(adsbygoogle = window.adsbygoogle || []).push({}); proof that the "n" brillouin zones are of equal areas?

i'm trying to find a way to prove that the brillouin zones are indeed of equal areas.

if i draw, for examle, the first 3 or 4 brillouin zones of a cubic 2-dimensional lattice, then it is relatively easy to show geometrically how the parts of each of the higher order zones can be combined and put together in order to form the 1st zone. but how can i prove that this is correct for all additional zones as well, up to the nth zone?

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# Proof that the n brillouin zones are of equal areas?

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