April 1, 1999
OLD CHALLENGE. It is well known that the circle provides the least-perimeter way to enclose given area in the plane. What is the least-perimeter way to enclose given area in the surface of the unit cube?
ANSWER. Al Zimmermann correctly finds that the best way to enclose small area is not a small circle on one of the sides of the cube, but better a "circle" around one of the corners, as shown in the first figure. This "circle" is actually composed of three quarter-circles. For the complimentary large area on the outside, this "circle" is also best. To enclose half (or close to half) the area of the surface of the cube, Zimmermann rightly uses a square as in the fourth figure. For intermediate areas, he proposes a rectangle about one of the edges, but it turns out that you can do better than that, using circular arcs enclosing two or three of the corners, as in the second and third figures.
Figures 1-4. The least-perimeter way to enclose various areas in the surface of the unit cube. Andrei Gnepp, Ting Ng, and Cara Yoder, SMALL Geometry Group 1998, Williams College.
The complete answer was discovered and, what is much more, proved correct just this past summer by a group of undergraduate students working with me at Williams College. The students, Andrei Gnepp, Ting Ng, and Cara Yoder, are currently preparing their paper for publication. Gnepp and Ng were just awarded graduate school fellowships by the National Science Foundation. The "SMALL" undergraduate research project at Williams is one of many Research Experiences for Undergraduates sponsored by the National Science Foundation.
Until this century, much of mathematics had considered only smooth surfaces, without "singularities" like the corners of the cube, which make the problem harder and more interesting. John Snygg submitted a proof for Figure 1 which involved cutting one edge and then unfolding the corner of the cube in the plane.
The analogous problem on the surface of a circular cylinder (including top and bottom) is still unsolved. If you have some ideas, talk to me or the students: Andrei Gnepp firstname.lastname@example.org>, Ting Fai Ng email@example.com>, Cara.M.Yoder@williams.edu.
NEW CHALLENGE. Ruth Leitschuck asks for years in the 1900s with the same calendar as the year 2000. She explains, "Just thought this year might be programmed into the old VCR on January 1st for correct day and date." (Also, what about years before 1900?)
Send answers, comments, and new questions by email to:
Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.
Copyright 1999, Frank Morgan.