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
• He had already published the first editions of Volume 1, 2, 3, and the second edition of Volume 1, by 1973. It was in 1977 when the publishers sent him galley proofs for the second edition of Volume 2, having switched to phototypesetting (away from hot-metal typesetting a la Linotype, though IIRC it was actually Monotype) that he was disappointed with the results. And he had some back-and-forth with them and they did improve their fonts (https://tex.stackexchange.com/a/367133/48), but he was still dissatisfied.
> I didn't know what to do. I had spent 15 years writing those books, but if they were going to look awful I didn't want to write any more.
• At this time he came to know of the existence of digital typesetters. Typesetting with computers had existed before, but it had always seemed a crude toy, rather than something suitable for “real books”. But he saw Patrick Winston's Artificial Intelligence that had been just published (I think he got an early proof copy to review or something), and he realized for the first time that digital typesetting was an option (apparently Winston's book was printed at >1000dpi, and Knuth later got his hands on a machine that claimed a resolution of 5333 dpi: see this wonderful comment from Knuth's student and “right-hand man”, David Fuchs: https://news.ycombinator.com/item?id=20009875)
• In fact it was the fonts that he was dissatisfied with rather than the typesetting, so METAFONT was in some sense the primary/motivating project and TeX was only written in order to be able to use METAFONT.
• Actually his first idea was to simply take the old fonts, get high-resolution scans of them (not easy to obtain at that time) and use them directly. He approached Xerox Research Center but:
> I asked if I could use Xerox's lab facilities to create my fonts. The answer was yes, but there was a catch: Xerox insisted on all rights to the use of any fonts that I developed with their equipment. Of course that was their privilege, but such a deal was unacceptable to me: A mathematical formula should never be "owned" by anybody! Mathematics belongs to God.
• So he went home and (after trying a bit with TV cameras) tried projecting photographs of the pages onto the wall and tracing the outlines, and it was while staring at these images that he realized that the shapes of letters were not arbitrary but there was some logic to them (e.g. in the font he was using, the spacing between the vertical strokes in 'm' was equal, and equal to that in 'n'), and he decided (as a computer programmer) to capture this design in code — something that had never before been done. The hardest letter to capture this way is S, hence the paper in the OP.
> Finally, a simple thought struck me. Those letters were designed by people. If I could understand what those people had in their minds when they were drawing the letters, then I could program a computer to carry out the same ideas. Instead of merely copying the form of the letters my new goal was therefore to copy the intelligence underlying that form. I decided to learn what type designers knew, and to teach that knowledge to a computer.
• This is also why METAFONT never really caught on among typographers: as Charles Bigelow (quoted by Richard Southall, https://luc.devroye.org/Southall-METAFONT1986.pdf) observed, “the designer thinks with images, not about images”. Knuth did not want crude “geometric” constructions of letters (as some prior 16th century typographers had attempted: https://www.ams.org/journals/bull/1979-01-02/S0273-0979-1979... and as some typographers only passingly familiar with METAFONT think!). He wanted actual real typographically beautiful shapes, but to be able to generate those shapes with code. This is obviously much harder than simply drawing the shapes using visual intuition, even if it enables variation. (See “The Concept of a Meta-Font”: https://gwern.net/doc/design/typography/1982-knuth.pdf — again, many people in the typography world confuse the abstract concept of a meta-font introduced in this paper with (their incorrect impressions of) the METAFONT program, and omit crediting Knuth for variable fonts).
• The second edition of Volume 2 was not printed with Linotype. Yes the machines still existed in Europe and he talked to typesetters (he mentions in particular a person from Belfast), but it was in fact published using TeX (the first version, TeX77 and MF78). He was still unhappy with the results, though, and spent a few more years learning more about typography and working with people like Bigelow and Hermann Zapf, before the rewrite into the current TeX82 and MF84 (and current version of Computer Modern). I think it's only with the third edition (1997) that he's finally satisfied.
Not so about the Linotype. Back in 1980, I personally ran the Alphatype CRS phototypesetter (bought by DEK for the purpose), in the basement of Margaret Jacks Hall, that produced the entire camera-ready copy of The Art Of Computer Programming, Volume II, Second Edition. The DVI files and Computer Modern fonts were created by the early, Sail-language, 36-bit versions of TeX and Metafont that were later redesigned and implemented to be more cross-platform. Knuth rewrote the firmware that resided on the Alphatype (in 8080 assembly language), and I wrote the code that translated from DVI and drove it from the DEC20 mainframe over a serial line (trickier than it sounds; see our joint paper "Optimal prepaging and font caching" ACM TOPLAS Vol 7 Issue 1).
