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What is the simplest (lowest order) real spherical harmonic that cannot be rotated into its own mirror image? I would like to make an illustration of a simple smooth chiral object, but cannot find a good example.

Edit: I should have clarified that I need a linear combination of spherical harmonics, not just one of them.

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    I'm not sure if I understand chirality correctly, but maybe you could take a linear combination of basic harmonics composed with incommensurate rotations?2011-08-26
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    Apologies for my incorrect answer; you're quite right that linear combinations of $l=1$ harmonics can be rotated into their mirror images. Adding an $l=0$ function won't help, so that shows you'll need at least $l=2$. An arbitrary linear combination of the nine functions up to $l=2$ is chiral (even three $l=1$ and one $l=2$ should suffice), so that raises the question how exactly you define "simplest", since "lowest order" doesn't distinguish between the various possibilities including $l=2$ functions.2011-08-26
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    Thanks for your comment. I don't think arbirary combinations are chiral, for exampel you can make the functions independent of z and then they are not chiral. But maybe most functions are indeed chiral. For example we can take the harmonic polynomial $x+y+z + xy$, can you prove that this is chiral? I am also interested in how to prove these things.2011-08-26
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    I've updated and undeleted my answer. About "arbitrary combinations": By "arbitrary" I meant something like "in general position". As it happens, your example is in the set of measure zero of achiral combinations :-)2011-08-29

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