Writing mathematics requires a rich collection of alphabets, special symbols, and typographic
styles.
Alongside the familiar Latin alphabet (\(a,\, f,\, X\)), one frequently encounters Greek
letters (\(\alpha,\, \pi,\, \Delta\)), specialized characters (\(\hbar,\, \aleph,\, \mathrm{III}\)),
arithmetic operators (\(\pm,\, !,\, \otimes\)),
and logical notation (\(\exists,\, \forall\, \Rightarrow\)). Added to these, there are further
symbols such as (\(\infty,\, \partial,\, \sim)\), as well as entire font families like Calligraphic
(\(\mathcal{A},\, \mathcal{V},\, \mathcal{C}\)),
or Blackboard Bold (\(\mathbb{N},\, \mathbb{R},\, \mathbb{C}\)). The list is extensive, but among
all of the members, the most important set of characters
is, without a doubt, the Greek alphabet.
\[
\begin{array}{c}
Α\, \alpha,\ Β\, \beta,\ \Gamma\, \gamma,\ \Delta\, \delta,\ Ε\, \epsilon,\ Ζ\, \zeta,\ Η\, \eta,\
\Theta\, \theta,\ Ι\, \iota,\ Κ\, \kappa,\ \Lambda\, \lambda,\ Μ\, \mu,\\[2px]
Ν\, \nu,\ \Xi\, \xi,\ Ο\, ο,\ \Pi\, \pi,\ Ρ\, \rho,\ \Sigma\, \sigma,\ Τ\, \tau,\ \Upsilon\,
\upsilon,\ \Phi\, \phi,\ Χ\, \chi,\ \Psi\, \psi,\ \Omega\, \omega
\end{array}
\]
If you engage with mathematics a lot, you'll pick up most of these 24 character pairs almost
automatically. Before long, however, you'll
realize that not all Greek letters are created equal. While some are completely free (at least if
your native language's writing system is
based on the Latin alphabet), others are a complete
The lowercase \(\phi\) is my favorite Greek letter. I even used it extensively in my university thesis, where defining the best linear prediction \(\widehat{Z}_h\) of a timeseries \(Z\) at time \(h \in \mathbb{Z} \setminus B\) based on observations \(Z_B\) at times \(B = \{b_1, \ldots, b_n\}\) as \[ \widehat{Z}_h := \phi_{b_1}Z_{b_1} + \cdots + \phi_{b_n}Z_{b_n} \] each \(\phi_{b_i} \in \mathbb{R}\) acts as a prediction coefficient. By grouping them as a vector \(\phi^{(h)} := (\phi_{b_1}\, \cdots\, \phi_{b_n})^T\) the prediction can be expressed as \(\widehat{Z}_h = \langle Z_B,\, \phi^{(h)} \rangle = (Z_B)^T \phi^{(h)}\). If you are interested in how \(\phi^{(h)}\) is computed, as well as why it actually yields the 'best' linear prediction, I highly encourage you to take a look at my thesis.
The main reason I like \(\phi\) so much is its flexibility. Unlike many other Greek letters, \(\phi\) isn't loaded with many conventions (most of them favor the variation \(\varphi\)). Hence, one is free to use it in a wide range of contexts: it feels great both as a scalar or as a vector, and it can even denote a function. In fact, \(\phi\) feels comfortable representing almost any abstract object one might need. Its only drawback is that, especially in handwriting, it can sometimes be confused with the empty set, although context usually resolves this nicely.
A short explanation of why \(\eta\) got its particular ranking.
A short explanation of why \(\vartheta\) got its particular ranking.