I just want to add that they are a gorgeous set of books and I am so happy he did this. While a good chunk of the content is above my pay grade it is still enjoyable to flip through them and read about things like MIX. Gorgeous typesetting. And his writing is so very engaging for such a dense topic.
All of his books have been great, and he is the next author whose oeuvre I am going to try to collect (and read) in its entirety --- will finish up Tolkien this year when the second Myths and Legends box set is released in the U.S. and I can replace my ratty photocopy of _The Book of Exodus_ which was sent to me the second time I requested it on Interlibrary Loan.
Note that they weren't directly printed with a hot metal casting machine (I believe Monotype was used, since it output discrete letters which did not need to be sawn apart as the Line of Type the competitor used), but rather what could be composed on a hot metal casting machine was, then additional spacing material and special characters and extensions which weren't available from that keyboard were sourced and the whole put together as a composed galley, then a proof was pulled and once approved, printed, then photographed to make a negative which was then used to make an offset plate for actually printing the book.
The other question is how much innovation is the result of spite? The professor who told someone they're never going to amount to anything, the business person that told someone "that's never going to work".
Butterfield [1] started a company to develop a game, but during their interactions the employees created some photo sharing code which ended up as Flicker, which he then sold off. Butterfield then started another company to develop a game, but to make internal company communications more efficient they developed some chat software which ended up as Slack, which he then sold off.
I have not heard if Butterfield has started a third company to develop a video game.
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).
⸻
1. Funnily enough this is the second time in two days that I’ve shared this article, albeit in different contexts.
> was Metafont the only outline-based font technology
Surely Karow's Ikarus was earlier than that.
One of the main innovations of Metafont was the use of "pen"s, so that one would describe a single path and the software would trace it and imitate the use of one or more pens, to end with an outline of something with thickness, and essentially more curves. It mimics how drawing and writing actually happens.
AFAIK, Zapf did not like this approach at all, as he was used to design typefaces the traditional way, by specifying all the curves. Richard Southall embraced the new paradigm and used Metafont as it was supposed to be used, but produced only a couple of demo typefaces (mainly the nmt family) and a handful of commercial ones (I can now only remember Colorado, with Ladislas Mandel, used in the phone directories of US West). I think he also implemented Melior, but of course this was never distributed as it was a proprietary Zapf design.
Note: all the above are based on recollections of my discussions with Zapf, Southall, and Knuth, in the distant past. All my relevant printed materials are in a different country right now, and I don't have easy access to them.
Yeah, _Digital Typefaces_ by Peter Karow was first published in 1986 (in German) and the company had been running for a long while before then, the software having been introduced at ATypI in Warsaw in 1975.
_TeX and METAFONT_ (the book) was published in 1979 (I still vividly remember checking a copy out of the local college library as a high school student in 1983) having its initial release in 1978, after being precipitated by the infamous second edition galley proofs on TAoCP 30 March 1977.
Ah, I suspected it wasn’t the first, it’d just been a long time since I dealt with some of this history. 30 years ago, I had the entirety of Gutenberg to Emigre at the front of my brain but much of it has vanished since.
Yeah, Barbara Beeton was kind enough to send me a copy of the AMS report for which I am _very_ grateful.
Didn't mean for my post to come across as cavalier --- it's a _very_ tough row to hoe, and even now, I don't think that there are good solutions in this space (but I haven't checked for a while, been out of the typography scene for a while now --- I'd love to be wrong). Ironically, my current project
is circling back to the underpinnings of this sort of thing (I need to make a single stroke font so as to make it easier to set text in CNC projects) and I'm hoping to approach this from the bottom up and eventually arrive at a visual and interactive version of METAFONT/POST which will also work as a general-purpose drawing program (so that I'll have one to use when I can no longer use Freehand/MX) --- hopefully that will then allow me to finish a compleat digital version of Warren Chappell's typeface designs as we discussed peripherally ages ago.
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.
I fear we'll never see another Donald Knuth... even if there were somebody else like him (and maybe there isn't!), there'd be nowhere for him to go in today's world.
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.
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
Actually:
• He had already published the first editions of Volume 1, 2, 3, and the second edition of Volume 1, by 1973. It was in 1977 when the publishers sent him galley proofs for the second edition of Volume 2, having switched to phototypesetting (away from hot-metal typesetting a la Linotype, though IIRC it was actually Monotype) that he was disappointed with the results. And he had some back-and-forth with them and they did improve their fonts (https://tex.stackexchange.com/a/367133/48), but he was still dissatisfied.
> I didn't know what to do. I had spent 15 years writing those books, but if they were going to look awful I didn't want to write any more.
