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Rare Greek Variables

gwern.net

124 pointswalterbell78 comments

One of John Nash's first papers deliberately used every Greek letter.

For the film "A Beautiful Mind" I used this paper for writing on his dorm room window. As luck would have it, a widely circulated publicity still showed Russell Crowe intent behind "0 < Pi < 1" taken straight from that paper.

Suffice to say this was divisive within the math community. Half of us can't imagine Pi meaning anything besides, um, Pi. The other half didn't even blink.

Someone shared with me a hilarious email exchange within the Berkeley math department, wondering if the math consultant was deliberately trying to make Russell Crowe look bad.

I got the chance to edit an interview with John Nash for the DVD extras, where he bragged to Ron Howard about using every Greek letter. I left that in.

Nice!

I've seen lowercase pi used to represent a permutation, also seen it used to represent a price in mechanism design.

For anyone else interested in the interview, it looks like it's this one (https://vimeo.com/308851217), and the comment about using every Greek letter is at 1:41 or so.

> As luck would have it, a widely circulated publicity still showed Russell Crowe intent behind "0 < Pi < 1" taken straight from that paper.

That's funny. I see this as the first image in the gallery here [1]. I wonder if Nash deliberately used pi for that variable name, just to have this little joke.

[1]: https://amblin.com/movie/beautiful-mind/

The Yoneda embedding is denoted in a few works by よ (the hiragana for the first syllable in Yoneda's name): https://ncatlab.org/nlab/show/Yoneda+embedding.

This is the only use of Japanese script in mainstream English-language mathematics that I'm aware of, although I'd love to be proven wrong.

A colleague of mine ran out of greek letters in his thesis. He added hebrew letters.

(pdf: http://www.andreas-lochbihler.de/pub/lochbihler06diplomarbei... look at the index at the end)

Aleph is a Hebrew letter I think? It's used for describing infinities: https://en.wikipedia.org/wiki/Aleph_number

Well, you'd skip that one just like you'd skip lower-case pi,

Ha, nice idea! But most of these have a problem for me.

varkappa -- there is already kappa, and one definitely shouldn't use both.

varsigma -- looks too much like zeta to me, others may differ. And there is already sigma, one probably shouldn't use both.

upsilon -- looks too much like nu to me, others may differ.

varpi -- looks like overbar omega, or some other variant of omega.

Upsilon -- looks like capital Y.

varrho -- there is already rho.

Xi -- seems like a good one!

vartheta -- this looks unique but again, there is already theta.

iota -- looks too much like i or 1.

Pi -- good one, I have seen this used for sets. In context there is rarely any confusion with product notation.

Generally agree, but should be supplemented with influence of hand-written math.

> Xi -- seems like a good one!

This one is fine to typeset, but is slow to write by hand because it has three disconnected strokes. (You have to pick up your pen twice.) People don't want to use it to do actual hand calculations, and so rarely use it in papers (both because they aren't used to it and, perhaps, because they know their readers won't want to write it by hand).

> Pi -- good one, I have seen this used for sets. In context there is rarely any confusion with product notation.

I think the rarity is indeed because of risk of confusion with product notation. Note that capital sigma is also very rarely used (15th), and my guess is that many of its appearances come from people mistakenly using it for a summation rather than \sum, e.g., \Sigma_{n=1}^N.

When typeset, the size of the summation symbol is pretty obviously bigger than the variable Sigma, but in handwriting it's not as clear. It's true that subscript/superscript notation for the variable being summed over (and that variable's limits) tends to disambiguate, but very often in handwritten math people just write a quick squiggle for the summation sign without bothering to write down the summed variable because it's obvious from context.

So overall, I think the rarely used Greek letters are rarely used for pretty good reasons! I'd say Upsilon is the most unfairly ignored. It's rendered like a Y in some fonts, but not the one that TeX uses for math, and it's easy to draw quite distinctly by hand.

Probably not widely known in the Latin-writing world, but capital xi has a handwriting variant that looks like two z's joined vertically at the hip; that's where the lowercase variant comes from.

The varkappa and the varpi are just calligraphy versions of the regular letters κ & π, so I would avoid them. Also, the ς is just the form σ takes when at the end of a word, so I'd avoid that too.

And I don't know why anyone would use the uppercase Ξ (Xi) when the lowercase is such a nice letter: ξ.

I like ξ! Have been using it in my paper. When presenting it, people have different ideas about how to pronounce it. Some say it like x-i, some say khi, and some try hard to say xi. I like the x-i version as it is similar to its latex notation \xi.

It seems okay to use both uppercase and lowercase versions at the same time. For example Ξ is a set and ξ is an element of the set.

