Re: [math-fun] A symmetrical rotation of Euclidean space
----- I don't exactly know what you're asking for, but maybe you are trying to make a jump off Berlekamp's complex transformations by restricting to reals? I think the answer to your questions just falls out of basic representation theory of a finite cyclic group. ----- I thought it was clear I was asking for the angles theta_j. ----- In some choice of coordinates, your T commutes with a cyclic permutation of basis vectors, call that transformation T'. T' is a regular representation of the generator of a cyclic group, so we know its eigenbasis, which is shared with T. ----- The transformation T is *defined* as a cyclic permutation of (orthonormal) basis vectors. I don't know anything about representation theory, and didn't knowingly use it here. —Dan
Okay, that makes things easier than T an arbitrary rotation around the same axis (it is possible to generalize though). For three dimensional space I guess the angle in the one remaining plane would be 2Pi/3. For five dimensional space, the angle in the first plane E_1 is 2Pi/5, and in general for E_m, m<=k, theta_{m,k} = 2*pi*m/(2*k+1). Is that the answer you got? The reason I would suggest representation is that we can say that Frobenius (1849-1917) probably knew something like this. Do you think the proof of this plane theorem was written any earlier? --Brad Also here's some helpful Mma code: init[n_, m_] := {Cos[2 Pi m/n #] & /@ Range[1, n], Sin[2 Pi m/n #] & /@ Range[1, n]}; fin[n_, m_] := RotateRight /@ {Cos[2 Pi m/n #] & /@ Range[1, n], Sin[2 Pi m/n #] & /@ Range[1, n]}; ang[n_, m_, k_] := ArcCos[init[n, m][[k]].fin[n, m][[k]]/ init[n, m][[k]].init[n, m][[k]]]; N@ang[7, 3, #] & /@ {1, 2} N[2 Pi (3/7)] On Mon, Sep 21, 2020 at 10:07 AM Dan Asimov <dasimov@earthlink.net> wrote:
The transformation T is *defined* as a cyclic permutation of (orthonormal) basis vectors.
I don't know anything about representation theory, and didn't knowingly use it here.
—Dan
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Dan Asimov