Five Disks Problem


Given five equal disks placed symmetrically about a given center, what is the smallest radius r for which the radius of the circular area covered by the five disks is 1? The answer is r=phi-1=1/phi=0.6180339..., where phi is the golden ratio, and the centers c_i of the disks i=1, ..., 5 are located at

 c_i=[1/phicos((2pii)/5); 1/phisin((2pii)/5)].

The golden ratio enters here through its connection with the regular pentagon. If the requirement that the disks be symmetrically placed is dropped (the general disk covering problem), then the radius for n=5 disks can be reduced slightly to 0.609383... (Neville 1915).

See also

Arc, Circle Covering, Disk Covering Problem, Flower of Life, Miquel Five Circles Theorem, Seed of Life

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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.Gardner, M. The Second Scientific American Book of Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 142-144, 1961.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.

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Five Disks Problem

Cite this as:

Weisstein, Eric W. "Five Disks Problem." From MathWorld--A Wolfram Web Resource.

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