Varsigma is also a bad idea because it is just a sigma, only used at the end of words. So it seems bad taste to use it as if it were its own letter.

We should also not forget that a lot of maths is spoken (in classes and seminars) so there the distinction between, say, \theta and \vartheta would be lost was they are pronounced the same...

Indeed. I once who had a lecturer who used p, rho, and "curly p" as different symbols -- the latter was the Cauchy Principal Part. Suffice to say my (handwritten) notes were full of little arrows saying "pee", "rho" and "squiggle".

Yes, I thought 'oh this has been done carefully' when I read the bit about considering `\var` uh.. variants, but I was surprised that that consideration wasn't grouping them together - counting sigma with 'var sigma' for example.

Are they ever treated as different? Here's a whole lot of \vartheta on Wikipedia for 'theta function', for example: https://en.wikipedia.org/wiki/Theta_function#Jacobi_theta_fu...

I always thought it was just like having both a 'double-storey' and 'single-storey' 'a' available in the same font.

I had a fantastic math professor in my bachelor of engineering program. Besides promising to do handstand in the last lecture if everybody submitted the teacher evaluation, he would tell math related anecdotes. One about a high school friend, whom just loved the concept of greek letters so much that he invented his own greek letters, upper and lower case. During his hand ins he would regularly define variables as his own invented letters, paw, py or ypricon, and alledgedly did so at his oral exam, causing the censor to be highly confused and embarrased to not be familiar with those characters.

For sure a think-out-of-the-box move, and a good prank for a math teacher

I had to re-read that to see that the objection was at an oral exam. I would think most math profs wouldn't even blink at a squiggle as long as it was used consistently.

That reminds me that I used xmas trees and reindeers as index marks (i.e. instead of prime and double prime) in a hand-in once, obviously around xmas time.

For some reason I could just never get zeta right, especially not quickly; so in relevant exams (EE) I wrote 'By <whatever-squiggle>, I mean zeta throughout'. I suppose I don't know for sure that that worked, (spoiler alert: didn't ace them!) but I assume it was ok.

One of my math teachers regularly abstracted formulas as houses and cigars. You know, 'cause (3x+5)² kind of looks like a house with an x as front door and a ² as a chimney. Cigars were used instead of ellipsoids. When translated they flew through space.

Extremely tangential, but when I was learning algebra I for some reason always envisioned it as a great battle between the forces on either side of the equals sign. Terms being added and subtracted were normal troops. Multiplication -- great leaders and generals. Division -- special forces who could lay waste to the other side.

The analogy is really not illuminating in any way whatsoever, but it did add a fun narrative to my problem sets.

There is a reason Ξ is not used, at least as a complex variable. Say you wanna take the complex conjugate of Ξ. What would that look like? Now say you wanted to divide that by, say, Ξ. What would that look like?

It looks like a mess of lines is what it looks like.

In before 王 becomes a thing.

主 - U+4E3B MATHEMATICAL XI BAR WITH GRAVE ACCENT

国 U+56FD CONTAINERIZED PRODUCTION READY MATHEMATICAL XI BAR WITH SUBPRIME INDICATOR

We haven't given the actuaries enough time, they'd slap an umlaut on there, and then we're good to go.

> complex conjugate of Ξ [...] divide that by, say, Ξ

This is not a real concern; it was done by Barry Mazur as a joke, hoping to provoke Serge Lang into using his catchphrase "this notation sucks", so Mazur could present him with a prepared "This Notation Sucks" tee-shirt. See the Paul Vojta section of http://www.ams.org/notices/200605/fea-lang.pdf

\varkappa is a disaster. I once spent way too much time zooming in on a pdf trying to puzzle out if the funny symbol was an x or a kappa. I was left in a state of confusion and dismay. Traumatized

Exactly, we should use symbols that people are likely to be familiar with. We've collectively decided to include the greek alphabet in this.

The Pi symbol has a conventional meaning of course -- Pi is to products as Sigma is to sums. So, I'm sure it is avoided as a variable name for the reason that Sigma usually is.

Agreed, definitely doesn't count. This nudges in \Theta which does surprise me a little

Yeah. Although -- I should note that the article did mention it as well, just toward the end. I was just a lazy reader on my first skim through and missed it.

I bet upper case theta is too close to lower case theta, and nobody wants to put their new variable so close to one with such a strong implicit meaning.

When I used to teach LaTeX classes, most of my students were math department secretaries, so I always made a point of explaining when someone would want \Pi versus \prod or \Sigma vs \sum.

Of course, being the callow youth that I was back then, I also tended to pronounce the names of the letters like a classicist so ψ is not pronounced "sigh" but rather "p-see", and similar ξ is not "zee" but "k-see." A childhood spent surrounded by people with Slavic surnames had my mouth well-trained to manage odd consonant clusters at the beginnings of words.

