4 ms·

The Letter S, by Donald Knuth (1980) [pdf]

304 points by bambax 5mo ago

bombcar 5mo ago
I just spent 30 minutes reading a detailed mathematical version of "draw an S; next draw a more different S".
- philringsmuth 5mo ago
  Trogdor!
  - herodoturtle 5mo ago
    Burninating the countryside. Burninating the peasants. Burninating all the peoples.
- jll29 5mo ago
  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.
  - commandlinefan 5mo ago
    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.
- trbleclef 5mo ago
  How would Knuth opine on consummate Vs, I wonder...
WillAdams 5mo ago
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 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).
- dhosek 5mo ago
  For the work on Euler, this article¹ (https://www.sciencedirect.com/science/article/pii/S240587262030068X 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.
  - WillAdams 5mo ago
    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 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.
adzm 5mo ago
(1980)
- dang 5mo ago
  Added. Thanks!
mrandish 5mo ago
Knuth is just a treasure.
adm4 5mo ago
Wonderful man, here is a lecture on the topic from Joint Mathematics Meeting, Étienne Ghys. https://www.youtube.com/watch?v=1OIxzewWilc https://www.youtube.com/watch?v=1OIxzewWilc
fraserphysics 5mo ago
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.
bananaflag 5mo ago
It's not clear to me why the S is more difficult than the others.
- antonvs 5mo ago
  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.
  - noufalibrahim 5mo ago
    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.
    - whatshisface 5mo ago
      As Knuth points out, applying math to art is applying artistic sense to a continuum of forms all at once.
      - noufalibrahim 5mo ago
tobr 5mo 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.
- hairywalt 5mo ago
  Maybe another old example of this scanner bug? https://www.dkriesel.com/en/blog/2013/0802_xerox-workcentres_are_switching_written_numbers_when_scanning https://www.dkriesel.com/en/blog/2013/0802_xerox-workcentres...
zhangdake 5mo ago
[dead]
wolfi1 5mo ago
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
- golem14 5mo ago
  Suddenly, I don't feel so bad about my own procrastinitis.
  - andyferris 5mo ago
    How many great inventions and discoveries were the product of yak shaving? I'd imagine quite a lot.
    - ChrisMarshallNY 5mo ago
      Wasn’t that the story behind Slack?
      - throw0101a 5mo ago
        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 https://en.wikipedia.org/wiki/Stewart_Butterfield
        5mo ago
czzprr 5mo ago
I would love to see Donald Knuth on Seasame Street..
dan-bailey 5mo ago
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.
kgwxd 5mo ago
Fender v. Knuth inbound.
NiloCK 5mo ago
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.
dannyobrien 5mo ago
For me, a modern descendant of METAFONT is probably Iosevka's build system, which has its own internal DSL, PatEL, for defining its font forms, based on decomposed sub-functions. PatEL's a Lisp-with-infixes-and-indentation that compiles to JS[1]. See the definitions of "O" and related glyphs for a good example[2]. [1] https://github.com/be5invis/PatEL https://github.com/be5invis/PatEL [2] https://github.com/be5invis/Iosevka/blob/main/packages/font-glyphs/src/letter/latin/o.ptl https://github.com/be5invis/Iosevka/blob/main/packages/font-...
akramachamarei 5mo ago
I am struck, in reading his paraphrasing of Torniello's instructions, by their potential fitness as a classroom geometry exercise. You could assign students each a letter or two; might be fun.

304 points by bambax 5mo ago

bombcar 5mo ago

I just spent 30 minutes reading a detailed mathematical version of "draw an S; next draw a more different S".

philringsmuth 5mo ago
Trogdor!
- herodoturtle 5mo ago
  Burninating the countryside. Burninating the peasants. Burninating all the peoples.
jll29 5mo ago
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.
- commandlinefan 5mo ago
  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.
trbleclef 5mo ago
How would Knuth opine on consummate Vs, I wonder...

WillAdams 5mo ago

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 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).

dhosek 5mo ago
For the work on Euler, this article¹ (https://www.sciencedirect.com/science/article/pii/S240587262030068X 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.
- WillAdams 5mo ago
  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 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.

adzm 5mo ago

(1980)

dang 5mo ago
Added. Thanks!

mrandish 5mo ago

Knuth is just a treasure.

adm4 5mo ago

Wonderful man, here is a lecture on the topic from Joint Mathematics Meeting, Étienne Ghys. https://www.youtube.com/watch?v=1OIxzewWilc https://www.youtube.com/watch?v=1OIxzewWilc

fraserphysics 5mo ago

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.

bananaflag 5mo ago

It's not clear to me why the S is more difficult than the others.

antonvs 5mo ago
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.
- noufalibrahim 5mo ago
  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.
  - whatshisface 5mo ago
    As Knuth points out, applying math to art is applying artistic sense to a continuum of forms all at once.
    - noufalibrahim 5mo ago

tobr 5mo 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.

hairywalt 5mo ago
Maybe another old example of this scanner bug? https://www.dkriesel.com/en/blog/2013/0802_xerox-workcentres_are_switching_written_numbers_when_scanning https://www.dkriesel.com/en/blog/2013/0802_xerox-workcentres...

zhangdake 5mo ago

[dead]

wolfi1 5mo ago

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

golem14 5mo ago
Suddenly, I don't feel so bad about my own procrastinitis.
- andyferris 5mo ago
  How many great inventions and discoveries were the product of yak shaving? I'd imagine quite a lot.
  - ChrisMarshallNY 5mo ago
    Wasn’t that the story behind Slack?
    - throw0101a 5mo ago
      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 https://en.wikipedia.org/wiki/Stewart_Butterfield
      - 5mo ago

czzprr 5mo ago

I would love to see Donald Knuth on Seasame Street..

dan-bailey 5mo ago

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.

kgwxd 5mo ago

Fender v. Knuth inbound.

NiloCK 5mo ago

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.

dannyobrien 5mo ago

For me, a modern descendant of METAFONT is probably Iosevka's build system, which has its own internal DSL, PatEL, for defining its font forms, based on decomposed sub-functions. PatEL's a Lisp-with-infixes-and-indentation that compiles to JS[1]. See the definitions of "O" and related glyphs for a good example[2]. [1] https://github.com/be5invis/PatEL https://github.com/be5invis/PatEL [2] https://github.com/be5invis/Iosevka/blob/main/packages/font-glyphs/src/letter/latin/o.ptl https://github.com/be5invis/Iosevka/blob/main/packages/font-...

akramachamarei 5mo ago

I am struck, in reading his paraphrasing of Torniello's instructions, by their potential fitness as a classroom geometry exercise. You could assign students each a letter or two; might be fun.