Suggestions and feature requests for the design of a font for math articles/books

Question

Edit: The project website, GitHub repository, and Discord server for Darwin are now up!

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A little context and introduction: I'm currently working on a family of fonts ("Darwin") designed for use in formal writing and optimised in particular for use in mathematics articles, books, etc. It is meant to fill the following requirements, which no currently available fonts for LaTeX fill:

1. It must be completely free and open-source.
2. It must have exceptional symbol coverage among many different languages, covering essentially all languages making use of the Latin, Cyrillic, and Greek alphabets.
3. It must come with a number of optical sizes, which are additional font families with size-specific optimizations―think an extra font for footnotes, a font for usual paragraph text, and an extra font for book covers.
4. It must come with a matching math font family to uniformise the style across the document. This includes all the usual math symbols as well as font families for \mathbb, \mathsf, etc.

Currently the regular style of the font looks like this (please ignore the currently uneven spacing):

I have also prepared PDF samples of articles in English, Russian, and Greek compiled using the current version of the font, though please note that these contain only the regular style of the text version, and currently have very uneven spacing:

1. English Sample: Bhatt–Lurie's The prismatization of $p$-adic formal schemes. Original: arXiv:2201.06124.
2. Russian Sample: Sergei Akbarov's Stereotype Dualities in Geometry. Original: arXiv:2311.05131.
3. Greek Sample: Grigorios Tachyridis's Krylov subspace methods for the solution of linear Toeplitz systems. Original: arXiv:2303.03223.

I should also note that these samples were compiled from sources of papers directly downloaded from the arXiv, and as such I bear no relation to their authors.

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Now, for the actual question: Throughout my own research as well as discussions with friends and colleagues, I've often found situations in math which require special symbols or features from fonts/LaTeX packages, as well as distinct preferences when it comes to certain choices of fonts. For example, these include:

1. The use of symbols that are specific to certain areas and not usually covered by the default LaTeX fonts, like:

   a) The hiragana よ for the Yoneda embedding in category theory.

   b) The Cyrillic letter Ш for the Tate–Shafarevich group in arithmetic geometry.

   c) The Cyrillic letter Л for the Lobachevsky function in hyperbolic geometry.

   d) The Cyrillic letter Э for evolutionary vector fields on a jet bundle in some articles, e.g. this one, on the geometry of PDEs, as pointed out by Igor Khavkine.

2. Better glyph support for \mathcal, \mathbb, \mathfrak, etc., such as:

   a) \mathbb for the lowercase alphabet.

   b) \mathbb for Greek.

   c) More exotic things like \mathcal or \mathfrak for Greek.

3. A preference of "empty fill" (txof style) over double-strucked blackboard bold (amsbb style) or vice versa, so that having both versions available in a font would be nice:

vs.

4. A bigger set of delimiters, as discussed here, for situations where one would e.g. need parentheses bigger than \big(...\big) but smaller than \Big(...\Big).

5. Variable size delimiters for doubled parentheses ((...)), doubled brackets [[...]] and doubled angle brackets << ... >>, a feature request by Willie Wong.

6. Native and easy handling of bold text and bold math for section and chapter headers, a feature request by Apoorv Potnis.

7. Built-in optimisation of microtype through a variable width axis, allowing for an even better spacing optimisation than microtype already wonderfully does.

8. Symbols for binary relations that are missing from current LaTeX packages, such as inequality versions of $\asymp$, a feature request by Iosif Pinelis.

I plan to address all these points and more in the development of Darwin, but I'm sure the above list is anything but exhaustive, and there ought to be countless more such features which would benefit the community. So, in the spirit of the items above, I'd like to ask: are there any font features you want to have as part of your work, but weren't available or required workarounds at the time you thought of them?

Here "features" should be understood as including things in the spirit of the above examples, but not limited to them.

