A quantum physicist’s views on type

A quantum physicist’s views on type: Nell May in conversation with CW Gardiner

A thing I have always felt is that support for the concepts of maths, mathematical thinking and mathematical typesetting in virtually all places is woefully inadequate. A journal like the NZ Listener can discuss Shakespeare in some depth, but be unable to typeset some of the simplest equations, such as would be taught in high school at the same time as Shakespeare. – CW Gardiner

When Microsoft Word proved too primitive to accommodate my overly realistic newspaper design for a school history assignment, it was my father that calmed my panic. He split my ‘heavy’ file over two floppy disks and transferred the data to his office Mac. I sat by his side as he proceeded to code in LaTeX my Who shot JFK? newspaper vision. ‘Anything is possible,’ he told me, ‘as long as you can describe it.’

The following is a conversation on mathematical typesetting I had with my father, Crispin Gardiner, a New Zealand quantum physicist, that took place in Pully, Switzerland, during May 2023.

Day 1

Can you describe your job?
I don’t have a job. I’m retired.

You were just in Innsbruck working with Peter Zoller.¹
I wasn’t getting paid. A job means you get paid.

Can you describe your past profession?
Well, you give lectures. These have to be thought out and prepared. That is quite strenuous actually.

I mean the topic.
What? Physics?

How do you describe what it is you do?
I do theoretical physics which involves… (laughing). Come to a theoretical physics course and I’ll tell you. Most people wouldn’t have a clue.

How do you describe it to someone who doesn’t know what theoretical physics is?
(Long pause). Well, I suppose there are certain theoretical structures which are very precise and accurate. There are two parts of physics: firstly, you can measure things. Now with modern equipment, that has enormous precision. Accurate to 10 decimal places, easily. And secondly, there is a set of theoretical frameworks. These are mathematical equations and you can write down the whole set of such equations which describe everything we fundamentally know about the universe. You could do it on two or three pages. Of course that is just writing it down, that is not to say that if I gave it to someone that they would have the slightest clue what it means.

We can talk of the physics of atoms, that might be the easiest thing to start with, where there is one single equation that will describe it all. You’ve got to set out the parameters of the atoms, you know how many electrons it’s got. But given that, with this one equation, if you had enough computing power you could calculate everything that could be measured about that atom. And at present with the techniques we get very good accuracy. That means we are fairly confident that anything we calculate by these very precise methods will have the answer exactly right.

Now, that essentially extends over all of physics. If you wanted to ask why this aluminium here is a metal, why does it have its particular properties, then the answer can be given. Some are very complicated and a bit hard to calculate. Essentially, what I am doing is working out ways to calculate things that we couldn’t calculate before.

I’ve done about five or six innovative things. I’ve invented methods that have solved complete classes of problems of how to actually describe things. And five or six of those is quite a lot, not many people have done as many of those which are completely new procedures. But they are procedures within a well defined framework. They are just saying that if you did this, and this, and this, and this, and that, and that, and that within this calculation, then it would all become very clear to understand how it goes. Like, if you wanted to ask the full physics of how quantum systems like atoms can interact with other atoms via light beams channelled down waveguides and go on forever in all sorts of different ways, making great big circuits of them, well, that is something that I invented, how that would be done. I wasn’t the only person involved, there were two other people. One of them was Matthew Collett,² if you know him, the other was Howard Carmichael.³ Howard and I simultaneously invented exactly the same formula independently in completely different ways. That was 1993. The other one was 1984 to 1985.

And then there are descriptions of how to describe the formation of Bose Einstein Condensates, this was essentially me. There was also Rob Ballagh and Peter Zoller; the question was… you have lots of very cold atoms, then with the coldest of them all the atoms somehow cooperate together and behave just like a fluid, purely from quantum effects and this one big quantum state. How does that actually form? That was the question.

People had done experiments and it did form. There were two principal estimates of the time scale: one of which said it happened instantly and the other of which said it was impossible. Both were considered respectable estimates. Essentially what it shows is that they were asking the wrong question. They were asking a very unrealistic question! Suppose the atoms were all completely unorganised and you cool them down, how can they enter this particular state? What those people didn’t take into account was that there would already always be some atoms in that state. The problem wasn’t how do they enter that state, the problem in the laboratory was that if you are in that state how did it grow to a certain amount and how fast did it grow? That was the real question that one wants to know the answer to. Theirs was a very theoretical question. So what you found is a curve that grew like this… (hand gesturing)… which unfortunately can’t be described without me waving my hands, it starts off slowly and then it comes up and grows.

