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3D 3-phase Lissajous figure
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Circlotron:
Turns out i've been thinking about this problem on and off for sixteen years... For a while I thought the path that would be traced was not circular but more of a twisted 3D figure 8. The seed for the whole thing was this thread six years ago ->

--- Quote ---Suppose we have a cube of magnetic material and we wind some wire around it. Apply a sine wave to the winding and the magnetic field points back and forth along the axis of the winding.

Then we add a second winding at right angles to the first and supply this second winding with a sine wave 90 degrees phase shifted from the first winding. The vector sum of the two resulting magnetic fields would now point back and forth between the axes of the two windings.

Now suppose we add a third winding on our cube at right angles to both previous windings, and we now supply our three windings with three sine waves each shifted 120 degrees as per a normal 3-phase AC power supply.

What path would the resultant magnetic field now follow?
Would this device be of any practical use?
--- End quote ---


From here -> https://www.eevblog.com/forum/projects/three-phase-magnetic-cube/ 

Looks like I can finally stop loosing sleep over it.
daqq:

--- Quote from: Circlotron on October 19, 2020, 11:01:03 am ---For a while I thought the path that would be traced was not circular but more of a twisted 3D figure 8.

--- End quote ---
I made an animation on the topic. Two sines are of the same frequency, the frequency of the third one is varied, ranging from 1x to 3x of the other two sines. There's a moment and a point of view where you get an 8-ish figure, but you need to play with the mutual phases, sizes, frequencies.
daqq:
And this also sweeps the phase of one of the sines. This proves nothing, it's only pretty to look at  ;)
Nominal Animal:

--- Quote from: Circlotron on October 19, 2020, 10:42:36 am ---
--- Quote from: Nominal Animal on October 19, 2020, 04:32:07 am ---This is the biggest circle you can fit in a cube with vertices at \$(\pm 1, \pm 1, \pm 1)\$ according to e.g this essay by Maris Ozols.

--- End quote ---
Wow! That's great! The maths in that reference makes my head spin. :o I don't even know which way up the page goes. The thing is, the other day I was laying on the couch half dozing and thinking about all this and I suddenly had this strong feeling about the size of the circle. Can't explain it.

--- End quote ---

An easy way would be to consider a cube centered at origin, with edge length 2, and its intersection with a plane through origin.  The cube planes are at \$x = \pm 1\$, \$y = \pm 1\$, and \$z = \pm 1\$.  Then ask yourself, what is the orientation of said plane that maximizes the minimum distance between origin and the surface of the cube (i.e., the minimum distance between origin and the cube planes).  It's obviously the one with normal \$(1, 1, 1)\$ (or equivalently \$(-1, -1, -1)\$).  In any other orientation the minimum distance is smaller.

If you consider a circle with a fixed center, it is just all the points that are at a fixed distance from the center and all on the same plane.  So, choosing the plane normal – remembering that because of cube symmetries, we only need to consider at most the positive octant in 3D – that maximizes the minimum distance between center and the intersection with the cube faces, maximizes the radius/diameter of the circle, too.  And we already know it is when the normal is diagonal to the cube.

The referred essay just extends this concept to any number of dimensions n.
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