In the post “Spirals” I have shown how to transform periodic Euclidean tilings into logarithmic spirals. A typical result looks like that:
The spiral has a center at the origin and goes out forever. Actually, it spirals not only around the origin, but around infinity too. We can see this better if we look at the Riemann sphere. It is a stereographic projection of the entire plane on a sphere placed above the origin. The image of the origin lies at the south pole of the sphere and points infinitely far away all go to the north pole. On this sphere, the spiral goes out from the south pole and grows until it comes to the equator. Then it crosses to the upper hemisphere and shrinks, going towards the north pole. Thus the spiral has two centers, one at the south pole and one at the north pole.
To see both centers of the sphere on the plane we can use the Cayley transform. With complex coordinates it is simply f(z)=(z-i)/(z+i). It maps the point (0,1) to (0,0) and (0,-1) to infinity. Applying first this mapping to the points of the plane and then doing the transform to get a spiral we have images of a spiral that has two centers, one at (0,1) and the other at (0,-1). The spiral shown above now looks like that:
Close to the two centers we get copies of the original spiral. The two spirals are linked together by a distorted copy of the basic square tiling.
You can find more details in the book “Indra’s Pearls”, written by David Mumford, Caroline Series and David Wright.