Another animated GIF made by Maple, the hypocycloid, where a smaller circle is running inside a bigger circle and comes back to its beginning after a finite number of steps. If the larger circle has radius R and the smaller circle has radius r, then parametric equations for the hypocycloid are x(t) = (R-r)cos(t) + r cos((R-r)t/r), y(t) = (R-r) sin(t) - r sin((R-r)t/r). If k = R/r, the equations can be written as x(t) = r(k-1)cos(t) + r cos((k-1)t), y(t) = r(k-1)sin(t) - r sin((k-1)t).
An animated GIF showing the transformation of a helicoid to a catenoid made by Maple.
I found this animated GIF on wikipedia, showing the special case of the hypocycloid above where R=3 and r=1, so k = 3, making a single triangle inside the big circle before coming back to its beginning. This is also called a deltoid, and I have tried making sculptures of knots where this is the cross section of the knot.
Another cool animated GIF of two tori made by Mathematica.
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