# Kakeya Needle Problem

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The Kakeya problem asks for the smallest convex region in which a unit segment can be moved back to itself but in the opposite direction. The answer is an equilateral triangle of unit height [3].

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Contributed by: Izidor Hafner and Burut Jurcic Zlobec (June 2016)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Given two parallel lines, a Pál joint lets you move a unit line segment continuously from one to the other through a set of arbitrarily small area.

A Kakeya set (or Besicovitch set) is one that can contain a unit segment in any direction.

A Kakeya needle set is a set through which a line segment can be moved continuously back to itself but turned 180°. (Thus a Kakeya needle set is a Kakeya set.)

Using Pál joints, a Kakeya needle set can be created from Perron trees, as indicated in this Demonstration [1, 3 pp. 129–130].

See Besicovitch's talk [5] and expository article [6].

References

[1] Wikipedia. "Kakeya Set." (Jun 20, 2016) en.wikipedia.org/wiki/Kakeya_set.

[2] K. J. Falconer, *The Geometry of Fractal Sets*, 1st ed., Cambridge: Cambridge University Press, 1990.

[3] D. Wells, *The Penguin Dictionary of Curious and Interesting Geometry*, London: Penguin Books, 1991.

[4] *Kakeya's Needle Problem* [Video]. (2015). Retrieved June 20, 2016, from www.youtube.com/watch?v=j-dce6QmVAQ.

[5] *The Kakeya Problem* [Video]. (1962). Retrieved July 11, 2016, from av.cah.utexas.edu/index.php?title=Category:The_Kakeya_Problem.

[6] A. S. Besicovitch, "The Kakeya Problem," *American Mathematical Monthly,* 70, 1963 pp. 697–706.

## Permanent Citation

"Kakeya Needle Problem"

http://demonstrations.wolfram.com/KakeyaNeedleProblem/

Wolfram Demonstrations Project

Published: June 21 2016