They were just looking at the first few microseconds, which in fact is very difficult unless you give the proper quantised description, which they were not doing. So that was another thing I invented. It turned out to not be as important as the optical stuff. Peter and I realise now that the optical stuff we did from 1984 or 1993 was really of much greater significance than we ever thought it was. People are starting to make machines like this now.

When did you start typesetting your own work?
The tale starts in 1986. I got my first royalties from the first book I had written which was done by completely old fashioned methods. And they were NZ$7,000 when the dollar was worth about three times what it is worth now and taxable at 60%. Therefore, it became essential to find some way of spending this money that was tax deductible, like on buying a computer to write another book. I had this idea of writing a book on quantum noise. I bought a computer, a Mac. I realised very soon that it was going to be utterly impossible using something like Microsoft Word to ever write a book.

Then I discovered TeX, which already existed for mainframe computers. It suddenly became available for the Mac, a thing called MacTeX, which I bought for NZ$700 on floppy disks. LaTeX also existed at that stage but not completely, the Classes for writing books were not yet there, and in the end I thought it would be better to use the plain TeX because if I used LaTeX I wouldn’t be able to write a book. I’d just be able to write various pages and I needed to do all sorts of other details. A book has got all sorts of things in it that a simple article hasn’t. Anyway, I made that decision. It might have been the wrong decision because the Book Class was probably available by the time I finished. The essential details are much the same. LaTeX is just a package which makes it a lot easier to do a whole lot of prearranged things. There were things I had to write in, like section commands. LaTeX had its own as well. The problem with this was that it was not easy to modify LaTeX at that stage. Decisions on most things had basically been made in advance and to change it you had to go and hack the code.

The way that TeX works is that there are a whole lot of primitive commands which are built into the software. Then there’s macro facilities for writing macros. Which means you can write commands of your own which do all sorts of complicated sets of instructions which you write. TeX itself is deliberately written as a primitive language so it has complete control of exactly where you put anything. The ability to spread letters out, paragraphs, pull paragraphs apart, put boxes here and so on all via the coding process. But it’s tricky to use.

Would somebody normally typeset this all themselves?
Well, yes. If you write mathematics there is no easy way to get it unambiguous other than to do it yourself. When I started doing this, working with Heidi Eschmann – who you must remember – I wrote things out by hand and she typed them into TeX. It worked well even though she was not really a programmer. You really have to be a programmer. In terms of doing mathematics, you really have to understand what you are doing.

How do the publishing houses control the standard of typesetting?
They don’t. You can get your thing laid out in LaTeX and it will be very precise but not necessarily laid out beautifully. You may have the page sizes wrong and so on, but all of that could be dealt with automatically. The system is extremely flexible. But this means that every person who writes something also invents some of their own methods, so it becomes very hard for somebody else to deal with. Most publishing houses prefer to just translate it all into xml, which can be done.

Has this happened for your work?
Only for the last book I did. And that was a nightmare. They translated it into xml which means the only way I could correct it was, well, the way they wanted me to correct it was just the old fashion method of pen on paper! Because I didn’t have access to their xml to process it.

They couldn’t give you access?
Maybe they could have, but they didn’t. It was like going back into the dark ages. For Quantum Noise, the first book I typeset, before they were really into electronic publishing, we essentially agreed that I would do it all and get it phototypeset in Hamilton and send the things over to them and they would print it. This worked.

The next thing I did was the second edition of Quantum Noise with Peter. And by then LaTeX was considerably improved and more flexible. We wrote two more chapters, one of which was transcripts of lectures which were based on papers we had written. Peter had given the lectures. It meant we already had the LaTeX source. I sort of hacked it together and put the whole thing into LaTeX. It is not a LaTeX anyone would like to read. And so we had this book, which is in the current form it now is with some things purely in LaTeX, and some things which you might say ‘they are LaTeX’… but they have all sorts of strange relics of the previous form which has been redefined so as to be useable again in LaTeX. It would be an enormous amount of work to change it.

