from Hacker News

Why A4? – The Mathematical Beauty of Paper Size

by casca on 9/5/22, 7:47 AM with 608 comments

  • by TacticalCoder on 9/5/22, 10:11 AM

    A4 is so ubiquitous around me (Europe) that, no matter its merits or not, it's really a major PITA when you get documents that are not A4. They don't line up nicely with the A4 stuff. They don't go gently in the A4 folders / sleeves / pockets / ring binders (files) / etc.

    Typical offenders in my case are medical bills printed on dot-matrix printers.

    I don't even know if there's perforated continuous stationery/form paper with tractor holes that, once you strip the holes, is A4 sized? I guess it's really uncommon if it exists for I don't ever recall seeing any actually used.

    Now don't get me wrong: hearing the song of a dot-matrix printer makes me happy (brings back lots of memories as my first printer was a dot-matrix one) and it's nice that that tech is still in use here and there. I just wish the dot-matrix printers in Europe would use A4 continuous paper (which may or may not even exist).

    And don't get me started on oversized continuous paper that is not perforated: this is the kind of stuff that can trigger obsessive-compulsive disorder!

    I guess all this rant to say that even in Europe, where A4 is ubiquitous, you still have places using other sizes than A3/A4/A5 etc.

  • by samwillis on 9/5/22, 10:14 AM

    The article sadly misses the root of the An sizes. A0 has an area of 1m^2, setting its dimensions and defining the rest of the sizes. Quite genius!

    The B sizes start at B0 with a length on the short side of 1m, then follow the same pattern. The C sizes, often used for envelopes, are defined as the geometric mean between the equivalent A and B size.

    As a child of an architect, when I was a kid my dad would bring old A0 drawings home as paper we could draw and paint on the back of. Loved it.

    - Edit: it does mention it, I skimmed it too fast…

  • by hazn on 9/5/22, 9:16 AM

    Beautiful (in an educational sense and in a visual sense) CGP Grey video on metric paper: https://youtu.be/pUF5esTscZI

  • by genezeta on 9/5/22, 9:25 AM

    There is a secondary series, B, in which the base size, B0, -instead of being defined for its area of 1 square meter as is A0- is defined to be 1 meter wide. A series is then 1, 1/2, 1/4, etc square meters, while B is 1, 1/2, 1/4, etc meters wide.

    It's not as common as the A series but it's used occasionally and it offers intermediate sizes between those of the A series.

  • by kasperni on 9/5/22, 9:10 AM

    A map of paper size usage throughout the world: https://vividmaps.com/paper-sizes-world-map/

  • by jcynix on 9/5/22, 12:16 PM

    The international paper size standard is ISO 216. It is based on the German DIN 476 standard for paper sizes. This standard was designed by the German engineer and mathematician Walter Porstmann in 1922.

    Cf https://de.wikipedia.org/wiki/Walter_Porstmann

  • by thrdbndndn on 9/5/22, 9:36 AM

    The fact A-series has a sqrt(2) radio is obviously neat on its own, but does it save material, or have much benefit in general in practice?

    My (very limited) experience in printing seems to be

    1) they still trim tons of bleed regardless;

    2) the fact you can cut A3 into two A4 etc. is great, but doesn't really matter when 99% of the printing paper will be just one size (A4) anyway?

    3) Lots of magazines and books use non-standard sizes.

    Disclaimer: I'm from a metric country. This is a genuine question, not in anyway trying to make excuses for The US' letter size or what not.

  • by gennarro on 9/5/22, 10:04 AM

    Here is another good explainer which might be a bit easier to parse: https://unsharpen.com/paper-sizes/

    Minor side point, I’ve never seen an x written as )( but always as / and then \ (or vice versa). I wonder if this is a regional thing ir just one person’s preference.

  • by jstanley on 9/5/22, 9:37 AM

    The article makes these 2 contradictory claims:

    > I can tell you now it is precisely 210mm by 297mm

    > So the only ratio that has this important property is the square root of 2, famously – and ironically in this case – not a ratio

    I suspect either:

    - A4 paper is not precisely 210x297 mm, or

    - the precision of the ratio has been sacrificed in order to allow integer numbers; but if we're willing to make that sacrifice, why not round up to 212x300 mm? and then we can have the nice round number that the classmate wanted in the first place, with a bonus that both dimensions are even-numbered so can be halved with mm-graduated rulers.

  • by frogpelt on 9/5/22, 12:24 PM

    I love this little fact but stating, "It is possibly one of the greatest innovations of the 18th century" is ridiculous hyperbole.

    Or maybe the 18th century was severely lacking in innovations.

  • by avnigo on 9/5/22, 9:16 AM

    I can't not share this excellent video on the subject by CGP Grey on "metric paper":

    https://www.youtube.com/watch?v=pUF5esTscZI

  • by solidsnack9000 on 9/5/22, 9:41 PM

    I hope we will use the A* family of paper sizes in the USA more widely. Many excellent notebooks, notepads, &c are available from Germany, Japan, France and England in the A* family of sizes, and they all work with the corresponding envelopes. There is so much paper correspondence -- in the form of utility bills, government notices, packing lists, and many other things -- that could fit on A5, which is a handy size for many things; and now manufactured in great quantities all over the world.

  • by dtx1 on 9/5/22, 9:24 AM

    A0 ≈ 1 m² was new to me but makes perfect sense. Imagine a world where the makers of USB used a similarly elegant naming convention.

