Nice! The author touches on the area properties and here's the most practical life hack derived from the standard I personally use. It uses the relationship between size and mass.
Because A0 is defined as having an area of exactly 1 square meter, the paper density (GSM or grams per square meter) maps directly to the weight of the sheet.
>A0 = 1 meter square.
>Standard office paper = 80 gsm
>Therefore, one sheet of A0 = 80 grams.
>Since A4 is 1/16th of an A0, a single sheet of standard A4 paper weighs 5 grams.
I rarely need to use a scale for postage. If I have a standard envelope (~5g) and 3 sheets of paper (15g), I know I'm at 20g total. It turns physical shipping logistics into simple integer arithmetic. The elegance of the metric system is that it makes the properties of materials discoverable through their definitions.
Oh nice, that is a neat trick! One small nitpick (that makes no difference): The side lengths of the ISO Ax formats are rounded to the next mm, so actually the A0-format has an area of 0.999949m^2
that reminds me of an old joke: how doe the postal services make their profit? I don't get it. - Ah, that's easy. How much wieght may letters have? - 20g - And how much weight do the average letters have? - About 6g. - See? That's their profit
> I rarely need to use a scale for postage. If I have a standard envelope (~5g) and 3 sheets of paper (15g), I know I'm at 20g total. It turns physical shipping logistics into simple integer arithmetic.
...was using a scale for postage a concern? If you're shipping things on the order of three sheets of paper, you're way below any conceivable threshold. USPS charges a flat rate on letters under 370 grams!
If you're sending 1,700 pieces of looseleaf paper in a box... just weigh the box.
German postage for letters is under 20g, under 50g and under 500g so I had this issue a few times so far when sending a few letters a day over a few weeks. You can see it here for international letters for example: https://www.deutschepost.de/en/b/briefe-ins-ausland.html
Thankfully I just had a scale, but I can see this being helpful when you don't.
As a European living in North America I developed a weird cognitive dissonance. When I'm in North America the regular printer page (US Letter) seems too squat. When I'm in Europe the A4 page looks too svelte. I now need an in-between format for it to look "right".
> When I'm in Europe the A4 page looks too svelte.
Off-topic but as a non-native English speaker, TIL what svelte means lol. I often get exposure of new words first from a product name. Same happened with Chrome.
I happened to be looking at ski boot fitting this morning and came across a web app from fischersports that allows you to measure your feet using your phones camera. Surprise surprise it uses a sheet of A4 paper.
Fun article. I liked the bit about how the size of A0 can be uniquely determined from abstract constraints. But I'm not convinced that the "Measuring Stuff" section involves anything more than memorising the exact dimensions of A4. I don't see how it applies the stuff about preserved ratios.
25. Of October 1786: Lichtenberg suggested his friend Beckmann a paper format in the aspect ration 1/√2.
»einen Bogen Papier zu finden, bey dem alle Formate … einander ähnlich wären. … Die kleine Seite des Rechtecks muß sich nämlich zu der großen verhalten wie 1:√2 oder wie die Seite des Quadrats zu seiner Diagonale. Die Form hat etwas angenehmes und vorzügliches vor der gewöhnlichen.«
Here’s a better tip to measure things without a proper measuring device: spread out your hand on a table and measure the distance between your pinky and thumb. Remember that. Now when you need to measure something just measure it in number of pinky-thumb-stretches. I can quickly get the dimensions to +/- an inch by doing a few quick walks with my hand.
It is hand to remember a few finger/knuckles/elbow/shoulder combinations for common measures. One of your phalanges should be ~1 inch, for example, and one of your finger nails is probably ~1 cm wide.
There's a reason that the English system of measurement had things like "hand" and "foot" - because when you're not measuring things exactly, close enough and commonly available is fine.
For small screws, in the millimeter range, the jump between metric sizes is too big. So, in addition to M1, M2, M3, M4, M5, etc. standard metric screws include M1.4, M1.6. M1.8, M2.5, and M3.5 (rare) to fill in the gaps.
Screw sizes and drill sizes should have been sized by a ratio, like resistor values. But that would have been a pain for manual machining.
Domestic drill sets don't seem to be designed for tapping holes but if you stick to M3, M6 and M10 the tapping sizes do correspond with the 2.5, 5 and 8.5mm drills[1].
I guess if it was based on a ratio system you would need special tapping drills for all of them.
e.g. M4 needs a special 3.3mm tapping drill already.
