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# Disk Covering Problem

Given a unit disk, find the smallest radius required for equal disks to completely cover the unit disk. The first few such values are

 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Here, values for , 8, 9, 10 are approximate values obtained using computer experimentation by Zahn (1962).

For a symmetrical arrangement with (known as the five disks problem), , where is the golden ratio. However, rather surprisingly, the radius can be slightly reduced in the general disk covering problem where symmetry is not required; this configuration is illustrated above (Friedman). Neville (1915) showed that the value is equal to , where and are solutions to

 (11) (12) (13) (14)

These solutions can be found exactly as

 (15) (16)

where

 (17) (18)

are the smallest positive roots of the given polynomials, with denoting the th root of the polynomial in the ordering of the Wolfram Language. This gives (OEIS A133077) exactly as

 (19)

where the root is the smallest positive one of the above polynomial.

is also given by , where is the largest real root of

 (20)

maximized over all , subject to the constraints

 (21)
 (22)

and with

 (23) (24) (25) (26) (27) (28) (29)

(Bezdek 1983, 1984).

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

 (30)

(OEIS A086089; Kershner 1939, Verblunsky 1949).

Circle Covering, Five Disks Problem

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## 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 Eőtvős Sect. Math. 27, 141-151, 1984.Finch, S. R. "Circular Coverage Constants." §2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 484-489, 2003.Friedman, E. "Circles Covering Circles." http://www.stetson.edu/~efriedma/circovcir/.Gardner, M. The Second Scientific American Book of Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 142-144, 1961.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.Sloane, N. J. A. Sequences A086089 and A133077 in "The On-Line Encyclopedia of Integer Sequences."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.

## Referenced on Wolfram|Alpha

Disk Covering Problem

## Cite this as:

Weisstein, Eric W. "Disk Covering Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiskCoveringProblem.html