Usefulness of non-upper case \mathcal symbols?

Question

I'd like to understand the purpose of mathcal for arguments that aren't upper-case letters. For example, \mathcal{2} gives ∈ and \mathcal{p} gives √.

I tested it for all numbers and lower-case letters, but didn't find any pattern behind them. Does this happen due to a bug or is there an automatic correspondence of sorts?

P.S.: I came across several questions complaining about the abscence of stylish lower-case letters in mathcal, but this isn't what I'm asking about (even though it makes no sense to me).

Answer 1

As Heiko noted in the comment, the same font used for \mathcal is used for other symbols. In fact, \mathcal essentially selects the \symsymbols font family (which is family number 2).

\documentclass{article}

\begin{document}
\verb!\mathcal!: $\mathcal{ABCabc012}$

\verb!\fam=\symsymbols!: $\fam=\symsymbols ABCabc012$
\end{document}