I actually wrote the files originally in Microsoft Word which had a facility for putting in Glossaries, that’s what they called them. I put in the whole LaTeX commands that way. I’d modified all the fonts so in the end it looked something like the mathematics you could read. You could read it!

You modified the fonts?
I changed the fonts. I made fonts of my own!

Can you introduce LaTeX to me?
Can I introduce LaTeX to you?

It would be helpful to see it.
Yes, well, I thought this would be the better way to start other than the historical way. (We look together at the LaTeX working environment).

See, these are all the Greek letters used, and they’ve all got codes and those codes are all just the Greek names. And these are things of which there are not many versions. This is the big limitation. Getting sets of the mathematical symbols that are compatible with the font you might want to use for the text. There are many fonts you might want to use for the text. It is not difficult to set up, but it is difficult to give them the matching mathematical characters.

Now, you notice \input{MasterDefs}. That is a lot of definitions and things I have made. This is the sort of thing you have to do to make it really work. For instance this sets out the size of the paper. That is a package someone else has written. If you have installed TeX it will be sitting on your computer ready to run.

• This
  \usepackage[cmex10]{amsmath}
  \usepackage{amssymb}
  says use a package from a set of fonts that the American Mathematical Society has defined, they are much more flexible than the LaTeX ones. These are other fonts.
• This
  \usepackage{theorem}
  is a particular package. . . I don’t know why it is there, I don’t think I need it.
• This
  \usepackage{graphicx}
  is important for including graphics inside that is part of the diagram.
• This
  \usepackage{psfrag}
  is a curious one, where, if you want to draw a diagram that has got symbols on it. Normally no drawing programme has got the mathematical symbols available. This means you can write code in PostScript, it has got to be PostScript, like in Adobe Illustrator. So you put a particular string of characters in the place where you want your mathematical expression to be, and it will substitute in a mathematical expression of your choice in the PostScript.
• This
  \usepackage{color}
  allows you to use colour.
• This
  \usepackage{cite}
  is for citations and so on.
• Now, this
  \usepackage{fourier}
  is the font.

Now, if I don’t use that, see, that is now a comment, and I go over to here and just reprocess it, it now has completely changed the font package. Now this is the standard font package you get.

This is like a pared-back version of inDesign?
It is a bit more than that. It’s non-visual. It has all the parameters you may wish. Some of it is shapes. Not all of it is. In TeX the typesetting is done using one of two modes.

Ordinary text is set in the “Text Mode”, which behaves more or less as in any other typesetter. It takes the text, and by adjusting the spaces between the words makes justified paragraphs, which are then arranged in one or more columns. The spacing between the letters in a word is determined by metric information about the font, this includes kerning. Blank lines (one of more) are treated as paragraph markers, while all other white spaces are interpreted as a single adjustable space. The rules governing the process can be modified in the style files, and to some extent within the input files. This can be necessary when a page turns out to be very difficult to get right, and some less than optimal choice must be made.

Mathematics is treated in “Math Mode”. Special Math Mode fonts are used for the equations, with metrics appropriate to mathematical use. White space is ignored, and layout is generated using the font metric information as well as some rather intricate rules related to how the character is used. Each ascii character is put in one of eight “classes” which determine this. Subscripts and superscripts, fractions, things like square roots, summation signs, etc are carefully (and automatically) adjusted. This complexity is an algorithmic representation of the empirical knowledge developed by traditional mathematical typesetters. Rules like, for example if you write x=y+z, the spacing around the + sign is different from and less than the spacing around the = sign. These rules are not hard wired, but can be modified in the style files. But this is something for experts in style creation, and is rarely done.

The overall TeX layout of a page is very powerful, but since it requires you to code everything, rather than adjust on a screen, it is not as easy to use as inDesign. But the math capabilities are unmatched by anything else. And the layout mechanism is capable of almost anything you can want, especially when augmented by the inclusion of PostScript and PDF graphics. From the very beginning TeX had a mechanism for inserting instructions for the typesetting device, for example PostScript code to be interpreted directly by the laser printer, and I did all the graphics in “Quantum Noise” using this mechanism.

