what letters go well together as sets of variables?

mostly you just vibe it out

It is very common to label coordinate axes (for example) with the variables $x$, $y$, and $z$. People working with $\mathbb{R}^4$ often add $w$ as a fourth. This is incredibly awkward. Yes, $w$ through $z$ is a line of adjacent letters, but you really want something that goes after $z$, not something that goes before $x$. But $z$, of course, is the last letter of the alphabet. So you’re left with this awkward decision between using the variable set ($w$, $x$, $y$, $z$) and using the variable set ($x$, $y$, $z$, $w$).

Relatedly, $z$ is often used as a variable for a complex number. I believe this is because people wanted to use $x$ and $y$ for the real and imaginary axes, so they wrote $z = x + iy$¹¹ I don’t know why $a + bi$ but $x + iy$ and $u + iv$. Claude suggests it’s because $b$ is a constant being multiplied by a basis element and $y$ is a variable being multiplied by a scalar, and also that $x + iy$ is used more often in contexts where you need to worry about not confusing it with $y_i$.. This means that it’s conventional to use $w$ as your second complex variable, $w = u + iv$ if you want variables for the real and complex parts. So we have a situation where $w$ and $z$ form a pair, even though they’re separated by two other letters, and also $z$ conventionally comes first in the pair even though it is the later letter.

Many other sets of letters are well-explained by adjacency (and visual resemblance), and are often just used straightforwardly in order:

• $a$, $b$, $c$, $d$

• $f$, $g$, $h$

• $i$, $j$, $k$

• $m$, $n$

• $p$, $q$, $r$

• $s$, $t$

• $u$, $v$, $w$

• $x$, $y$, $z$

Often as you go later into these sets you start hitting slightly less favored letters. For instance, ($a$, $b$, $c$) is a much more classic list than ($a$, $b$, $c$, $d$). You often see $p$ and $q$ paired together, and less often see $r$ thrown in. On the other hand, even though there’s a strong ($x$, $y$) pair individually, $z$ doesn’t particularly feel like a third wheel when it shows up. Similar is true for ($i$, $j$) and $k$.

It’s not too uncommon to see $h$ paired with $k$, or even to see something like $g$, $h$, $k$ used as elements of a group. This happens in contexts where people want to avoid $i$ and $j$. The sorts of variables it feels natural to use $i$ and $j$ for are pretty narrow, compared to other variables. They’re really common for indices, you see as basis vectors, and they don’t love being used elsewhere so much. I think sometimes you see ($h$, $k$, $\ell$), when you start on $h$ and need a third letter, even though $\ell$ is kind of annoying because you need to use an alternate form of it or else it looks like a one.

Presumably $e$ is often avoided because the symbol is used for the number. You still sometimes see ($a$, $b$, $c$) paired with ($d$, $e$, $f$), though. Maybe less awkward is when you do ($A$, $B$, $C$) with ($D$, $E$, $F$).

One can also consider Greek letters. ($\alpha$, $\beta$, $\gamma$) of course makes sense. A few other sets echo common Roman sets, like ($\mu$, $\nu$) or ($\sigma$, $\tau$). But you see ($\lambda$, $\mu$) much more often than you’d ever see ($\ell$, $m$), presumably because $\lambda$ is a much more distinct letter than $l$ is. Strangely, it’s typically ($\varepsilon$, $\delta$) even though this is the inverse of the alphabetical order. This is because $\varepsilon$ is mnemonic for error, so it’s the primary entry in the pair, and then $\delta$ comes second instead of $\zeta$ because it feels less weird to pair a letter cognate to $e$ with one cognate to $d$ than one cognate to $z$.

The weirdest set is probably ($\theta$, $\varphi$, $\psi$), used for angles (or ($\varphi$, $\psi$) used without $\theta$ for homomorphisms). I think they just have sort of similar vibes or something? Other groups sometimes skip a few inconvenient letters, but $\theta$ and $\varphi$ are just nowhere near each other in the alphabet! I suppose $\varphi$ and $\psi$ only have $\chi$ between them, though, and I can see why you’d skip $\chi$.

Also, shoutouts to when people use $\text{よ}$ for the Yoneda embedding.