My partner (an architect) does something similar, plus - when she holds her arm out straight - the distance from the tips of her fingers to the opposite shoulder blade is almost exactly a meter.
Some Moleskine cahier notebooks are wrapped in a paper sleeve that has a ruler printed on the back. Inches on one edge; centimetres on the other (a slight improvement for non-American users over Field Notes notebooks, with a ruler on the inside back cover that’s inches only - also the sleeve is rather longer, 28cm in fact).
I’ve started to determine the right package of paper towels to purchase according to the cents per square meter value. You can discern the quality of a deal at the grocery by referring to the ‘cents per X’ market located on price tag next to the marked price.
I’m beginning to turn sour on the ‘2 Jumbo-Mega-Rolls are the equivalent of 8 Super rolls’ scheme that’s en vogue. Are there retractable roll holders to accommodate for all of this?
It doesn’t help that many of these packages are priced and then marked down in ways to entice the buyer toward purchasing them instead of more reasonably priced and proportioned ones.
With paper towel I have been thinking that the area might not be important, but the number of sheets would be. As long as they do not get too small. And then there is also the quality and if they are half or full. For some uses you just want the full.
It is complicated area. Not to even get to loo roll. Where I noticed that the ecological one I bought feels quality wise inferior to normal one. And this is premium type of stuff. So it sits between the premium and cheap, but more on premium end.
The worst is that the assumption that the greater quantity is cheaper per unit, but for some reason that's not always true so you have to sit there and do the math in order to get the best deal.
Maths Youtuber Noel Friedrich recently made a video about A4 paper[0]. It turns out that since the ISO specification rounds the side lengths down to whole millimeters, with compounding errors, more than 2^10 A10s (smallest paper size in the standard) fit into one A0 (largest size in the standard).
As an American I have done this with 8.5 x 11 "letter" paper. I wonder if there's some way one can take advantage of the special properties of A[n] paper.
1000% yes! An 8.5x11" paper is effectively a 12" ruler accurate to 2 decimal places.
Fold an 8.5" into a square (right triangle) and the long edge is exactly 12.02"
Fold that in half and you can measure 6.01", and 3.005" (exactly). You get 1.5" for free, and can fairly accurately get exactly 1" by rolling the other 3" side into thirds.
If you want to get an exact 1", you can technically get there via 11"-8.5"-1.5", and that gives you the full imperial (fractional) measurement basis, all from folding a (presumably accurate) 8.5x11" piece of paper.
A[n] sizes are useful when enlarging or shrinking documents. Enlarge or shrink by muliples of sqrt(2) and there's always a fitting paper size available. Or you can put two A5s together on an A4, or two A4s on an A3.
As a long time European I never thought I’d come to see the sense of American ways, but having lived here now for a couple of years, it actually is easier for it to just be straight up 8.5 x 11 rather that a ratio that includes a square root.
Everyone makes paper the same hypothetical way, by producing large sheets and cutting them in half, and ANSI E (34"x44" or 864mm x 1118mm) isn't that different than A0 (841mm x 1189mm), but the slight starting difference means that there are two aspect ratios for ANSI (17/22 and 11/17). On the one hand, they're simple fractions and not irrational numbers; on the other, they're different, so you can't just double the size of something printed on ANSI A/letter sized to fill ANSI B/tabloid size, the way you can go from A4 to A3.
Only a small subset of users will ever want to do that (since if you're printing text you probably need to re-typeset it to keep the type a good size for reading), but only a small subset of users actually care about the aspect ratio or exact dimensions of their paper at all, so whether it is 8.5 or 8.11 or 8.314159... inches doesn't really matter.
I've been working with paper sizes a lot for the last year, and I've rarely thought about the square root of two ratio and when I've, it has been just to amuse myself. However, knowing that to get an A5 piece of paper I just need to cut/fold in half an A4, and that I can get to A3 and A2 by adding A4s, has been really useful. If I were in USA, didn't have that, and instead would have to install yet another new size system in my head[^1], I would despair.
What bothers me mostly about American papersizes (I’m also a European immigrant) is that the ratio is not consistent between sizes. So if I design a poster, but want a couple of letter sized printouts for some reason, I have to create a whole new design, rather then just shrink everything down. Otherwise the margins get all wonky.
One nice thing with Letter size is you can fit 80 columns of 10 dpi text with standard LaserJet margins. With A4 you have to squeeze the characters together slightly to make that fit.