What font package are we looking at?
Those all are called Computer Modern Roman fonts. Designed by TeX inventor Donald Knuth⁴ using Metafont. Computer Modern is a design based on a set of fonts designed sometime around 1900 for mathematical publishing. It was a Monotype setup. They were metal types in those days. So, this was a set of metal typefaces where mathematicians had been consulted, and a great deal of care had been taken to get it all matching. Tastes have changed since then. But this is what Knuth felt was the best to start with. Because he knew the decisions of a professional were made in it, and he felt the results looked correct.

Would you use Computer Modern?
No. It looks old fashioned. If you see it printed it has very very fine lines. It is not legible for me. The actual letters and the equations as well. I think everyone thinks it is a bit ugly. (We look through various papers hosted on arxiv.org)

What font would you use?
This is one I have used for several books, Utopia, it is from a package called Fourier, which is pretty good.

This type of situation here (pointing at the screen) with that gap in the line spacing, is this normal?
Well that is a consequence of having that rather big mathematical expression there. And those are things that turn up if you want to use it. Look at this symbol here: that ‘a’ has got a ‘superscript 3’, meaning it is ‘a cubed’. And a ‘subscript s’. And this square root symbol, you can see it could be squeezed down a bit. And that would require a lot of manual work to do that. That is a problem you run into. It would be best to write it in such a way that it wasn’t there. I wasn’t the person who put this together. And there are things like this here, if you want a comma after your equation, I would normally put a bit more space after a fraction like that. It looks sort of funny sitting down there.

How would you have typeset this paper differently?
I think I would have written that equation completely differently. I probably would have rewritten it in such a way that it was written as a displayed equation. That would have been the only solution to that problem. I know exactly what I would do!

And the bottom part of the square root, could that be higher?
No, that would be impossible. That is the symbol.

So there is no flexibility to respond to a situation like this?
You make your package decision and that is it. You can use as many fonts as you like. In terms of the mathematics there’s normally one set of fonts that have been chosen, but you can hard-wire other fonts in for particular purposes. I’ve always felt that the square root is a bit big. That’s the defect of this font, you might say.

Could the square root symbol be variable?
Well, it is variable in a sense. But not continuously.

And this is the problem, to fit all this in it has got to go to the second size set and that size is a bit too big for it. Depending on what is there it will expand to different shapes. We might find another example. I think we have lots of square roots in this paper. This was a very important paper really. But it took so long to solve the physics issues that nobody cared by the time we had it.

(We look at various work typeset by CW Gardiner)

That’s my original Quantum Noise. I’ve taken the original PostScript and turned it into a PDF. That was a bit tricky to do. It is Times Roman there. But there were maths fonts that were available as PostScript fonts which matched Times Roman quite well. In fact, this was the only choice that was available. And the Times Roman was in fact a bit too narrow to match it. So that’s expanded horizontally by 5%.

You modified the font?
Yes. It required essentially creating a new font, done in PostScript. At that time there were PostScript versions of only the maths fonts, and not of the other Computer Modern fonts. These were in fact based on an earlier version of the Computer Modern fonts, called the ‘Almost Modern fonts’, and this version was significantly different from Computer Modern. The difference was that the font was not nearly so fine, and the characters were narrower. As you can see, this is a different font altogether from this. It is sort of the right thickness. As you can see, it matches the Times Roman reasonably well.

The Times package that is available nowadays, as far as I remember, simply replaces the text font with Times, it doesn’t do anything to the maths.⁶ If I go back to here and reprocess it, you see that the maths fonts don’t match. They are too broad for the narrow Times Roman. And you see there, it looks sort of spidery sitting in the middle of a rather bold thing.

There are packages, like in the Fourier package where the basic text font is Utopia. The maths fonts use the Utopia fonts where they might exist. For example, they would use an italic p from Utopia, or possibly one that has been especially designed to fit it. But then you’ve got all the ones that don’t even turn up, like the Greek letter gamma and so on, they are not there. The Fourier ones are just the maths fonts but written to match with Utopia. I have written several books with it. I don’t think there are really many improvements it needs. But the range of packages… you see, if I go back here, you can see that list of packages, see here, there’s one I think is awful.

