Joel David Hamkins shared a beautiful animation by Matthew Henderson on twitter which shows a method of drawing infinitely many touching circles. I can't embed the whole video here, so I have picked several frames below.
It shows that if one fixes a vertex of a rectangle of constant area $1$, and moves around one adjacent vertex by drawing circles, then the trace of the other adjacent vertex to the fixed one will be circles as well; this gives a method of drawing infinitely many touching circles. See the correspondence between two sets of circles below. One may play around with Python to reproduce the video.
A latter tweet by Matt seems to relate to the phenomenon above: "the construction with the rectangle of constant area is circle inversion in disguise". This seems to be related to the topics of circle packing and inversive geometry. But I know nothing but the two links to the Wikipedia articles.
Question: What is the underlying mathematics of the animation? In particular, what is the "magic" in the constant area requirement for the rectangles?