astrophysics at the University of Yerevan, and became an international expert in radiative transfer theory, all the while continuing to develop his powerful geometric methods. Mamikon eventually published a paper outlining them in 1981, but it seems to have escaped notice, probably because it appeared in Russian in an Armenian journal with limited circulation. (Reference [59] in the Bibliography.)

This passage perfectly captures the spirit of the book New Horizons in Geometry, by Tom Apóstol and Mamikon Mnatsakanian. In a remarkable display of mathematical versatility and imagination, the authors present us with a wealth of geometrical gems. These beautiful and often surprising results deal with a multitude of geometric forms, their interrelationships, and in many cases, their connection with patterns underlying the laws of nature. Lengths, areas and volumes, of curves, surfaces, and solids, are explored from a visually captivating perspective. The preponderance of results discussed by the authors are new, and when not new, are presented with unusual insights and unexpected generalizations.

The exposition is uniformly lucid and delightful, with a heavy emphasis on dynamic visual thinking. Some derivations that might ordinarily be carried out using methods of calculus are accomplished with ingenious visual arguments. For instance, an amazing variety of results are derived visually for cycloids, epicycloids and hypocycloids, general roulettes, pursuit curves, traces and envelopes of trammels, conic sections, and so forth. Classical constructions and characterizations of the conics are generalized, with focal points replaced by focal disks, and Dandelin spheres inside cones replaced by tangent spheres inside twisted cylinders.

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.