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• ## field harmonics in triaxial coordintes

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• Back a while I found that if I put spherical coordinates in a SO(3) like matrix and squared it I got a new projective plane out as the harmonics and they
Message 1 of 1 , Sep 1, 2011
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Back a while I found that if I put spherical coordinates in a SO(3) like
matrix
and squared it I got a new projective plane out as the
harmonics and they looked like d orbitals besides ( Legendre functions).
The success with making a 4d sphere out of the triaxial
coordinates and projecting them as a triaxial tube to
get the neat two holed torus with the little loop in the middle
made me think that they could be acting as harmonics.
So I put them into SO(4) which is code for an electromagnetic field
in 3d. I got a pretty torus out.
When I increase the magnetic field torus tube radius,
the electric field ( excited ) becomes a regular torus.
The magnetic field becomes a torus with two twists at higher radius.
So if we think in terms of U(1)*SU(2) (the two hole torus):
the triaxial coordinates are the SU(2) and the Dirac
U(1)*SU(2)*SU(2) is the SO(4) with thecsecond SU(2) being
the magnetic tube torus.
So these are excited electron like tori...
or just geometry if you like.

Mathematica:
(* An SO(4) like matrix with the electric field as triaxial*)
(* and the magnetic field as a triaxial tube trajectory*)
m = {{0, Cos[t],
Cos[t + 2*Pi/3], 1 + Cos[p]},
{-Cos[t], 0, Cos[t - 2*Pi/3], 1 + Cos[p + 2*Pi/3]},
{-Cos[t + 2*Pi/3], -Cos[t - 2*Pi/3], 0, 1 + Cos[p - 2*Pi/3]},
{-1 - Cos[p], -1 - Cos[p + 2*Pi/3], -1 - Cos[p - 2*Pi/3], 0}}
(* here the transpose product*)
mm = m.Transpose[m]
(* picking the new transformed coordinates: space or electric field like*)
x0 = mm[[1, 2]]
y0 = mm[[1, 3]]
z0 = mm[[2, 3]]

ParametricPlot3D[{x0, y0, z0}, {t, -Pi, Pi}, {p, -Pi, Pi},
Axes -> False, Boxed -> False]
(*here the square product*)
mm1 = m.m
(* picking the new transformed coordinates*)
x = mm1[[1, 2]]
y = mm1[[1, 3]]
z = mm1[[2, 3]]

g1 = ParametricPlot3D[{x, y, z}, {t, -Pi, Pi}, {p, -Pi, Pi},
Axes -> False, Boxed -> False, PlotPoints -> 60,
PlotStyle -> {Magenta, Specularity[White, 10]}]
Export["Pantlegtorus.3ds", g1]
(* picking the new transformed coordinates: time or Magnetic field like*)
x1 = mm1[[4, 1]]
y1 = mm1[[4, 2]]
z1 = mm1[[4, 3]]
g2 = ParametricPlot3D[{x1, y1, z1}, {t, -Pi, Pi}, {p, -Pi, Pi},
Axes -> False, Boxed -> False, PlotPoints -> 60,
PlotStyle -> {Pink, Specularity[White, 10]}]
Export["tritoralfieldprojection2nd.3ds", g2]
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