Electrical response matrix of a regular 2𝑛-gon

Blair-Stahn, Nathaniel; Wilson, David

doi:10.1090/S0002-9939-09-09734-2

Electrical response matrix of a regular $2n$-gon
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by Nathaniel D. Blair-Stahn and David B. Wilson PDF

Proc. Amer. Math. Soc. 137 (2009), 2015-2025 Request permission

Abstract:

Consider a unit-resistive plate in the shape of a regular polygon with $2n$ sides, in which even-numbered sides are wired to electrodes and odd-numbered sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular $2n$-gon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests.

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