I also want to point out that suggestions without technical details are more than welcome. Part of the reason I wanted to ask this question on MO instead of somewhere like TeX.SE is that I believe mathematicians not familiar with type design or the technicalities of LaTeX surely also have extremely valuable feedback to offer. So, for example, suggestions like "this symbol sketched by pencil below would work great for doing [x] in area [y], but I'm not aware of an implementation of it, or find the current ones flawed because of [reason]" would also be exceedingly welcome.

Answer 1

Here is my own personal wish list (or rather a partial one, consisting of the items I can come up with off the top of my head). Some of them have existing workarounds, and some of them are perhaps not directly solvable by creating a new font, but I figure it's better to err on the side of making suggestions that may not be useful, than to not say anything at all.

• Regarding blackboard bold, I personally prefer a style with vertical rather than curvy lines, such as in the title of this old paper of mine.
• You didn't mention Hebrew letters; these are used to denote infinite cardinalities. As mentioned here, the default $\aleph$ and $\beth$ do not have a uniform appearance.
• There does exist a symbol for the wreath product of two groups: $G\wr H$, but I think I'm not alone in finding it unsatisfactory, having a somewhat nondescript appearance. When I first encountered the wreath product symbol in Nathan Jacobson's book Basic Algebra I, it looked a lot more like a reverse integration symbol. See page 79: I don't know what other people think of that version, but I think it's better than $\wr$. Some circumstantial evidence that people don't really like $\wr$ includes: (1) I often see people writing $\text{wr}$ for the wreath product, which is pretty weird if you think about it (offhand, I can't think of any other product that people represent using two Latin characters), and (2) in Richard Stanley's book Enumerative Combinatorics Volume 2, he uses the symbol $\S$ instead (which I don't particularly like either, but at least it's more distinctive than $\wr$).
• There are three graph products that are closely related to each other: $G\times H$, $G\boxtimes H$, and $G\mathbin{\square} H$. I had to kludge the last one using \mathbin\square (as of this writing, Wikipedia doesn't even bother with \mathbin and so it looks even worse). Admittedly, this kludge works reasonably well, but I think it would look nicer if the three symbols all had the same size (compare $\times \otimes$ with $\times \boxtimes$), which they currently don't.
• Sometimes one wants a modified summation symbol, e.g., $$\mathop{{\sum}'}_{m,n} {1\over (mz + n)^{2k}}.$$ This is discussed in Exercise 18.44 of Knuth's original $\rm\TeX book$, where he points out that the above solution \mathop{{\sum}'} centers the limits on the primed summation sign rather than the original summation sign. The workaround for that is much more complicated. I don't think that it's necessary to create separate symbols for every possible accent that someone might want to decorate the summation symbol with; offhand, I think I've only ever seen $\sum'$ and $\sum^*$. An example of the former may be found in Jean-Pierre Serre's book A Course in Arithmetic, Chapter VII, Section 2.3.
• Sometimes I want to put an arrow over something to indicate that it's a vector. If I just want a rightward-pointing arrow then \vec is usually fine; e.g., $\vec v$. But if I want an arrow going in the opposite direction, then the only easy solution I know is to use \overleftarrow, as in $\overleftarrow{v}$, which is clumsy. Using a harpoon instead of an arrow looks a bit more elegant to me, but seems to be complicated to implement. It would nice if extensible arrows or harpoons in both directions were easily implemented.
• There are many different ways that people like to write the symbol for a disjoint union. Some are well supported; others are less so. I don't have particularly strong feelings on the topic but thought I would mention it because it does seem to come up a lot.

Answer 2

Here are the suggestions I can think of

• Legible mathfrak capitals, i.e. easily distinguishable glyphs that resemble their latin counterparts enough that an uninitiated reader can easily tell which one it's meant to be. Like the Mathematica mathfrak I guess. Obviously a balance needs to be struck between legibility and tradition, but I find the illegibility of the default mathfrak adds an unnecessarily barrier to understanding sometimes.

• For mathematical writing it would be nice if punctuation such as parentheses wouldn't be slanted in the textit font. Punctuation doesn't get slanted in math mode, so in theorem-like environments that use italic text you often see a mixture of slanted and upright punctuation and it looks weird, e.g.

