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  "body": "\n## Summary\n\nSpherical harmonics are the restrictions to the unit sphere of [[homogeneous polynomials|2C3PAQ7H50]] that are [[harmonic|0DEB0JMN0W]].\nThose of degree $\\ell$ form a finite-dimensional space, and these spaces give an orthogonal decomposition of the square-integrable functions on the sphere.\nThey 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.\nOn the sphere in $\\R^3$ they are built from the [[associated Legendre polynomials|VPJCSVRY6S]].\n\n## Explanations\n\n### L1 human\n\nAny reasonable function on a circle can be written as a sum of sine and cosine waves of increasing frequency.\nSpherical harmonics are the analogue for a sphere: basic wave patterns from which every function on the sphere can be built.\nDegree $0$ is the constant pattern.\nDegree $1$ is a pattern that is high at one pole and low at the other, like the function $z$.\nHigher degrees have more and more lobes.\nThey describe the shapes of atomic orbitals, the gravity field of the Earth and the temperature map of the cosmic microwave background.\n",
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