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Pál Joints

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.

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DETAILS

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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