• For specific symbols, conditional independence is represented by something usually written as \perp\!\!\!\perp, as in

and in category-theoretic probability an arrow with a dot on it is often used for stochastic morphisms, as in

I guess ideally the dot would be a bit smaller than the \bullet I used here. I don't think either of these exist as a single symbol in any font.

Answer 3

1. Bold versions of the blackboard bold letters. It's common to want $\mathbb{R}$ or $\mathbb{C}$ as part of a section heading, e.g.

   $\textbf{Conjugacy classes in $\textbf{GL}_{\boldsymbol{n}}(\mathbb{R})$}$

   and the $\mathbb{R}$ looks out of place because of the stroke weight.

2. If it's possible to make \overline align well with characters, that would be helpful. The default overline alignment is very poor, e.g. $$ \overline{f},\quad \overline{j}, \quad\overline{s}, \quad\overline{A}, \quad\overline{B}, \quad\overline{V} $$ Using \bar only fixes some of these, doesn't match the look of \overline, and has its own problems with some other letters: $$ \bar{f},\quad \bar{j},\quad \bar{s},\quad \bar{A}, \quad\bar{B},\quad \bar{V}, \quad \bar{d}, \quad \bar{m}, \quad \bar{M}, \quad \bar{T} $$ There are similar issues with \widehat. All of this can be fixed with well-chosen \mkern's, but it should be automatic.

Answer 4

The theme of this answer is "incomplete math fonts". Most of the special math fonts currently don't even cover the full alphanumeric Latin alphabet (let alone other alphabets):

• \mathbb is missing support for lowercase letters (as mentioned in the question) and digits (as mentioned by user @indicator in the comments, linking to the discussion here).
• \mathcal is missing support for lowercase letters (TeX.SE discussion) and digits (TeX.SE discussion).
• \mathscr is missing support for lowercase letters (TeX.SE discussion) and digits.
• … (This answer is community wiki, so others are invited to add any additional examples that fall within this category.)

Arguably, all of these can be considered to be separate fonts, and probably have alternatives in Xe(La)TeX, but it would be nice to have them somehow belong to the same font family and be available in LaTeX.

Answer 5

Unlike the somewhat obscure scripts you mentioned for particular/unique mathematical symbols, Greek is used constantly in math for all sorts of things. LaTeX has lowercase and uppercase versions of many Greek letters: $\pi$ and $\Pi$, for instance. However, when the uppercase of a Greek letter is identical to its Latin counterpart, there is no special uppercase version in LaTeX: for example the uppercase of $\alpha$ looks like $A$ (although possibly it should not be italicized) so there is no "\Alpha" command.

This is understandable. However, in many cases I would like a "bigger" version of my Greek letter symbol that is distinct from its Latin counterpart. The one example that really comes to the top of my mind is $\chi$. Chi is used for many things in math/physics/statistics, and it looks particularly bad with subscripts because of how "low" the default letter is. Take a look at how the chromatic polynomial of a graph $G$ is written: $\chi_G(x)$. The $G$ and the $\chi$ sit at almost the same level, even though the $G$ is supposed to be a subscript here. This problem is discussed also at https://tex.stackexchange.com/questions/191551.

It might be a little weird, but you could consider somehow incorporating bigger versions of the lowercase Greek letters that don't have unique uppercase forms. Or at least try to fix chi.

Answer 6

Not sure if this has been mentioned, but many mathematicians are missing a \bigtimes similar to \bigotimes = $\bigotimes$.

Answer 7

This may not be an answer to your questions, but it is also too long for a comment. I hope it can be useful, so I leave it here.

First of all, your font looks nice, so I hope you keep up the hard work!

The most important point: I see nobody commenting on the format. If I were to make a math font today, it would certainly be a Unicode math font. There are several reasons for this, one being that most of the symbols people ask for (and many that nobody ask for) do indeed have their slots there. Let me give a few examples in images below, where I have used the Stix Two Math font, which is rather complete.