• At this time he came to know of the existence of digital typesetters. Typesetting with computers had existed before, but it had always seemed a crude toy, rather than something suitable for “real books”. But he saw Patrick Winston's Artificial Intelligence that had been just published (I think he got an early proof copy to review or something), and he realized for the first time that digital typesetting was an option (apparently Winston's book was printed at >1000dpi, and Knuth later got his hands on a machine that claimed a resolution of 5333 dpi: see this wonderful comment from Knuth's student and “right-hand man”, David Fuchs: https://news.ycombinator.com/item?id=20009875)
• In fact it was the fonts that he was dissatisfied with rather than the typesetting, so METAFONT was in some sense the primary/motivating project and TeX was only written in order to be able to use METAFONT.
• Actually his first idea was to simply take the old fonts, get high-resolution scans of them (not easy to obtain at that time) and use them directly. He approached Xerox Research Center but:
> I asked if I could use Xerox's lab facilities to create my fonts. The answer was yes, but there was a catch: Xerox insisted on all rights to the use of any fonts that I developed with their equipment. Of course that was their privilege, but such a deal was unacceptable to me: A mathematical formula should never be "owned" by anybody! Mathematics belongs to God.
• So he went home and (after trying a bit with TV cameras) tried projecting photographs of the pages onto the wall and tracing the outlines, and it was while staring at these images that he realized that the shapes of letters were not arbitrary but there was some logic to them (e.g. in the font he was using, the spacing between the vertical strokes in 'm' was equal, and equal to that in 'n'), and he decided (as a computer programmer) to capture this design in code — something that had never before been done. The hardest letter to capture this way is S, hence the paper in the OP.
> Finally, a simple thought struck me. Those letters were designed by people. If I could understand what those people had in their minds when they were drawing the letters, then I could program a computer to carry out the same ideas. Instead of merely copying the form of the letters my new goal was therefore to copy the intelligence underlying that form. I decided to learn what type designers knew, and to teach that knowledge to a computer.
• This is also why METAFONT never really caught on among typographers: as Charles Bigelow (quoted by Richard Southall, https://luc.devroye.org/Southall-METAFONT1986.pdf) observed, “the designer thinks with images, not about images”. Knuth did not want crude “geometric” constructions of letters (as some prior 16th century typographers had attempted: https://www.ams.org/journals/bull/1979-01-02/S0273-0979-1979... and as some typographers only passingly familiar with METAFONT think!). He wanted actual real typographically beautiful shapes, but to be able to generate those shapes with code. This is obviously much harder than simply drawing the shapes using visual intuition, even if it enables variation. (See “The Concept of a Meta-Font”: https://gwern.net/doc/design/typography/1982-knuth.pdf — again, many people in the typography world confuse the abstract concept of a meta-font introduced in this paper with (their incorrect impressions of) the METAFONT program, and omit crediting Knuth for variable fonts).
• The second edition of Volume 2 was not printed with Linotype. Yes the machines still existed in Europe and he talked to typesetters (he mentions in particular a person from Belfast), but it was in fact published using TeX (the first version, TeX77 and MF78). He was still unhappy with the results, though, and spent a few more years learning more about typography and working with people like Bigelow and Hermann Zapf, before the rewrite into the current TeX82 and MF84 (and current version of Computer Modern). I think it's only with the third edition (1997) that he's finally satisfied.
Not so about the Linotype. Back in 1980, I personally ran the Alphatype CRS phototypesetter (bought by DEK for the purpose), in the basement of Margaret Jacks Hall, that produced the entire camera-ready copy of The Art Of Computer Programming, Volume II, Second Edition. The DVI files and Computer Modern fonts were created by the early, Sail-language, 36-bit versions of TeX and Metafont that were later redesigned and implemented to be more cross-platform. Knuth rewrote the firmware that resided on the Alphatype (in 8080 assembly language), and I wrote the code that translated from DVI and drove it from the DEC20 mainframe over a serial line (trickier than it sounds; see our joint paper "Optimal prepaging and font caching" ACM TOPLAS Vol 7 Issue 1).
This guy Fuchs.
I just want to add that they are a gorgeous set of books and I am so happy he did this. While a good chunk of the content is above my pay grade it is still enjoyable to flip through them and read about things like MIX. Gorgeous typesetting. And his writing is so very engaging for such a dense topic.
All of his books have been great, and he is the next author whose oeuvre I am going to try to collect (and read) in its entirety --- will finish up Tolkien this year when the second Myths and Legends box set is released in the U.S. and I can replace my ratty photocopy of _The Book of Exodus_ which was sent to me the second time I requested it on Interlibrary Loan.
Note that they weren't directly printed with a hot metal casting machine (I believe Monotype was used, since it output discrete letters which did not need to be sawn apart as the Line of Type the competitor used), but rather what could be composed on a hot metal casting machine was, then additional spacing material and special characters and extensions which weren't available from that keyboard were sourced and the whole put together as a composed galley, then a proof was pulled and once approved, printed, then photographed to make a negative which was then used to make an offset plate for actually printing the book.
