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Physics > Classical Physics
Title: Symmetry criteria for the equality of interior and exterior shape factors
(Submitted on 27 Mar 2024 (v1), last revised 7 Apr 2024 (this version, v2))
Abstract: Lienhard (2019) reported that the shape factor of the interior of a simply-connected region ($\Omega$) is equal to that of its exterior ($\mathbb{R}^2\backslash\Omega$) under the same boundary conditions. In that study, numerical examples supported the claim in particular cases; for example, it was shown that for certain boundary conditions on circles and squares, the conjecture holds. In the present paper, we show that the conjecture is not generally true, unless some additional condition is met. We proceed by elucidating why the conjecture does in fact hold in all of the examples analysed by Lienhard. We thus deduce a simple criterion which, when satisfied, ensures the equality of interior and exterior shape factors in general. Our criterion notably relies on a beautiful and little-known symmetry method due to Hersch (1982) which we introduce in a tutorial manner.
Submission history
From: Kyle McKee [view email][v1] Wed, 27 Mar 2024 21:58:19 GMT (4662kb,D)
[v2] Sun, 7 Apr 2024 17:42:00 GMT (4662kb,D)
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