Equal Zones and Tautochrones

Abstract

We describe surfaces of revolution with the parallel equal zones property, where equally-spaced planes parallel to the axis of rotation cut the surface into zones with equal surface areas. The sphere is one such surface, but there are many others, some symmetric, most not. They all have a connection with round-trip tautochrones.

• MSC: Primary 51N20
• Secondary 26A06

Acknowledgments

I would like to thank the referees for their helpful suggestions and my wife Mei Chun Chang for her constant support.

Additional information

Notes on contributors

Fred Kuczmarski

FRED KUCZMARSKI received his Ph.D. in mathematics from the University of Washington. He enjoys baking bread and listening to nature in Washington’s Cascades.