1. INTRODUCTION. A spectacular landmark in the history of mathematics was the discovery by Archimedes (287-212 B.C.) that the volume of a solid sphere is two-thirds the volume of the smallest cylinder that surrounds it, and that the surface area of the sphere is also two-thirds the total surface area of the same cylinder. Archime-des was so excited by this… (More)
1. INTRODUCTION. Given a finite set of fixed points in 3-space, what is the lo-cus of a point moving in such a way that the sum of the squares of its distances from the fixed points is constant? The answer is both elegant and surprising: the locus is a sphere whose center is at the centroid of the fixed points (if we allow the empty set and a single point… (More)
1. INTRODUCTION. Two incongruent solids with remarkable properties are shown in Figure 1. One is a slice of a solid hemispherical shell with inner radius r and outer radius R cut by a plane parallel to the equator and at distance h < r from the equator. The other is a cylindrical shell with the same radii and altitude h. The surface of each solid consists… (More)
1. INTRODUCTION. The centroid of the boundary of an arbitrary triangle need not be at the same point as the centroid of its interior. But we have discovered that the two centroids are always collinear with the center of the inscribed circle, at distances in the ratio 3 : 2 from the center. We thought this charming fact must surely be known, but could find… (More)
1. MAMIKON'S THEOREM. Areas and volumes of many classical regions can be determined by an intuitive geometric method that does not require integral calculus or even equations for the boundaries. The method and its many applications was first announced by Mamikon A. Mnatsakanian in , but this paper seems to have escaped notice, probably because it… (More)
1. INTRODUCTION. Although it is well known that every tetrahedron circum-scribes a sphere, the following two simple consequences apparently have not been previously recorded. First, any plane through the center of the inscribed sphere divides the tetrahedron into two smaller solids whose surface areas are equal if and only if their volumes are equal.… (More)
1. INTRODUCTION. Conics have been investigated since ancient times as sections of a circular cone. Surprising descriptions of these curves are revealed by investigating them as sections of a hyperboloid of revolution, referred to here as a twisted cylinder. We generalize the classical focus-directrix property of conics by what we call the focal… (More)
1. INTRODUCTION. In his delightful book Mathematical Snapshots, Steinhaus  describes the simple, engaging construction illustrated in Figure 1. Wrap a piece of paper around a cylindrical candle, and cut it obliquely with a knife. The cross section is an ellipse, which becomes a sinusoidal curve when unwrapped.