Suddenly, I don't feel so bad about my own procrastinitis.
How many great inventions and discoveries were the product of yak shaving? I'd imagine quite a lot.
The other question is how much innovation is the result of spite? The professor who told someone they're never going to amount to anything, the business person that told someone "that's never going to work".
Wasn’t that the story behind Slack?
Butterfield [1] started a company to develop a game, but during their interactions the employees created some photo sharing code which ended up as Flicker, which he then sold off. Butterfield then started another company to develop a game, but to make internal company communications more efficient they developed some chat software which ended up as Slack, which he then sold off.
I have not heard if Butterfield has started a third company to develop a video game.
[1] https://en.wikipedia.org/wiki/Stewart_Butterfield
You can still hear the music written for the game 'glitch' if you join a slack audio room with nobody else in it.
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_:
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).
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).
⸻
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.
> was Metafont the only outline-based font technology
Surely Karow's Ikarus was earlier than that.
One of the main innovations of Metafont was the use of "pen"s, so that one would describe a single path and the software would trace it and imitate the use of one or more pens, to end with an outline of something with thickness, and essentially more curves. It mimics how drawing and writing actually happens.
AFAIK, Zapf did not like this approach at all, as he was used to design typefaces the traditional way, by specifying all the curves. Richard Southall embraced the new paradigm and used Metafont as it was supposed to be used, but produced only a couple of demo typefaces (mainly the nmt family) and a handful of commercial ones (I can now only remember Colorado, with Ladislas Mandel, used in the phone directories of US West). I think he also implemented Melior, but of course this was never distributed as it was a proprietary Zapf design.
Note: all the above are based on recollections of my discussions with Zapf, Southall, and Knuth, in the distant past. All my relevant printed materials are in a different country right now, and I don't have easy access to them.
Yeah, _Digital Typefaces_ by Peter Karow was first published in 1986 (in German) and the company had been running for a long while before then, the software having been introduced at ATypI in Warsaw in 1975.
_TeX and METAFONT_ (the book) was published in 1979 (I still vividly remember checking a copy out of the local college library as a high school student in 1983) having its initial release in 1978, after being precipitated by the infamous second edition galley proofs on TAoCP 30 March 1977.
Ah, I suspected it wasn’t the first, it’d just been a long time since I dealt with some of this history. 30 years ago, I had the entirety of Gutenberg to Emigre at the front of my brain but much of it has vanished since.
Yeah, Barbara Beeton was kind enough to send me a copy of the AMS report for which I am _very_ grateful.
Didn't mean for my post to come across as cavalier --- it's a _very_ tough row to hoe, and even now, I don't think that there are good solutions in this space (but I haven't checked for a while, been out of the typography scene for a while now --- I'd love to be wrong). Ironically, my current project
https://github.com/WillAdams/gcodepreview
is circling back to the underpinnings of this sort of thing (I need to make a single stroke font so as to make it easier to set text in CNC projects) and I'm hoping to approach this from the bottom up and eventually arrive at a visual and interactive version of METAFONT/POST which will also work as a general-purpose drawing program (so that I'll have one to use when I can no longer use Freehand/MX) --- hopefully that will then allow me to finish a compleat digital version of Warren Chappell's typeface designs as we discussed peripherally ages ago.
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.
Maybe another old example of this scanner bug?
https://www.dkriesel.com/en/blog/2013/0802_xerox-workcentres...
I just spent 30 minutes reading a detailed mathematical version of "draw an S; next draw a more different S".
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.
I fear we'll never see another Donald Knuth... even if there were somebody else like him (and maybe there isn't!), there'd be nowhere for him to go in today's world.
How would Knuth opine on consummate Vs, I wonder...
Trogdor!
Burninating the countryside.
Burninating the peasants.
Burninating all the peoples.
This is wonderful.
Having models attempt an SVG letter S remains one of my personal/informal LLM benchmarks. They are still pretty bad at it.
Wonderful man, here is a lecture on the topic from Joint Mathematics Meeting, Étienne Ghys. https://www.youtube.com/watch?v=1OIxzewWilc
Knuth is just a treasure.
I would love to see Donald Knuth on Seasame Street..
I used to design fonts back in the 90's — I always designed the S first, because if I couldn't get the S to work, there was no fuckin' point.
It's not clear to me why the S is more difficult than the others.
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.
As Knuth points out, applying math to art is applying artistic sense to a continuum of forms all at once.
That's a nice quote.
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
+1 for "S has a bit of reputation"
How does 's' compare to other languages' letters - e.g. arabic or chinese - in terms of difficulty?
Fender v. Knuth inbound.
(1980)
Added. Thanks!