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Kakeya Needle Problem

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The Kakeya needle problem asks for the plane figure of least area in which a line segment of length 1 can be freely rotated (where translation of the line segment is also allowed). Surprisingly, there is no minimum area (Besicovitch 1928). Another iterative construction which tends to as small an area as desired is called a Perron tree (Falconer 1990, Wells 1991).

When the figure is restricted to be a convex set, the smallest region is an equilateral triangle of unit height. Wells (1991) states that Kakeya discovered this, while Falconer (1990) attributes it to Pál.

If the requirement of a convex set is replaced by the weaker assumption of being simply connected, then the area can still be arbitrarily small, but if the set is required to be star convex, then pi/108 is a known lower bound (Cunningham 1971).

Cunningham and Schoenberg (1965) construct star convex domains in which a unit line segment can be rotated, with areas approaching

1/(24)(5-2sqrt(2))pi=0.284258...

(OEIS A093823; Le Lionnais 1983). This gives an upper bound for the infimum of the areas of star convex domains with this property.

See also

Curve of Constant Width, Kakeya Set, Lebesgue Minimal Problem, Perron Tree, Reuleaux Polygon, Reuleaux Triangle

Portions of this entry contributed by Laura Wisewell

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 99-101, 1987.Besicovitch, A. S. "On Kakeya's Problem and a Similar One." Math. Z. 27, 312-320, 1928.Besicovitch, A. S. "The Kakeya Problem." Amer. Math. Monthly 70, 697-706, 1963.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 81-82, 2003.Cunningham, F. Jr. "The Kakeya Problem for Simply Connected and for Star-Shaped Sets." Amer. Math. Monthly 78, 114-129, 1971. https://doi.org/10.1080/00029890.1971.11992708.Cunningham, F. Jr. and Schoenberg, I. J. "On the Kakeya Constant." Canad. J. Math. 17, 946-956, 1965. https://doi.org/10.4153/CJM-1965-090-x.Falconer, K. J. The Geometry of Fractal Sets, 1st pbk. ed., with corrections. Cambridge, England: Cambridge University Press, 1990.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 24, 1983.Littlewood, J. E. Littlewood's Miscellany. Cambridge, England: Cambridge University Press, p. 38, 1986.Ogilvy, C. S. A Calculus Notebook. Boston, MA: Prindle, Weber, & Schmidt, 1968.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 147-153, 1990.Pál, J. "Ein Minimumproblem für Ovale." Math. Ann. 88, 311-319, 1921.Sloane, N. J. A. Sequence A093823 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 151-152, 1999.Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 50-52, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 128-129, 1991.

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Kakeya Needle Problem

Cite this as:

Weisstein, Eric W., with contributions by Laura Wisewell. "Kakeya Needle Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KakeyaNeedleProblem.html

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