Legendre Polynomials and Multipoles

Far from any lump of charge, its potential settles into a tidy series, the multipole expansion, and the angular shape of each term is a Legendre polynomial $P_l(\cos\theta)$. The pieces are familiar by name: the monopole is the same in every direction, the dipole is $\cos\theta$ with a single belt of zero around the equator, the quadrupole adds another, and each higher multipole carves the sphere into more zones. The reason is in the polynomial: $P_l$ has exactly $l$ zeros between $-1$ and $1$, and each zero is a nodal cone, a direction where that multipole contributes nothing, separating $l+1$ lobes that alternate in sign. The scene draws those lobes as a polar diagram around the vertical axis, red where $P_l$ is positive and blue where it is negative, with the nodal cones marked; the whole figure is what you get spinning this cross-section about the axis, since the pure multipoles do not depend on the azimuth. A probe sweeps the polar angle and ties the radius of the lobe to the value of the polynomial in the panel below, where $P_l(x)$ with $x = \cos\theta$ is plotted with its $l$ roots, the cosines of the cone angles. Every $P_l$ starts at $1$ when $\theta = 0$ and ends at $\pm 1$ at the south pole, and any two different ones are orthogonal, which is exactly why this set is the natural alphabet for expanding a field on a sphere. Step the multipole and watch a lobe and a cone appear with each increment of $l$.