New Horizons in Geometry Dolciani Mathematical Expositions Series

The collaborative work of Tom Apostol and Mamikon Mnatsakanian has been lauded for its clarity and originality. In this volume the authors present an impressive collection of geometric results that reveal surprising connections between lengths, areas and volumes in various shapes, and allow one to compute difficult integrals, all using intuitive visual calculations. One noteworthy idea that the reader will encounter is Mamikon's Sweeping Tangent Theorem from which the authors obtain a visual derivation of the property that the length of an arc of a catenary is proportional to the area under the arc. This is one of many 'proofs without words' contained within. In addition, a variety of results are derived visually for cycloids, conic sections, and many more geometric objects. As befits a book that emphasises visual thinking, the text is beautifully illustrated. This is a book that will inspire students and enrich any geometry or calculus course.

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. Sums of squares; 14. Appendix.

Tom Apostol is an Emeritus Professor of Mathematics at the California Institute of Technology.
Mamikon Mnatsakanian is Project Associate at 'Project Mathematics!' at the California Institute of Technology.