Visualizing the spherical harmonics

Visualising the spherical harmonics is a little tricky because they are complex and defined in terms of angular co-ordinates, $(\theta, \phi)$. One way is to plot the real part only on the unit sphere. Matplotlib provides a toolkit for such 3D plots, mplot3d (see Section 7.2.3 of the book and the Matplotlib documentation), as illustrated by the following code.

import matplotlib.pyplot as plt
from matplotlib import cm, colors
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from scipy.special import sph_harm

phi = np.linspace(0, np.pi, 100)
theta = np.linspace(0, 2*np.pi, 100)
phi, theta = np.meshgrid(phi, theta)

# The Cartesian coordinates of the unit sphere
x = np.sin(phi) * np.cos(theta)
y = np.sin(phi) * np.sin(theta)
z = np.cos(phi)

m, l = 2, 3

# Calculate the spherical harmonic Y(l,m) and normalize to [0,1]
fcolors = sph_harm(m, l, theta, phi).real
fmax, fmin = fcolors.max(), fcolors.min()
fcolors = (fcolors - fmin)/(fmax - fmin)

# Set the aspect ratio to 1 so our sphere looks spherical
fig = plt.figure(figsize=plt.figaspect(1.))
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x, y, z,  rstride=1, cstride=1, facecolors=cm.seismic(fcolors))
# Turn off the axis planes

The spherical harmonic Y_32