> I wonder if there's some way one can take advantage of the special properties of A[n] paper.
Not as a consumer. As a paper producer, you take advantage of it by cutting large sheets of paper in half to produce smaller sheets. Since you never sell any sizes that aren't clean multiples of each other like this, you've minimized the amount of paper you waste. That's the "advantage".
This was once the standard way of making paper; I don't know if it still is.
As a consumer I used to use it all the time, though it matters a lot less these days. Two A4 pages at 50% zoom (A5) fit on one A4. You could cut your printing cost for drafts in half by doing that, back when we had to actually print to check the layout. Same went for posters etc, and since the aspect ratio was preserved it was really handy to preview at home on A4 sheet before taking it to the print shop.
I’m sure you can do that on other size systems, but ISO paper sizing gives you accurate scaling.
Same goes for photocopies, photocopiers can scale copies so two A4 sheets copy to one, if you don’t need the same size.
Pythagoras would appreciate her usage of his theorem, but I'd have just laid my papers diagonally across the screen to measure it directly without computing any square roots. Since I'm a yank, 11 + 11 + 5.5 would do quite nicely.
I use my phone when I want to measure stuff. Not an app, just the physical phone as a ruler. Almost always the dimensions of whatever phone I've got is published on the internet. It's a quick hack and better than carrying around A4 papers ;)
Ha - I have made so many measurements using an A4 with great accuracy that this monitor-story might as well have been me :-)
I never understood the US paper size system while living there (or since...!), don't get me started with feet and inches and 16'ths etc - ISO, metric and base10 is just so much more logical and easy to use...
Elegant, sure, but... fold a sheet of US Letter (8.5 x 11.0 inches) in half, and you're on your way to a pirate hat. Fold and pull it several more times, and you have a boat.
At one point on an international project I had to fly a box of UK A4 to the USA in my luggage so the Americans could check their software could cope with the different size. It did, but lugging it around was a pain - paper is heavy!
I don’t know what specialty stores you’re talking about but A4 is readily available at most stationary stores, or anything related to letter writing, pens, or paper. I got the A5 notebook I’m currently using at Barnes and Noble, they also have A4.
Heck, I’m pretty sure you could get a sheaf of it at any number of office supply stores right now if you wanted.
Kodak used to have an industrial printer partnership with Heidelberg. They would test their printers with pallet loads of A4. Most I've ever seen in the US.
I buy A4 notebooks all the time. I use fountain pens, so many of the notebooks and even loose paper with the proper sizing (coating, that is) usually come in EU sizes. Tomoe River... Clairfontaine... etc.
I had the reverse, we had to get a ream of US Letter and corresponding envelopes sent over so we could ensure the layouts printed properly. Also some chequ… “checks” which were fascinating.
An interesting question, but I think it would be very hard to do it accurately. Also, some of the reports the needed to print during testing were looong.
Hint: The aspect ratio sqrt(2) should be quite convenient for foldable phones. Current ones seem to be more or less square if unfolded - what use case is that good for?
Yeah, I’ve used a sheet of paper as a ruler too...
As regards metric/A* paper sizes, it seems like just a coincidence that this scheme resulted in a standard size that is useful for everyday documents, since it only works for powers of 2 and starts with the definition of 1 square meter. If a meter were 1.5x smaller or larger, then I don’t think there would be a standard size that works so well.
EDIT: Being curious about this, I did some more reading, and discovered there is a “B” series of paper sizes that maintain the same ratio relationship, but are exactly in between all the A sizes! That’s useful.
The creators of metric weren't above buggering it to fit human scale needs.
Take the length/weight relationship.
Definitionally, it'd be way more elegant for the unit of mass to be based on the unit of length directly, a cubic meter of something, but having your base unit of mass be a ton wasn't going to fly.
So they instead tried for 1/100th of the meter and landed on the gram, but it turns out they misjudged and now your standard unit of weight is the prefixed kilogram instead, because everyone used kilograms instead.
Which is to say, if you didn't get a pretty good paper size out of the definition used for A0, someone would have found a different definition which did produce a pretty good paper size, and then declare it was the only natural one.
I don't think anybody loses sleep over the kilogram issue because, well, it's metric after all. A kilogram is exactly 1000 grams, so the gram is just as perfectly well defined. Nothing would really change if they were to promote the gram to be the standard unit of mass (not weight!) someday.
