---
tag: 3F32Y21HP4
kind: concept
name: Spherical harmonics
msc: [33C55]
sources:
  - url: https://en.wikipedia.org/wiki/Spherical_harmonics
same_as:
  wikidata: Q877100
  wikipedia: Spherical harmonics
---

## Summary

Spherical harmonics are the restrictions to the unit sphere of [[homogeneous polynomials|2C3PAQ7H50]] that are [[harmonic|0DEB0JMN0W]].
Those of degree $\ell$ form a finite-dimensional space, and these spaces give an orthogonal decomposition of the square-integrable functions on the sphere.
They are the eigenfunctions of the [[Laplace–Beltrami operator|F4NWZNTRMD]] of the sphere, and they play for the sphere the role that $e^{in\theta}$ plays for the circle.
On the sphere in $\R^3$ they are built from the [[associated Legendre polynomials|VPJCSVRY6S]].

## Explanations

### L1 human

Any reasonable function on a circle can be written as a sum of sine and cosine waves of increasing frequency.
Spherical harmonics are the analogue for a sphere: basic wave patterns from which every function on the sphere can be built.
Degree $0$ is the constant pattern.
Degree $1$ is a pattern that is high at one pole and low at the other, like the function $z$.
Higher degrees have more and more lobes.
They describe the shapes of atomic orbitals, the gravity field of the Earth and the temperature map of the cosmic microwave background.