Left pointing arrows as accents:

Wreath product (not the symbol asked for, but this one is the one from the official slot U+2240, you could in your font design a better symbol, or having the standard one with an alternative one that people can use):

Cyrillic can be there (as well as all capital greeks):

Less than or equivalent to:

Independent symbol:

Various binary symbols (note here that Stix (top) is not having them uniform while Latin Modern (bottom) do)

One more reason to stay within the bounds of Unicode math is accessibility. If one wants to have the possibility to copy and paste from pdf files it is crucial. If one only cares about the output in the pdf, this is not a big problem.

There are also draw backs with Unicode math. Many alphabets exist, but some are lacking (bold sans greek exist, but sans greek does not, to give an example). It is still possible to use private ranges to make them available for macro packages, but then one should remember that one will not be able to copy and paste and keep their meaning. Another bad thing is that there are gaps in the alphabets. So, italic h is not between italic g and italic i, but somewhere else, because it goes as the Planck constant and the Unicode consortium decided against more than one slot that look the same, or something like that. Bad.

When it comes to implementation, the different existing Unicode math fonts do things a bit differently, and therefore some combinations render a bit bad sometimes. One thing is italic correction. An excessive use of it can make important characters like italic f to be hard to use:

The top line here is (f) in TeX Gyre Bonum Math. It relies heavily on italic correction. On the second line it is how a fixed version can look like. In the third line we see the same expression in Lucida Bright Math. In that font no italic correction at all was used for these kind of symbols, and it still looks very good. Thus, a suggestion is to not make your font depend heavily on italic correction.

When it comes to optical sizes, I would probably not go through the hassle of having and maintaining several versions. The same probably goes through when it comes to bold math. Maintaining a complete bold font seems to be a lot of work. One can already today fake the original font into a slightly bolder one, and for the rare usage in titles, I think it comes out good enough:

A few general (perhaps obvious) suggestions:

• Use many variants for delimiters before going extensible. This is in particular important for the ones that are difficult to have as extensibles (angular brackets for example).
• Make sure that all the accents that can be made to grow do so.
• Add style alternatives for characters that need them. For example the wreath product seems to be one of them. It is then up to the user to choose.
• There are no slots for the calligraphic alphabet, only for the script one. Make sure to add both a calligraphic and a script alphabet. Make one of them a style alternative of the other (this is how it is done in Stix two math). Then the macro packages can make both available, one via \mathcal and the other via \mathscr, for example.
• Be careful where you set the accent anchors, and do set it for all alphabets, not only for the latin italic ones.
• Make the look consistent. The best way to test is probably by using available articles that use various symbols already.

There is much more to say, but I stop here. I look forward to see your new font!

Answer 8

One point which I want to clarify is that of special and unusual symbols for variables or objects: besides the relatively common ones I mentioned in the question body, such as ш for the Tate–Shafarevich group or よ for the Yoneda embedding, there are a few articles that use even more specific characters, such as:

• Fiorenza–Loregian's Recollements in stable $\infty$-categories, which uses the Georgian letter რ for "a (donné de) recollements";
• Fiorenza–Loregian–Marchetti's Hearts and towers in stable $\infty$-categories, which uses the kanji 切 for the "class of slicings of stable $\infty$-categories";
• Some articles relating to the work of Satake, such as those involving Satake compactifications, the Satake isomorphism, etc. use the katakana サ, e.g. Namikawa's Toroidal Compactification of Siegel Spaces;
• Some articles use Hebrew letters that are not conventionally used in mathematics, such as Tsadi (צ) and Mem (מ);
• I've used the kanji 忘 (from 忘れる, to forget) in my own work to denote forgetful functors.

I plan to support such usage as well in the form of requests in cases where such unusual notation is reasonable/beneficial. Such requests could either be opened as issues in the GitHub tracker for the font (a link will be added here as soon as it's ready), or by just communicating it to me by email, Discord, etc.