A Geometric Interpretation of an Infinite Product for the Lemniscate Constant
by Aaron Levin
A new infinite product for the lemniscate constant bearing a striking similarity to Viète's product for π was recently discovered by the author. In this article we strengthen the similarity between the two products by showing that they admit analogous geometric interpretations. Viète's product for π can be interpreted in terms of areas of regular 2n-gons inscribed in the unit circle, and with the proper definitions, the lemniscate product can be similarly interpreted, replacing the unit circle by the plane curve defined by x4 + y4 = 1. In constructing the analogy we will explore how to define “sine” and “cosine” functions for the curve defined by x4 + y4 = 1.
Solids Circumscribing Spheres
by Tom M. Apostol and Mamikon A. Mnatsakanian
Every tetrahedron circumscribes a sphere (called its insphere, with corresponding inradius and incenter). Polyhedra with more than four faces may or may not circumscribe spheres. Those that do are examples of what we call circumsolids, each with inradius and incenter. They include tetrahedra, regular polyhedra, some irregular polyhedra, some nonconvex polyhedra (such as stellated polyhedra), and many other solids whose faces can be cylindrical, conical, or spherical, as well as planar. All circumsolids share a common property: the ratio of volume to outer surface area is one-third the inradius (well known for a sphere, but not for a circular cone or a tetrahedron). Also, the volume centroid of any circumsolid and the centroid of its outer boundary surface are collinear with the incenter, at distances in the ratio 3:4 from the incenter.
These extend to 3-space corresponding planar results for circumgons—figures circumscribing circles—discovered in an earlier paper. More extensive applications are possible in space, as shown by examples that include star-like circumsolids such as stellated dodecahedra, and intersections of circumsolids.
One application of the volume-surface area ratio shows that any plane through the incenter of a circumsolid divides it into two smaller solids whose surface areas are equal if and only if their volumes are equal. Another yields (without integration) the volume of the solid of intersection of a right circular cone and an orthogonal circular cylinder having the same insphere. A limiting case is the classical Archimedean result on intersecting cylinders.
The paper also treats circumsolid shells—solids lying between two similar circumsolids with the same incenter. They have constant thickness, and the ratio of volume to mixed average surface area is one-third this constant thickness. This implies a far reaching extension to nonplanar surfaces of the classical Egyptian and prismoidal formulas.
On Goldbach’s Conjecture for Integer Polynomials
by Filip Saidak
Moment Sums Associated with Binary Linear Forms
by Peter Shiu
A Fundamental Theorem of Calculus for Lebesgue Integration
by J. J. Koliha
The Broken Spaghetti Noodle
by Carlos D’Andrea and Emiliano Gómez
Contractions of the Real Line
by Alan F. Beardon
The Constructive Mathematics of A. A. Markov
by Boris A. Kushner