Adam's proof is very slick! Does the nth Brillouin zone converge in shape to a circle as n goes to infinity? I suspect one can show without too much work that the nth Brillouin zone encircles a disk of radius r_n and lies in a disk of radius R_n, with R_n - r_n bounded. Is it possible that R_n - r_n goes to zero? Jim Propp On Thu, Mar 8, 2018 at 2:05 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Let Z(n, m; i, j) be the region such that i is the nth closest point and j is the mth closest point. Then I claim there is an isometry of the lattice which interchanges i and j, so we have:
Area[Z(n, m; i, j)] == Area[Z(n, m; j, i)]
Summing over all lattice points i, we get:
Area[mth Brouillin zone around j] == Area[nth Brouillin zone around j]
Best wishes,
Adam P. Goucher
Sent: Thursday, March 08, 2018 at 5:21 PM From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Brillouin zones
I just came across
http://skepticsplay.blogspot.com/2011/10/lot-of-brillouin-zones.html
Does anyone see how to prove that all the zones have the same area?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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