You’ve labelled the package as ‘awful’.
I can’t remember why I said it was awful. But I’m sure if I said it was awful then it is. Let’s look. Yes. The maths is sans but not the text. It is maths that looks wrong to me.

How does this look wrong to you?
It is just not how we do maths. For instance, the Greek is the wrong density compared to the roman. It is too fine and narrow. It is just awful. The text comes out perfectly legible, but in terms of how to write maths, I don’t think that is how it should be done. If you are going to do a sans serif maths thing, you have to do a lot more work than this. You have to actually design some sans serif Greek fonts of the right sort.

And are there not sans serif Greek fonts of the right sort?
If there are then the person who designed this package hasn’t put them in.

The thing I want to emphasise, going back to here, Look! (we look at a list of around eight packages). That is about as many choices as there are that really work. There are not many and some of them don’t work very well. The people who do this are not very expert in terms of actual font and text design. It’s all individuals, open software, none of it is commercial. There aren’t many people that do mathematical fonts. There was a major project to get a very good set, called Stix. It was designed by the American Mathematical Society. I don’t think it is very successful. I think that is awful. (pointing to the Greek capital sigma).

That sigma symbol is not meant to be so black?
Well, that is the blackest I’ve ever seen it. Stix is sort of a ‘Times Roman look’, but with a complete set of mathematical fonts to match it. I don’t like it.

The reason Times became popular in computer typesetting is that the first laser writers had Times, Helvetica, Courier and Symbol. There were very few fonts, and these were the first that came out and made it seem like the standard. They were built into PostScript. It seems to me the decision to make a font, Stix, that looks sort of like Times was not a good decision. I think this was an enormous project – it went on for years, with all the symbols you could want. But I don’t think it really works. Whenever I’ve used it, I thought, I think it would look better in Utopia.

Utopia is your preferred font?
For this sort of thing, yes. I always check when things come out, but it is really limited by the maths fonts. All the fonts I’m using are in some sense free. But Utopia is a commercial font, they have given the right to use it to the LaTeX community.

I think we need a visual illustration of the various package options.
How big do you want this project to be?

Not big. It’s an opportunity for you to express your thoughts. In that list you only have about eight different package options, that’s not much.
That’s a list of typeface choices, not layout styles.

Let’s stick with typefaces. Could we do a comparison? So people can see them, everyone at ECAL is visual.
Well, you’d only need a few to get the whole point. If it was me… (pause) I would take the standard default Computer Modern, because that is what everyone does. Then there is the one I prefer, the Fourier package with Utopia. And then there is like the Stix one… you look at that and you think, this is the result of an enormous project and yet it looks ugly. And then there is the one where they are trying to do it in sans serif… And maybe there is one more example worth showing.

Five examples?
Yes.

Day 2

CW Gardiner generates the first pages of a paper currently in progress, ‘Non-Lindblad Master Equations’, in five different LaTeX font packages.

Package 1: Fourier

Now this one is Fourier. It uses the font Utopia for the text, and the mathematics is rendered with the Fourier fonts, which are a set of fonts specially designed to match Utopia.

This is the standard package I would use, I’ve written three books in it. I can’t think of anything I would say are defects. It is relatively bold compared to some fonts and it is very easy to read on screen. It is relatively dark. It is a thick sort of a font, you don’t have to find things that get lost.

Now, we can look through and see if there are any other things we care to comment on. This Fourier package is well made. You can see the mathematical fonts and the Fourier have almost, as far as I can tell, exactly the same style, even though one is italic and one is roman. You can see them there. They match perfectly.

There is one thing I do object to in the default version of this package, which is not present here, because I haven’t used it. But if you look for calligraphic mathematical capitals, that is fancy curly Ds and so on… Hmm how would you write a calligraphic ‘D’? Can you draw a calligraphic D, a curly fancy D?

(Nell draws a capital calligraphic ‘D’).

That is not nearly fancy enough. I will show you in LaTeX, it is really a quirk of the LaTeX itself. Here, this is the kind of calligraphic style they use. Now, other packages interpret calligraphic fonts very differently. Some interpret them as copperplate script.

Why don’t we have an example of the calligraphic letters here?
I can put it in. We will just put it at the end as an Appendix.