21cm is close to 20 which means A5 has a side close to 10.
29.7 is close to 30. So why not use A4 sheets to install kitchen cabinets? A friend of mine advised me to buy a laser level. A tiny level was quite enough. A laser meter is nice though although I don't understand why it's inaccurate sometimes. Maybe it depends on the surface matter/paint.
Sqrt(2) being halfway between 1 and 2 multiplicatively, leads to interesting stuff. For example consider two integers A and B. They have a dual in the pair A+B and A-B. Well, not quite. You need to scale them down by 1/sqrt(2). If you do that to the duals, you get the originals again.
As the ratio used is a rational approximation of an irrational, I would guess that the ratio-preserving feature breaks down well before you get to atomic sizes, though I have neither proved that to be so nor figured out how to calculate the divergence.
If you have a Pro or Pro Max model of iPhone from the last several years, it has a LiDAR that allows the pre-installed Measure app to measure lengths/heights, etc., using the camera. Several higher end Android phones may also have the same.
> "Hold on. I think I hear another heckle. What is that? There are mobile phone apps that can measure things now? Really? Right. Security. Where's security?"
While this is nice, it is not inherently related to an A4 paper.
You could have memorized the length of a cheapo Bic pen if that is common in your area; Or a Parker or a Monte-Blanc if you carry one of those.
All recent iPhones (regular models since `03) have a width of 71.5mm. Remember that, and as long as someone near you has an iPhone, you are good to go using it as a ruler. (And people will definitely be, um, impressed).
We have in my kitchen several brands of small forks, all are 19.2mm (just checked. The large forks have a range of sizes). Next time I need to measure something I could just request a fork...
Trying not to be too negative but this is quite a long blog post for what effectively just comes down to using a piece of paper to measure something, which I imagine lots of us have done in a pinch. Also, nit: one of the calculations on paper size seems duplicated due to a typo.
Personally I make sure to always carry around a string and some scissors, so any time I need an arbitrary measurement I can just cut the string to the length of the thing I want to measure (making sure to label it in case of multiple required measurements), then measure the cut string later. Simple.
Although I still haven't figured out the best way to do that in reverse (when someone wants a specific measurement and I cut the string from that number), though I was considering a scheme where I start with strings of known length up-front then repeatedly cut successive halves until I converge on the desired number, accounting for cut accuracy and require precision.
Maybe you could start off with a long string and then mark off its midpoint, the midpoints of the halves, and so on. Then you wouldn't have to cut a new string every time you want to measure something.
The span from my thumb to my pinky in a “measuring position” is 20 cm (and is easily repeated by moving the thumb to the pinky and then stretching out the pinky again). The length of my “thumbs up” hand is 16 cm. The width of my fist is 10 cm. The length of my pinky is 6 cm. The width of my thumb is 2 cm. This allows me to estimate distances between ~2 m and 2 cm pretty well. Knowing your foot/shoe length also comes handy sometimes.
Nice! The author touches on the area properties and here's the most practical life hack derived from the standard I personally use. It uses the relationship between size and mass.
Because A0 is defined as having an area of exactly 1 square meter, the paper density (GSM or grams per square meter) maps directly to the weight of the sheet.
>A0 = 1 meter square.
>Standard office paper = 80 gsm
>Therefore, one sheet of A0 = 80 grams.
>Since A4 is 1/16th of an A0, a single sheet of standard A4 paper weighs 5 grams.
I rarely need to use a scale for postage. If I have a standard envelope (~5g) and 3 sheets of paper (15g), I know I'm at 20g total. It turns physical shipping logistics into simple integer arithmetic. The elegance of the metric system is that it makes the properties of materials discoverable through their definitions.
The 5 grams per sheet of common printer paper has certainly proven handy once or twice in some of my interactions in the informal economy.
Same for the US 5 cent coin. Defined mass of 5 grams.
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Paper's uniform mass per area makes it useful calibrating very tiny scales. 1mm² of 80 gsm paper will weigh about 80 micrograms.
"Measure the mass of an eyelash with a DIY microbalance" by Applied Science https://youtu.be/ta7nlkI5K5g
TIL GSM is just g/sq m. Like duh, feel so stupid xD
It's not you who should feel stupid.
The person deciding to use nonstandard "GSM" as a unit instead of the proper "g/m²" needs to feel stupid...