Everyone I know pronounces ξ as "double squiggly" and ς as "squiggly"

Not 'fancy c' and 'fancy epsilon,' disrespectively? (which is a new word but one I think we should spread).

Back in physics grad school it surprised everyone else that I not only pronounced 'ξ' as "ksi" rather than "cascade" or "squiggle" but, more amazingly, could actually write it like ξ rather than a squiggle... I still don't really get why that's impressive.

The trick to writing ξ is to notice that it is a fast/sloppy cursive way of writing three parallel horizontal bars (Ξ).

Well, Cascade is the name of the particle, not the symbol... (and it would normally be upper-case, no?)

for non-physicists: https://en.wikipedia.org/wiki/Xi_baryon

Also, for this reason, surprised it's one of the more rarely used Greek symbols...

yes... I hated it when professors used them on the board because I don't know how to write them other than making a squiggly...

The way I remember how to draw xi is horizontal line, epsilon, tail.

We had one prof who wrote these as ς with one, two, or even three squiggles below. I never figured out what letters they were supposed to be (zeta, sigma, xi?), but read them as something translating to "zig-zag".

ς and ζ look similar to their handwritten Roman equivalents.

ς is end sigma, the letter σ (S) when written at the end of the word.

ζ is zi (Z), it's not unusual to write a lower case Roman z with a similar tail. My mum was taught to do this, which confused me when I was learning to read.

ξ is xi (X).

ζ and ξ are “squigma” while ς really is sigma.

Xi is used non-stop in the probability / Monte Carlo community! I wonder if this is a sampling bias of arxiv versus AMS (pun intended).

My understanding of the utility of Greek letters in notations was their correspondence to first Roman letter of the respective concepts being represented [1].

Just as Roman letters could be stand-ins for concepts in the context, say, v for volume or speed (vitesse), R for resistence, L for length, d for diameter, r for radius etc.

So, for example, quantity /variable of [l]ength could be represented by [l]ambda, [a]ngle by [a]lpha, [d]ifference by [d]elta, [s]um by [s]igma, [e]rror by [e]psilon, [ph]ase by [ph]i etc. Yet [e]fficiency is eta (not epsilon), maybe it's due to the [h]eat energy context. Perhaps azimu[th] by [th]eta, as in spherical coord (r, phi, theta) or maybe zeni[th]...

Not sure why density was expressed with rho or why pi for circle's L/d ... Maybe it was just a natural choice in Greek?

Of course, there is no 1:1 correspondence, it all depends on context and anyone is free to choose anything, especially where it does not need to mean anything, simply a stand-in.

So not feeling like using a,b,c or 1st, 2nd, 3rd? How about alpha, beta, gamma? Here's a handy table [2].

Btw, omicron is 15th.

[1]: https://en.wikipedia.org/wiki/Greek_letters_used_in_mathemat...

[2]: https://www.rapidtables.com/math/symbols/greek_alphabet.html

The sometimes quoted reason for π is for (p)erimeter, but maybe it's just a mnemonic.

Rho probably stands for rheology. Not a very common phrase outside of physics research.

I see digamma is mentioned, but no love for qoppa or sampi?

https://en.wikipedia.org/wiki/Koppa_%28letter%29

https://en.wikipedia.org/wiki/Sampi

The problem is that these letters are not available in the repertoire of symbols provided by default in LaTeX, even with ams symbols loaded.

I don't see the problem. It's just letters of the alphabet.

(I'm Greek).

Where do the "var" versions of the letters come from? Just different handwriting styles?

I think the varsigma (ς) is at least a special case - it's only used when sigma is at the end of a word, is that right?

We (using the Latin alphabet) have different forms of at least lowercase a, q, w, x, z, and uppercase A, I, J, M, Z. Also the numbers 1, 4, 7 and 9.

Looking at something I wrote yesterday, I use one of the alternative forms consistently, and another only when writing my name.

> I think the varsigma (ς) is at least a special case - it's only used when sigma is at the end of a word, is that right?

Correct.

>> Looking at something I wrote yesterday, I use one of the alternative forms consistently, and another only when writing my name.

I've been known to change styles in the middle of a word, in the latin as well as greek alphabet. So much for graphology...

And thanks be to the gods for keyboards.

Yes, they're just different styles of hand-writing those letters. Usually they're more calligraphic-y, but for example I've always wrote \vartheta instead of \theta and my dad always writes \varpi instead of \pi. And while I just have awful, sloppy handwriting and I find \vartheta easier to write (easier to not try and hit the other end of the circle) my dad is just being posh :)

With the exception of the ς which is as you say, the form used at the end of words (I don't know why- probably something to do with ancient writing styles).

