American Mathematical Monthly -June-July 2006

June-July 2006

Ten Problems in Experimental Mathematics
by David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor and Eric W. Weisstein
dhbailey@lbl.gov, jmborwein@cs.dal.ca, vkapoor@math.ubc.ca, eww@wolfram.com
This article presents and solves ten symbolic/numeric challenge problems, in the spirit of the recent SIAM hundred-digit, hundred-dollar challenge problems. Our goal in this article is to provide solutions to these ten problems, and at the same time, to present a concise account of how one combines symbolic and numeric computation, which may be termed "hybrid computation," in the process of mathematical discovery.

A Geometric Interpretation of an Infinite Product for the Lemniscate Constant
by Aaron Levin
adlevin@math.brown.edu
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
apostol@caltech.edu, mamikon@caltech.edu
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.

Notes

On Goldbach’s Conjecture for Integer Polynomials
by Filip Saidak
fillip@math.missouri.edu

Moment Sums Associated with Binary Linear Forms
by Peter Shiu
P.Shiu@lboro.ac.uk

A Fundamental Theorem of Calculus for Lebesgue Integration
by J. J. Koliha
j.koliha@ms.unimelb.edu.au

The Broken Spaghetti Noodle
by Carlos D’Andrea and Emiliano Gómez
cdandrea@math.berkeley.edu, emgomez@math.berkeley.edu

Contractions of the Real Line
by Alan F. Beardon
afb@dpmms.cam.ac.uk

Evolution ofÂ…
The Constructive Mathematics of A. A. Markov
by Boris A. Kushner
boris@pitt.edu

Problems and Solutions Reviews> Musings of the Masters
Edited by Raymond G. Ayoub
Reviewed by Steve Kennedy
skennedy@carleton.edu