  • by samloveshummus on 9/5/22, 4:06 PM

    I had a major epiphany deriving this myself as a teenager, because it was the first time I used algebra to calculate something "real". The equation was trivial, but the principle of mathematical modelling struck me like lightning. In an instant the world felt more comprehensible and tractable, I suddenly felt like Neo seeing the Matrix.

  • by spaintech on 9/5/22, 10:33 AM

    Sure there is math behind it, I was at a paper factory once, I asked them about the whole A0 and so forth sizing, I was told in prefect german and translated that originally they made 1M^2 sheet (alle) and they would them take those and cut them by halfs, so when an order came in, it would read Alle/2 Alle/4 a so forth since they would then say take this pile to A, A2 and A4 cutting station. Who knows, but that is what the Aschaffenburg guide tolde me back in 1998, been telling everyone that story since… :/

  • by w4rh4wk5 on 9/5/22, 9:50 AM

    While I really appreciate the fact that you can split an A4 sheet and get two A5 sheets. Even as a European, I always felt like the American letter format feels 'nicer'. I feel like the letter format's ratio works better for taking notes in a notebook or browsing through printed pages -- less tall, but wider than A4.

    Anyone made a similar experience?

  • by c7b on 9/5/22, 9:38 AM

    Beautiful! One more thing to add re A4 vs letter size: A4 is also a better approximation of the 'golden ratio', so supposedly more aesthetically pleasing (though neither is a particularly good approximation, which shouldn't surprise since A4 is an approximation of sqrt(2) not (1+sqrt(5))/2).

  • by cylinder714 on 9/5/22, 6:07 PM

    I cannot believe no one has posted Markus Kuhn's excellent page on the subject, first appearing in 1996:

    https://www.cl.cam.ac.uk/~mgk25/iso-paper.html

  • by CHB0403085482 on 9/5/22, 12:02 PM

  • by pottertheotter on 9/5/22, 5:45 PM

    If you moved from Europe to the US (or vice versa), I'm curious what your thoughts are on using A4 vs US letter? And I mean once you were settled and transitioned to using the other standard, did you find yourself thinking one was better than the other?

    Another way of asking this is, if you could magically make it so that everyone had been using and today used one of the two (A4 or US letter), would it matter to you?

  • by dbelford on 9/5/22, 4:15 PM

    Are there fields (architects, engineers?) that take advantage of the the technical pen scaling? Or scanning software that is aware of the pen/paper ratios? This seems like a great way to have some consistency in analog/digital conversions. It would be nice to have 10 years of digitized notes that all have consistent sizing after scanning.

  • by jeff-davis on 9/5/22, 2:31 PM

    The article never mentions why folding in half is a good thing. It is, of course, but why half, and not fifths?

    The reason is that two is a good factor to work with. Multiplying and dividing by two is more common than by five in a lot of domains. Three is pretty useful also.

    But the metric system is based on ten, which has factors two and five. Five is not such a great factor and often creates awkward numbers when dividing by two or three.

    The US measurement system, while inconsistent and arbitrary in many ways, but handles factors of two a lot better. Two cups in a pint, two pints in a quart, four quarts in a gallon.

    Better would be to have a consistent measurement system in base 12. Two twos and a three are a nice set of factors to work with. And twelve is still a nice "order of magnitude" that gives a good sense of scale.

    When it comes to distances, feet are actually pretty good in this regard. Degrees and time are base 60, which includes an extra five factor, so I guess it's combining all of them.

    The non-US world is smug about A4 and the metric system, but the very reasons that A4 is good shows why the metric system is not quite as brilliant as people like to think it is.

    (Obviously standardization and consistency have value, too. The metric system is good in that regard.)

  • by 2143 on 9/5/22, 4:38 PM

    > 80 gsm (grams per square metre) paper at A0 size weighs precisely 80g. A4 paper at that density therefore weighs 5g since it’s been halved 4 times

    So when they mark the gsm number onto a ream of A4 sheets, is 80 the gsm of A0 or A4?

    Excellent article by the way. Thank you.

  • by davelondon on 9/5/22, 4:33 PM

    The same size relationship is for Euro stacking containers: https://en.wikipedia.org/wiki/Euro_container

  • by jbverschoor on 9/5/22, 1:58 PM

    US legal is way better suited for reading. Or the comic-book format

  • by ktzar on 9/5/22, 9:06 AM

    The metric system is such a beauty, that it's shocking that is not used everywhere. The fact that a liter of water has a relation with the size of the paper we use is just poetry.

  • by Aeolun on 9/5/22, 12:16 PM

    I never realized this, but I’m glad I learned why this is.

    It’s the kind of thing that you always wonder about but never seriously investigate.

    You could go a whole lifetime without knowing…

  • by rvba on 9/5/22, 1:30 PM

    I wish EU would force the manufacturers to advertise phone and TV screen sizes in centimeters, not inches.

  • by _HMCB_ on 9/5/22, 6:29 PM

    As an American, I can say A4 feels just perfect to me. Not too large or small.

  • by z3t4 on 9/5/22, 2:53 PM

    I'm still waiting for a thin low weight A4 size eink reader

  • by m-p-3 on 9/6/22, 12:49 PM

    Canadian here, I wish we switched to A* paper format :|

  • by kazinator on 9/5/22, 3:44 PM

    210x297 offers a better approximation than 212x300.

  • by WesolyKubeczek on 9/5/22, 9:37 AM

    [curses in Tschichold]

  • by mavu on 9/5/22, 9:43 AM

    This is very cool.

  • by anomas01010 on 9/5/22, 10:42 AM

    anomas01010