Now those are the sorts of fonts that LaTeX produces naturally, just capitals, and only used in the mathematics. [Appendix: B ii)] This is what the Fourier package gives by default. And that one I find is not very good for mathematics. They take up too much space.

There is something else. I will take some text and put this in, this will come out in the text interpreted as italics, which you will see looks quite strange. [Appendix: B i)] The spacing and even the shapes of italic letters are optimised for mathematics. Now I want to finish illustrating the point. I will put it now as it would be in text. [Appendix: A ii)] See, that gives you exactly the same thing in ordinary italics, and you can see, there is considerable difference. Even though they match in terms of density pretty well, these are designed for reading, and these are designed to be used as symbols.

This difference always happens across every possible package. The maths fonts are always different from the text fonts for italics. There are roman fonts used in maths now and again, and there is a maths roman option, but it is not so obviously different. Typically you don’t use roman much in equations, you might use one or two. The maths italics are set wider and their widths are variable in a way that is different.

Is there anything you would want to be different in the Fourier package?
Something that had a more elegant look would be nice. It has an absolutely consistent package that is very legible, but it gives the book a certain character and it is not always the character that you necessarily want.

Do a lot of other people use this package?
I find that very hard to say. You will find published books using Fourier, but most people don’t bother to use anything but the default. In general they are preparing manuscripts which will be reprocessed in one way or the other to be published. LaTeX is the undisputed best way to present exactly what the maths should look like, and to them that is enough. They don’t do things that are meant to be published as they are created, as I do.

Package 2: Stix

Now this one as you know is designed to be sort of like Times. In Times, you will find things, like the xs you see in [Sect.1 equations 4 and 5] the mathematical fonts are designed to match the text ones, and they have shapes that I find are not quite appropriate for maths. They tend to be a bit extravagant with using space. It tends to produce a very wide line of maths, it uses a lot of space, whereas Utopia tends to be much more compact. And these equations can get very long very easily. You are constantly faced with the fact, how do I get this to look exactly right in the space available? You do not want nearly as much space between the characters as they have here.

It should be borne in mind as a parenthetical fact that in TeX there are four different spacing versions used. This is the broadest one, it is the full displayed equation, things are spaced out a bit more. This here is inline, and the spacing tends to be less. You won’t see it much in that.

Here are the results – they use the same code, but it is interpreted differently:

It also tends to make the symbols smaller vertically. Like integrals and so on. That’s standard for all packages. Then there are a couple of different ones which are slightly similar but more cramped again. But certainly the distinction between Displayed and Inline is significant. If you were to put, say, something like pi squared over two to the n, it would reduce the font size and make it smaller, and I think that is a bad decision, but that is what it does.

Do you mean just in this package?
Do you mean just in this package?No, that is always the case. But exactly how much it does is defined in the package. When I put equations like that Inline example, I would compose the fraction differently in a way that has the same mathematical meaning but is formatted more smoothly.

Is this a package that is often used by other people?
The number of people that use anything other than the default (Computer Modern) is quite small. I think the Sigma is too thick in [Sect. 2 i) c] I also do not like the shape of this partial derivative sign, this curly thing here in equation (1). I think it does not look like the right shape. This may be a personal judgement as several fonts do put it with this shape. In the Fourier package it was the right shape.

Are there circumstances where you would use this package?
I have tried it. It tends to produce longer files. And I have always come to the conclusion that I don’t like it.

Is there anything else you would like to say about Stix?
It is a completely comprehensive solid package but I don’t think its aesthetics are quite right. Getting these vertical bars outside of brackets. [Sect. 2 ii) b] I think that Stix does do that better than Fourier. In Fourier they look not properly spaced, somehow.

Package 3: default⁷

The default choice of fonts is Computer Modern, which provides a standard set of text fonts. That is roman, bold, bold extended, italic, even slanted, two different kinds of numbers, a modern style and old style. Everything you might want for printing in a consistent set of fonts of ordinary text. Then, there are three maths fonts: CMSY, which is symbols; CMMI, which is just letters; and then there is the CMEX which is all sorts of integrals and brackets, bits and pieces. But I think it looks spidery and old fashioned.⁸

And this is the default and therefore what most people would use?
It is the default, yes. This was done in the 1980s, when people started making computer software with which they could also typeset their own book. So probably, the decisions were simply not informed enough by typographical opinion. I’m not entirely sure.