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Oh nice, that is a neat trick! One small nitpick (that makes no difference): The side lengths of the ISO Ax formats are rounded to the next mm, so actually the A0-format has an area of 0.999949m^2
Not to the next, to the nearest, otherwise it would have to be slightly larger than 1m^2.
that reminds me of an old joke: how doe the postal services make their profit? I don't get it. - Ah, that's easy. How much wieght may letters have? - 20g - And how much weight do the average letters have? - About 6g. - See? That's their profit
I really don't get it.
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> I rarely need to use a scale for postage. If I have a standard envelope (~5g) and 3 sheets of paper (15g), I know I'm at 20g total. It turns physical shipping logistics into simple integer arithmetic.
...was using a scale for postage a concern? If you're shipping things on the order of three sheets of paper, you're way below any conceivable threshold. USPS charges a flat rate on letters under 370 grams!
If you're sending 1,700 pieces of looseleaf paper in a box... just weigh the box.
German postage for letters is under 20g, under 50g and under 500g so I had this issue a few times so far when sending a few letters a day over a few weeks. You can see it here for international letters for example: https://www.deutschepost.de/en/b/briefe-ins-ausland.html
Thankfully I just had a scale, but I can see this being helpful when you don't.
Given that we are talking about A4 papers and grams, I'd bet this wasn't in the US.
In Europe, the typical flat rate is up to 100g for standard letters. And that's 20 sheets, which is not a particularly unusual letter to send.
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French "La Poste" sets the first threashold at 20 g.
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>USPS charges a flat rate on letters under 370 grams!
In the UK the limit for a letter is 100 grams:
https://www.royalmail.com/sending/uk/1st-class
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In Sweden, the lowest postage (one stamp, 22 SEK or around $2) is for max 50 grams.
A first class forever stamp only covers 1oz (28g).
CGP Grey has a video [0] that goes into some, let's just say deeper, detail into metric paper that is well worth a watch if you haven't seen it.
0: https://www.youtube.com/watch?v=pUF5esTscZI
edit: fat-fingered CPG, thanks @ProllyInfamous
You beat me in posting this (I searched for "CGP," first — you mispelled so I didn't see your comment).
----
My favorite CGP Grey video is Metric Paper..., which explores the vast (but limited) world we live in, from plancs to observable universe.
[•] <https://www.youtube.com/watch?v=pUF5esTscZI>
Before generative AI videos, this had been what I considered "the most psychedelic experience one can have without doing drugs." It's still a trip...
As a European living in North America I developed a weird cognitive dissonance. When I'm in North America the regular printer page (US Letter) seems too squat. When I'm in Europe the A4 page looks too svelte. I now need an in-between format for it to look "right".
> When I'm in Europe the A4 page looks too svelte.
Off-topic but as a non-native English speaker, TIL what svelte means lol. I often get exposure of new words first from a product name. Same happened with Chrome.
B5
Assuming "squat" and "svelte" refer to the aspect ratio, isn't B5 going to look just as "svelte" as A4?
B series paper sizes is a thing I knew but wish I didn't.
See also C series. Thankfully largely a moot point now.
Also sqrt(2) isn't the golden number. Not far from it but still...
The golden ratio makes for even less balanced paper than sqrt(2)
I happened to be looking at ski boot fitting this morning and came across a web app from fischersports that allows you to measure your feet using your phones camera. Surprise surprise it uses a sheet of A4 paper.
App is about halfway down this page, https://www.fischersports.com/rc4-podium-rd-worldcup-strd/U0... under 'find your size' and is powered by https://volumental.com/
Fun article. I liked the bit about how the size of A0 can be uniquely determined from abstract constraints. But I'm not convinced that the "Measuring Stuff" section involves anything more than memorising the exact dimensions of A4. I don't see how it applies the stuff about preserved ratios.
Nitpick: typo in the dimensions for A3.
The article declines to mention how precise paper is. The corners are very, very square and the lengths are very, very precise.
Better than an average square, better than an average ruler.
On one hand, you could do the measuring section with any standardized rectangle. On the other hand, any excuse to talk about metric paper
Letter paper, credit card, banknote, business card, etc.
