Steiner's Porism in finite Miquelian Möbius planes

2015/07/06 by Norbert Hungerbühler, Hungerbühler, Norbert, Katharina Kusejko +1
Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1507.01377

19 pages, 6 figures

arxiv created 2015/07/06 · arxiv updated 2015/07/07

Abstract

We investigate Steiner's Porism in finite Miquelian Möbius planes constructed over the pair of finite fields GF(p^m) and GF(p^2m), 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(p^m). 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.

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