David Richeson

POSITIVELY EXPANSIVE HOMEOMORPHISMS OF COMPACT SPACES

We give a new and elementary proof showing that a homeomorphism f:X→X of a compact metric space is positively expansive if and only if X is finite.

The Flaw in Euler's Proof of His Polyhedral Formula

(2007). The Flaw in Eulers Proof of His Polyhedral Formula. The American Mathematical Monthly: Vol. 114, No. 4, pp. 286-296.

SYMBOLIC DYNAMICS FOR NONHYPERBOLIC SYSTEMS

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.

Centers of the United States.

David Richeson (richesod @dickinson.edu; Dickinson College, Carlisle, PA 17013) received his B.S. from Hamilton College and his Ph.D. from Northwestern University. His current research interests include dynamical systems, topology, and the history and applications of the Euler characteristic. He enjoys spending his free time with his wife and young son and on the air at WDCV-FM, the Dickinson College radio station.

A Trisectrix From a Carpenter's Square

Summary In 1928, Henry Scudder described how to use a carpenters square to trisect an angle. We use the ideas behind Scudders technique to define a trisectrix—a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.

Circular Reasoning: Who First Proved That C Divided by d Is a Constant?

Summary We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant. He stated neither result explicitly (in surviving material), but both are implied by his work. His proof required the addition of two axioms beyond those in Euclids Elements.