In terms of its actual use, for example, if we look at these probability things. [Sect. 2, equations 10–12], those are right, those bars and so on. That is exactly how it should be. If you look at any mathematical decisions, they are very good. It is certainly not a good font for the screen, but when you see it printed on paper in its original form, it looks pretty good. In the first copies of Knuth’s book on this, the TeXbook, the Metafont book and the TeX programme done by Addison Wesley using those fonts, it all looks quite elegant. But in everyday use it doesn’t seem to achieve that same elegance. It must be a package suited to his style.

Do you ever use this package?
No. I find it too hard to read. But it’s guaranteed that it has every feature you need. When you go to the other ones sometimes you find they haven’t implemented something.

Package 4: sfmath, sfdefault

There are two packages here. sfmath replaces serif fonts with sans serif fonts in the math only. The other package, sfdefault, is one I made up to replace the serif fonts with sans serif fonts in everything other than the math. In fact it is very simple, only one line of code, which arises from the rather clever way the whole LaTeX package is structured.

When I look at this, their maths fonts don’t really match. The maths Greek doesn’t match the sans. That is not a sans Greek. These are clearly just the ordinary Computer Modern fonts for Greek. It is a lazy package job.

If there was a package made to the standard you require, would you want to use sans?
They can be quite useful for presentations in a theatre on a big screen. Sans tends to be more legible for small bits of text.

And for this context as an article, would you use a well-made sans?
No. I think for reading sans fonts are not the best thing to use. You get a better feeling from proper fonts with serifs. That is my opinion. But I don’t think this is a very good sans font. I think it might be the Computer Modern sans font.

Are you able to explain why it is not good?
No, I can’t.

It is just a feeling you have?
When I use a sans font, which sometimes I do, I prefer Helvetica. I haven’t been particularly fussy about that. I don’t tend to use sans much. But this here has the look of an old fashioned sans font. This is probably my prejudices. So, you can say it can be done, but to me it doesn’t look like correct mathematics, even though it is of course. It is just that nobody would do mathematics that way. There are also different conventions when you go to different languages. The French conventions are different from English, and there are Spanish conventions which are different too.

Do you mean in the mathematics?
Yes. Which sort of font you use for which sort of thing. French tends to use upright Greek whereas English tends to use sloping Greek. There are other things too. There was a man called Yannis Haralambous⁹ who wrote a complete TeX implementation of mathematics¹⁰ for the French Mathematical Society for them to use in their published journals. He was just basically taking what they traditionally used with metal type and trying to see if they could make it work with this sort of thing, and he succeeded—the result is very elegant.

Package 5: concrete

These concrete fonts are supposed to look like what a mathematician would do on a blackboard if they had very good handwriting. I can’t quite see it myself. But this package does yield the correct calligraphic fonts I use. You look at that and you think, well, if you made a presentation of that on the screen that might be quite good. It is sort of upright maths there. Of course, when you just write your maths on the blackboard you don’t use italics. It’s got a certain swish to things. I should point this out I suppose.

So I always make it this way now so it is not confused. When you are writing handwriting you have to be careful to distinguish which is which. And you will find that is true in the mathematics fonts done properly. There is some care to make sure they really are distinctive. You’ll find they use that sort of ‘g’ in the italics, the printed ‘g’ that is common in text. This is not commonly used in mathematics but Fourier does it, and it is ok. And of course italic ‘a’ is usually that one for mathematics, but some do use a double story ‘a’.

Going back to Concrete again. You wouldn’t want to write a book in this, or even a paper. It is really something for presentations.

What was it specifically designed for?
It’s hard to say. When it was designed in the 1980s, people weren’t giving presentations with Powerpoint. They used overhead transparencies which they produced by hand. I’ve not seen this package used much at all. It’s quite legible, almost like a typewriter font. It does have differences in thickness. But not serious differences in thickness. The thing you can say about this font is that its maths is correct. Like all the things Knuth did in mathematics, he got them right.¹¹

I think we have reached the end.
I think so too.¹²