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The Helen of Geometry

by John Martin (Santa Rosa Junior College)

Award: George Pólya

Year of Award: 2011

Publication Information: College Mathematics Journal, vol. 41, no. 1, January 2010, pp. 17-27.


Just as the ancient beauty Helen of Troy is said to have launched a thousand ships, this article describes how a beautiful, classical curve launched, if not a thousand theorems, then certainly scores of interesting results and just as many interesting disputes among many of the leading mathematicians of the 17th century.

As is well known, the cycloid is the curve traced by a point on the circumference of a circle as it rolls along a straight line without slipping. The author engagingly describes how much that is understood about the cycloid now was determined more than three hundred years ago: construction of its tangents, its quadrature, its arclength, that its involute is another (congruent) cycloid, and that it solves both the tautochrone and brachistochrone problems. Moreover, what is remarkable is how much of this was discovered without any calculus or even analytic geometry, only synthetic techniques and very rudimentary notions of limits.

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About the Authors (From the Math Fest 2011 Prizes and Awards Booklet)

John Martin was born and raised in Riverside, California where he learned to dislike smog and congested freeways. He did his undergraduate work at Humboldt State University where he found joy in learning about the history of mathematics. He began his teaching career in Glendale California and later earned his M.A. in mathematics from the University of Southern California. In the fall of 1981 he was hired to teach full time in the mathematics department at Santa Rosa Junior College, where he remains.

Subject classification(s): Geometry and Topology | Plane Geometry
Publication Date: 
Tuesday, August 23, 2011