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I recently came across the wonderful proof of quadratic reciprocity given by Zolotarev in the $1800$'s and have seen a wonderful visualisation of this given by a "card trick" (see Jerry Shurman's nice explanation at http://people.reed.edu/~jerry/361/lectures/qrz.pdf).

I was wondering if there are such visual "tricks" for explaining other reciprocity laws, for example cubic reciprocity?

Of course I realise that it can't be anything to do with the signs of permutations anymore since the cubic symbols give cube roots of unity as outputs. However maybe there is some other representation/character of a symmetric group that does work?

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