Geometric Understanding of Octopole Moment Beyond Quadropole

[shdrums9]

Generalizing past the quadropole moment-- geometric understanding of the octopole+

I'm having a bit of trouble articulating my question, but I hope the explanations will help you to understand the source of my confusion:

The mono, di, and quadropole moments are all geometrically understandable.

When looking at a dipole moment (say, of two opposite charges) we usually calculate by placing our origin half way between the two charges. If we place the charges along the y-axis (one charge is some distance above the origin, the other is an equal distance from the origin below the origin), we can draw a line of zero potential along the x axis, since at any point on the x axis, the charges are equidistant.

Similarly, I can see the symmetry in a quadropole moment by placing 4 charges in a square array in the x-y plane and then measuring along the z axis from an origin at the center of the square.

However, we run out of dimensions when looking for a line of zero potential with the octopole moment. Although the point at the middle of an octopole moment has a zero potential, there is no other point with the same sort of symmetry.

So I guess my question is, am I prescribing too much 'geometric significance' to the mono, di, and quadropole moments, or do octopole terms (and those of higher order) fundamentally differ from the first three since we are bound by three dimensions?

 

[fresh_42]

It depends on the arrangement I'd say. E.g. the picture here seems to be generalizable to higher dimensions:
https://www.researchgate.net/figure/Plots-showing-the-real-part-of-the-monopole-dipole-and-quadrupole-terms-n-0-1-2_fig2_322696643
https://www.researchgate.net/figure/a-Magnetic-field-profile-of-SEM-C-generated-from-an-octopole-arrangement-of-coils-with-a_fig4_5552263