Let f(x)=x2
Let g(x)=sin(x)
f(g(x))=(sin(x))2 Which is usually written sin2 (x)
g(f(x))=sin(x2 )
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gregorian chant
No, one example would be wüstite. Please see the link.
It depends on what kind of art, of course. For example, math plays a huge role is the musical arts. In composition of musical pieces, math is used in order for the composition to flow properly and having correct rhythmic qualities.
There is no homophone for "composition".
The composition of your nails is yes skin.