Student handout: Spherical Harmonics on a Balloon

Central Forces 2023
Students set up a spherical coordinate system on a balloon, draw a spherical harmonic, and use the balloon as a prop to describe the main features of their spherical harmonic to the class.
What students learn

Spherical harmonics are continuous functions on the surface of a sphere.

The \(\ell\) and \(m\) values tell us how the function oscillates across the surface.

Spherical harmonics are complex valued functions.

Each small group is assigned a spherical harmonic from the list below:

  1. \(Y_1^1\)
  2. \(Y_1^0\)
  3. \(Y_1^{-1}\)
  4. \(Y_2^1\)
  5. \(Y_2^0\)
  6. \(Y_2^{-1}\)
  7. \(Y_3^1\)
  8. \(Y_3^0\)
  9. \(Y_3^{-2}\)

Using a tiny Argand diagram, represent the value of the spherical harmonic at:

  • at the equator (\(\theta = \pi/2\)) for \(\phi = 0, \tfrac{\pi}{4}, \tfrac{\pi}{2}, \tfrac{3\pi}{4}, \pi, \tfrac{5\pi}{4}, \tfrac{3\pi}{2}, \tfrac{7\pi}{4}\)
  • repeat for \(\theta = \tfrac{\pi}{6}, \tfrac{\pi}{3}, \tfrac{2\pi}{3}, \tfrac{5\pi}{4}\)

Tip: Make reference marks in black and draw the complex value of the spherical harmonic in a different color.


Keywords
Spherical Harmonics Kinesthetic Balloon Complex Numbers
Learning Outcomes