I’ve often wondered about packing circles and have always assumed that it would get into messy numbers very quickly. Your article is a charming revelation.
It says that if a, b, and c are integers, d will be one, too. I think this is true only if a, b, and c bear some relationship to each other. Try 1, 2, and 3. Yes. Try 2, 2, 5. No dice. On the other hand, tests indicate that if d does turn out to be an integer, you’re off and running. Apparently e, f, g, and so forth will also be integers.
Old Lyme, Conn. You’re correct. If the curvatures of the four initial circles are integers, the curvature of every smaller circle is also an integer.—I. Peterson “Circle game” includes in its closing paragraph the observation by mathematician Jeffrey C. Lagarias that the research on packing circles within a circle has “no foreseeable application in mind.” One practical application that occurred to us a few years ago was tightly packing a metal tube with small wires. Our U.S. Patent 5,962,790 describes our interest in filling the tube with the maximum number of wires of one or more given sizes. The packed tube acts as an ultrasonic waveguide in industrial flow meters. Larry Lynnworth and Yi Liu