N.B. A detailed on-line essay by S. Finch was the starting point for this entry.
Given a Unit Disk, find the smallest Radius required for equal disks to completely cover the
Unit Disk. For a symmetrical arrangement with (the Five Disks Problem),
, where is the Golden Ratio. However, the radius can be reduced in the
general disk covering problem where symmetry is not required. The first few such values are
(1) | |
(2) | |
(3) | |
(4) |
(5) |
(6) |
(7) |
(8) | |||
(9) | |||
(10) | |||
(11) | |||
(12) | |||
(13) |
Letting be the smallest number of Disks of Radius needed to cover a
disk , the limit of the ratio of the Area of to the Area of the disks is given by
(14) |
See also Five Disks Problem
References
Ball, W. W. R. and Coxeter, H. S. M. ``The Five-Disc Problem.''
In Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 97-99, 1987.
Bezdek, K. ``Über einige Kreisüberdeckungen.'' Beiträge Algebra Geom. 14, 7-13, 1983.
Bezdek, K. ``Über einige optimale Konfigurationen von Kreisen.'' Ann. Univ. Sci. Budapest
Eötvös Sect. Math. 27, 141-151, 1984.
Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/circle/circle.html
Kershner, R. ``The Number of Circles Covering a Set.'' Amer. J. Math. 61, 665-671, 1939.
Neville, E. H. ``On the Solution of Numerical Functional Equations, Illustrated by an Account of a Popular Puzzle and of its Solution.''
Proc. London Math. Soc. 14, 308-326, 1915.
Verblunsky, S. ``On the Least Number of Unit Circles which Can Cover a Square.'' J. London Math. Soc. 24, 164-170, 1949.
Zahn, C. T. ``Black Box Maximization of Circular Coverage.'' J. Res. Nat. Bur. Stand. B 66, 181-216, 1962.
© 1996-9 Eric W. Weisstein