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a question about the SH

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i have a simple question about the SH, there was a sentence: "Projecting an environment map into 3rd order spherical harmonics..." what the word "3rd order" mean?

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Spherical Harmonics are a form of polynomial, the 'order' here refers to the order of polynomial used, the number of basis functions (and therefore size of spherical harmonic vector you get is dependant on the order) so '3rd order' results in 9 basis functions, '4th order' - 16 basis functions, '5th order' 25 basis functions.

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