The Deltoid is a Kakeya Set

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A circle of radius 1 rolls inside a fixed circle of radius 3 (the fixed circle is shown when "labels" is selected); a point on the circumference of traces out the green curve, called a deltoid (or tricuspoid). Let the tangent to the deltoid at meet the deltoid again at and . Then the midpoint of lies on the circle of radius 1 with center at origin. The length of is 4, so the deltoid is a Kakeya set: a set through which a line segment can be moved back to itself but turned 180°.

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


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The Kakeya needle problem asks whether there is a minimum area for a region in the plane such that a needle of unit length can be turned through 180° [3].

The deltoid is a hypocycloid of three cusps. It was first studied by Euler in 1745. The curve is also called a Steiner curve [4].

References

[1] D. G. Wells, The Penguin Dictionary of Curious and Interesting Geometry, New York: Penguin Books, 1991 p. 52 and p. 129.

[2] E. W. Weisstein. "Deltoid" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/Deltoid.html (Wolfram MathWorld).

[3] Wikipedia. "Kakeya Set." (Jun 13, 2016) en.wikipedia.org/wiki/Kakeya_set.

[4] Wikipedia. "Deltoid Curve." (Jun 13, 2016) en.wikipedia.org/wiki/Deltoid_curve.



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