This article was in Springer's The Mathematical Intelligencer in 1980. The next article in that volume was "Strange Attractors" by David Ruelle. When I read Ruelle's article in the early 1980s, I noticed Knuth's article. By the time I got to writing my third paper on strange attractors in 1988, I was using TeX.
His book _TeX and METAFONT_ (about the initial public release) goes into these difficulties in greater detail and includes the charming response by his wife when shown some initial efforts:
>Why don't you make them _S_ shaped?
To some degree, this problem was eventually solved, c.f., the five volume set _Computers and Typesetting_:
but then one had the effort to create a new typeface set for math equations by the AMS, eventually named Euler as written up in "AMS Euler — a new typeface for mathematics". _Scholarly Publishing_ and so forth, but arguably, things went awry in that rather than capture the ductus of Prof. Zapf's pen, and model based on that stroke and a pen shape, the expedient approach of simply modeling the outline was arrived at and implemented due to the difficulty and lengthy time required for the idealized approach.
Another consideration may have been that there doesn't seem to be an available algorithm which is robust and accurate and automatic for determining the curves which describe the union of arbitrary Bézier curves (some projects get around this by making high resolution pixel images and tracing them).
For the work on Euler, this article¹ (https://www.sciencedirect.com/science/article/pii/S240587262...) goes into the whole Digital Typography program at Stanford for whom, one of the projects was creating those outlines for the Euler math fonts. It’s worth remembering that at the time, not only was Metafont the only outline-based font technology,² but things like scanners were rare to nonexistent and the bitmaps that were used to determine coordinates for the curves of the fonts were hand-drawn on fine-lined graph paper (and sent to Hermann Zapf for approval).
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1. Funnily enough this is the second time in two days that I’ve shared this article, albeit in different contexts.
Knuth touches the reason for writing TeX et al.- briefly in this paper. He wanted the second edition of his TAOCP to be printed with exactly the same typography as the first edition, but the publishers told him at first they wouldn't have the Linotype machines anymore, with which they printed the first edition. But Knuth wanted to preserve the typography for the other volumes and other editions, so he set aside the TAOCP and began researching typography and writing TeX et al. Took him a long time before he could return to TAOCP. Btw, the second edition got finally printed with Linotype as these machines still existed in Europe
I was just reading about Metafont the other day, so this was quite lovely to come across.
Fig 9 stood out to me as obviously wrong. The two glyphs on the left are pixel by pixel identical, as are the three middle ones, and the two on the right. Quite mysterious though considering this PDF appears to be a scan.
To spend nine pages full of mathematical formulae just to write a single letter (more) nicely shows the rigor/perfectionism/OCD that is the hallmark of Donald Ervin Knuth.
Thanks for giving us beautiful layout and better-looking fonts.
Every time I write a new paper when I press "compile" in Overleaf I'm greatful that he made our work more beautiful, and it motivates me to make the content matter, too.
Among the uppercase letters, "S" is the only one that's not either based on a single primary ellipse, or combined with straight lines that provide structural constraints. The entire glyph is governed by continuously changing curvature with no stabilizing axis or primitive shape to enforce proportions.
This results in a more complex and less obvious mathematical definition.
Also, a naively symmetrical "S" tends not to look good, probably because of these same issues, so needs further adjustment to match our visual expectations. This complicates the definition further.
If you do lettering by hand, S has a bit of reputation. It's hard to get right. Small mistakes stand out even to an untrained eye. However, once you do get it (subjectively) right, it's an extremely beautiful letter. This is even more true in hands like cancellaresca corsiva (the so called "italic font") where the letters are somewhat smooth flowing rather than built with rigid lines. More interestingly, because of all these parameters, you can play a lot with the letter especially if you want to do it as a drop cap or ornament it. As an example, making the lower bowl bigger as the the above comment invites gives the letter some personality.
This is all fine. What fascinates me with Knuth's work is how he applies mathematical rigour to concepts like these which are generally considered "artistic" and subjective. It underlines how mathematical ideas of symmetry etc. play a role in making the world we live in beautiful.
>Why don't you make them _S_ shaped?
To some degree, this problem was eventually solved, c.f., the five volume set _Computers and Typesetting_:
https://www-cs-faculty.stanford.edu/~knuth/abcde.html
but then one had the effort to create a new typeface set for math equations by the AMS, eventually named Euler as written up in "AMS Euler — a new typeface for mathematics". _Scholarly Publishing_ and so forth, but arguably, things went awry in that rather than capture the ductus of Prof. Zapf's pen, and model based on that stroke and a pen shape, the expedient approach of simply modeling the outline was arrived at and implemented due to the difficulty and lengthy time required for the idealized approach.
Another consideration may have been that there doesn't seem to be an available algorithm which is robust and accurate and automatic for determining the curves which describe the union of arbitrary Bézier curves (some projects get around this by making high resolution pixel images and tracing them).
⸻
1. Funnily enough this is the second time in two days that I’ve shared this article, albeit in different contexts.
2. As far as I know, although I could be wrong.
Fig 9 stood out to me as obviously wrong. The two glyphs on the left are pixel by pixel identical, as are the three middle ones, and the two on the right. Quite mysterious though considering this PDF appears to be a scan.
https://www.dkriesel.com/en/blog/2013/0802_xerox-workcentres...
Thanks for giving us beautiful layout and better-looking fonts. Every time I write a new paper when I press "compile" in Overleaf I'm greatful that he made our work more beautiful, and it motivates me to make the content matter, too.
Burninating the peasants.
Burninating all the peoples.
This results in a more complex and less obvious mathematical definition.
Also, a naively symmetrical "S" tends not to look good, probably because of these same issues, so needs further adjustment to match our visual expectations. This complicates the definition further.
This is all fine. What fascinates me with Knuth's work is how he applies mathematical rigour to concepts like these which are generally considered "artistic" and subjective. It underlines how mathematical ideas of symmetry etc. play a role in making the world we live in beautiful.
I remember seeing an animated documentary as a child called "Donald in Mathmagic Land" which ends with a quote attributed to Galileo.
> Mathematics is the language with which God has written the universe