Most of those are just handwriting variations of the regular Greek letters though. I know I usually use a lot of those variants in handwriting (apart from varpi) to distinguish them from the Latin letters (e.g. varkappa is preferable to kappa). It seems it would be more interesting to ignore variants like that.

I remember a lecturer who didn’t like saying or drawing xi and so called it squiggle and just scribbled about a bit on the blackboard. I think I also had a time when a lecturer used both phi and varphi for different things in one equation.

I also don’t understand why upsilon gets considered little-used but omicron doesn’t get that treatment. Maybe there isn’t even a \omicron in TeX.

I’m a little surprised that iota is so little-used.

Finally an anecdote from high school: my classmate once became very confused with a derivation on the board involving both k and K because he thought there was also a “middle k” being used as well as big k and little k.

varpi is a very old-school way of writing π (think 100 years ago), the modern letter is, of course, π. Also varsigma is the "final sigma" (ς), not the regular sigma (σ). vartheta also looks funky (θ normally).

I tend to write vartheta on paper as it can be done in a single stroke. I don’t write a final sigma on paper.

In college I had a physics professor who sometimes replace single-letter variables like X with pictograms like a football. It was a nice reminder that the symbol is completely arbitrary and meaningless.

Therefore I semi-jokingly propose math papers use emoji for their symbols.

I more seriously propose, every math paper should include a god damn table of symbol definitions. I never got “good at math” because not only is the symbology dense, it’s incredibly hard to find definitions. I’m pretty good at teaching myself new things from books and papers. But I just can’t grok math. Math on Wikipedia is a nightmare. It makes me sad because it’s unnecessary. It doesn’t actually have to be so hard.

My son is rather algebra-phobic, so to explain concepts requiring variables I usually use a coloured dot or (if no coloured pens are are to hand) a little picture. People who grok maths tend to be blind to the confusion that variables can engender, viz if I say "2x = 6" then "x" is to be understood as representing a specific value, which is in contrast to e.g. an area formula like "A = bh", so breaking away from letters altogether is sometimes helpful.

Very odd, I used some of these rare symbols a lot and thought they were more frequently used:

\varrho (ϱ) - for density \varsigma (ς) - I don't remember for what, but the regular sigma was already used \vartheta (ϑ) - In spherical coordinates

I think some of these are because these letters look closer to handwriting on a blackboard (the way it was written at my university at least).

I've also used a different kappa, because the regular kappa looks to much like a k, and the variant doesn't look like a k at all. Also, I've sometimes used a variant latin letter v, because the regular v in latex looks to much like a u.

For maximum troll points, use "variant pi" and "lowercase omega with macron" for two different variables in the same paper.

I really wonder how often omicron is really used as a Greek variable and who decides. Fun fact: Big O notation is actually big omicron notation.

But of course, the latin O is what's come to be used in typography, since in the US it's conventional to leave Greek capitals unitalicized and that's not the case with big O.

I have personally used \varsigma in one of my papers, where I had two different things that needed short and memorable glyphs and both started with "s". \varsigma turned out to be surprisingly different enough both from \sigma and s, but I would just give up an urge to make everything a single letter if I were rewriting that paper today.

I was just (tangentially) complaining about this from the student angle on "Ask HN: How did you learn math notation?"

https://news.ycombinator.com/item?id=29341620

The real hipster move would be to use Capital Sigma or Pi for your equations.

Only sigma males hold this power.

The WHO skipping Xi (Ξ ξ) in its COVID-19 variant classifications, for whatever reason cough, means that at least one of these will remain rarer than it should.

> "'Nu' is too easily confounded with 'new,' and 'Xi' was not used because it is a common last name," the WHO said, adding that the agency's "best practices for naming disease suggest avoiding 'causing offence to any cultural, social, national, regional, professional or ethnic groups.'"

> Those best practices were outlined in a May 2015 document issued by the agency. The organisation said at the time that it wanted to "minimise unnecessary negative effects on nations, economies and people" when naming infectious diseases.

source: https://www.nzherald.co.nz/world/omicron-why-who-skipped-nu-...

Ignoring the - what I thought was obvious - joke, why didn't they skip eta then? I'd also say that confusing nu with new to be just as nebulous.

In mathematics, the Greek alphabet is almost just as popular as the Latin alphabet. I think the programmer’s keyboard should include Greek letters.

I like to use glyphs from Stargate which chevrons lock onto when a Stargate is being dialled.

varpi is the Lemniscate constant, cf. https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions#...

McIlroy strikes again!

ϖ is not a separate letter it's just ω with circumflex.

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