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A hypocycloid is the path of a point fixed to the green circle rolling inside the white circle. If the ratio of the two radii is an integer, we get an ordinary star. If the ratio is rational and not an integer, we get a self-intersecting star. If the ratio is irrational, the curve does not close.
Contributed by: Tomas Garza (December 2019)
Open content licensed under CC BY-NC-SA
A hypocycloid  is the curve generated by tracing the path of a fixed point on a circle that rolls inside a larger circle. When the ratio of the radius of the larger cycle to that of the smaller one is an integer (), the curve obtained is an -cusp star. Otherwise, the curve obtained is a multi-spiked star, with spikes.
This Demonstration gives a few examples of this amusing behavior. Move the slider to observe the development of the curve as the inside circle rolls on.
 Wikipedia. "Hypocycloid." (Dec 10, 2019) en.wikipedia.org/wiki/Hypocycloid.