# Understanding a method of drawing infinitely many touching circles

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?

• Do you know en.wikipedia.org/wiki/Apollonian_gasket ? – kimchi lover May 3 '20 at 14:09
• @kimchilover: thanks for that. I'm aware that it is the first link of "see also" in the Circle packing article I mentioned. – Mars May 3 '20 at 14:17
• Choose a coordinate system where the fixed vertex is origin. Identify euclidean plane with complex plane. Circle inversion with respect to unit circle centered at origin corresponds to the map $\displaystyle\;z \mapsto \frac{1}{\bar{z}}$ over comple numbers. Notice the four points $0, z, z + \frac{i}{\bar{z}}, \frac{i}{\bar{z}}$ form a rectangle of area $1$. The point $\frac{i}{\bar{z}}$ is the "other vertex" your see. It is basically the point obtained by a inversion with respect to circle $|z| = 1$ followed by a counterclockwise rotation of $90^\circ$ with respect to origin. – achille hui May 3 '20 at 14:56