Steiner's Porism in finite Miquelian Möbius planes

Abstract

We investigate Steiner's Porism in finite Miquelian Möbius planes constructed over the pair of finite fields GF(pm) and GF(p2m), for p an odd prime and m≥1. Properties of common tangent circles for two given concentric circles are discussed and with that, a finite version of Steiner's Porism for concentric circles is stated and proved. We formulate conditions on the length of a Steiner chain by using the quadratic residue theorem in GF(pm). These results are then generalized to an arbitrary pair of non-intersecting circles by introducing the notion of capacitance, which turns out to be invariant under Möbius transformations. Finally, the results are compared with the situation in the classical Euclidean plane.