A hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle.

If the radius of a large circle is \(R\), and the radius of a small circle is \(r\), the following properties are shown depending on the value of \(k = \frac{R}{r}\).

- If \(k\) is an integer number, the curve becomes a closed curve and has \(k\) curves.
- If \(k\) is a rational number, it has \(p\) curves if it can be simplified to \(k = \frac{p}{q}\).
- If \(k\) is an irrational number, the curve is not closed, filling all the space between the large circle and the circle with radius \(R-2r\).

We made a tool to draw a hypocycloid with a 3D printer. Please refer to the link below.