This posting is a simple addendum to the much longer article, Notes on Calculating the Spherical Harmonics. Because it had already grown so long, I left out making any plots of individual eigenmodes, so here I simply present plots of the first 50 degrees of the pure spherical harmonic eigenmodes, which can be useful to help visualize some of the symmetry properties that were discussed in the longer article.

## Introduction¶

As a reminder, the spherical harmonics eigenmodes (or eigenfunctions) are the set of
functions
`\begin{align} Y_{\ell m} (\theta, \phi) &\equiv \lambda_\ell^m(\cos \theta) e^{im\phi} \end{align}`

where the functions are indexed by two integers — the degree `$\ell$`

and order `$m$`

—
and are functions of the colatitude-azimuth angular position `$(\theta,\phi)$`

on the
surface of the sphere.
The functions `$\lambda_\ell^m(x)$`

are the spherical-harmonic pre-normalized Associated
Legendre Polynomials, which are the subject of my
`Legendre.jl`

article series.

The following figure is an interactive “figure pager”^{1} which shows the real part of
a spherical harmonic eigenmode (for an orthographic projection of the sphere) and the
corresponding degree `$\ell$`

and order `$m$`

labeled at the top of the image and in the
textbox to the left of the image.
You can either use the arrow buttons to step through the eigenmodes, or you can
type in a degree and order into the textbox — formatted like “(50,25)” where the first
number is the degree and the second is the order — to jump to any specific eigenmode
within the range `$0 \le \ell \le 50$`

and `$0 \le m \le \ell$`

.

For instance, these plots help make the naming of the
zonal, tesseral, and sectoral
spherical harmonics a bit clearer:
the zonal harmonics are those where `$m = 0$`

and the eigenmode is azimuthally symmetric
around the poles; the sectoral harmonics are those where `$m = \ell$`

and the plot looks
a bit like the sectors of an orange (no latitude lines of zero); and the
tesseral harmonics for all other values of `$m$`

where the pattern looks more like a
checkerboard.
(I suggest you try out some of these combinations in the pager above.)

I also created a movie which runs through all 1326 eigenmodes in increasing order and degree.