The Deltoid is a Kakeya Set

​
move
8.1366
labels
A circle
C
1
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
C
1
traces out the green curve, called a deltoid (or tricuspoid). Let the tangent to the deltoid at
T
meet the deltoid again at
A
and
B
. Then the midpoint
M
of
AB
lies on the circle of radius 1 with center at origin. The length of
AB
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°.

Details

The Kakeya needle problem asks whether there is a minimum area for a region
D
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.
[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.

External Links

Rolling a Coin inside a Circle
Rolling a Coin around a Coin

Permanent Citation

Izidor Hafner
​
​"The Deltoid is a Kakeya Set"​
​http://demonstrations.wolfram.com/TheDeltoidIsAKakeyaSet/​
​Wolfram Demonstrations Project​
​Published: June 14, 2016