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.