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Pál Joints

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This Demonstration shows a method called Pál joints that lets you move a segment to a parallel segment through an area approaching 0. If the height is , then the area swept out by the segment is . Therefore, given two parallel lines, a unit line segment can be moved continuously from one to the other sweeping out a set of arbitrarily small measure.

Contributed by: Izidor Hafner and Borut Jurcic Zlobec (June 2016)
Open content licensed under CC BY-NC-SA

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This construction allows the continuous motion of a unit segment back to itself but rotated by 180°, again in a set of arbitrarily small measure [1].

References

[1] Wikipedia. "Kakeya Set. (Jun 9, 2016) en.wikipedia.org/wiki/Kakeya_set.

[2] K. J. Falconer, The Geometry of Fractal Sets, Cambridge, England: Cambridge University Press, 1990.

[3] D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin, pp. 128–129, 1991.

[4] C. Fefferman. Kakeya Needle Problem [Video]. (Jun 9, 2016) www.youtube.com/watch?v=j-dce6QmVAQ.

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