25. Of October 1786: Lichtenberg suggested his friend Beckmann a paper format in the aspect ration 1/√2.
»einen Bogen Papier zu finden, bey dem alle Formate … einander ähnlich wären. … Die kleine Seite des Rechtecks muß sich nämlich zu der großen verhalten wie 1:√2 oder wie die Seite des Quadrats zu seiner Diagonale. Die Form hat etwas angenehmes und vorzügliches vor der gewöhnlichen.«
Here’s a better tip to measure things without a proper measuring device: spread out your hand on a table and measure the distance between your pinky and thumb. Remember that. Now when you need to measure something just measure it in number of pinky-thumb-stretches. I can quickly get the dimensions to +/- an inch by doing a few quick walks with my hand.
It is hand to remember a few finger/knuckles/elbow/shoulder combinations for common measures. One of your phalanges should be ~1 inch, for example, and one of your finger nails is probably ~1 cm wide.
There's a reason that the English system of measurement had things like "hand" and "foot" - because when you're not measuring things exactly, close enough and commonly available is fine.
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Or be like the mythbusters guy and get a ruler tattooed on your arm!
Metric is beautiful.
I remember when I first got into metal work and wanted to get some tapping drills.
There are a plethora of standards when you start looking into it. For what I make though if I use metric I really only need one, ISO Coarse.
Metric is just well thought out and easier.
For small screws, in the millimeter range, the jump between metric sizes is too big. So, in addition to M1, M2, M3, M4, M5, etc. standard metric screws include M1.4, M1.6. M1.8, M2.5, and M3.5 (rare) to fill in the gaps.
Screw sizes and drill sizes should have been sized by a ratio, like resistor values. But that would have been a pain for manual machining.
Yep, as it is some sizes are easier to work with.
Domestic drill sets don't seem to be designed for tapping holes but if you stick to M3, M6 and M10 the tapping sizes do correspond with the 2.5, 5 and 8.5mm drills[1].
I guess if it was based on a ratio system you would need special tapping drills for all of them.
e.g. M4 needs a special 3.3mm tapping drill already.
1. According to my trusty Zeus tables.
I use my fingers:
When I spread my index finger and middle finger, not entirely as far as they can go, but rather far, that's 10 centimetres.
Thumb to pinky is 22 centimetres. These two are often precise enough for me.
My partner (an architect) does something similar, plus - when she holds her arm out straight - the distance from the tips of her fingers to the opposite shoulder blade is almost exactly a meter.
Some Moleskine cahier notebooks are wrapped in a paper sleeve that has a ruler printed on the back. Inches on one edge; centimetres on the other (a slight improvement for non-American users over Field Notes notebooks, with a ruler on the inside back cover that’s inches only - also the sleeve is rather longer, 28cm in fact).
Paper Towel stories:
I’ve started to determine the right package of paper towels to purchase according to the cents per square meter value. You can discern the quality of a deal at the grocery by referring to the ‘cents per X’ market located on price tag next to the marked price.
I’m beginning to turn sour on the ‘2 Jumbo-Mega-Rolls are the equivalent of 8 Super rolls’ scheme that’s en vogue. Are there retractable roll holders to accommodate for all of this?
It doesn’t help that many of these packages are priced and then marked down in ways to entice the buyer toward purchasing them instead of more reasonably priced and proportioned ones.
With paper towel I have been thinking that the area might not be important, but the number of sheets would be. As long as they do not get too small. And then there is also the quality and if they are half or full. For some uses you just want the full.
It is complicated area. Not to even get to loo roll. Where I noticed that the ecological one I bought feels quality wise inferior to normal one. And this is premium type of stuff. So it sits between the premium and cheap, but more on premium end.
Since not all paper is of the same thickness, shouldn't you compare "weight per price"?
The worst is that the assumption that the greater quantity is cheaper per unit, but for some reason that's not always true so you have to sit there and do the math in order to get the best deal.
Maths Youtuber Noel Friedrich recently made a video about A4 paper[0]. It turns out that since the ISO specification rounds the side lengths down to whole millimeters, with compounding errors, more than 2^10 A10s (smallest paper size in the standard) fit into one A0 (largest size in the standard).
[0] https://www.youtube.com/watch?v=zDKBCIMkDbw
As an American I have done this with 8.5 x 11 "letter" paper. I wonder if there's some way one can take advantage of the special properties of A[n] paper.
1000% yes! An 8.5x11" paper is effectively a 12" ruler accurate to 2 decimal places.
Fold an 8.5" into a square (right triangle) and the long edge is exactly 12.02"
Fold that in half and you can measure 6.01", and 3.005" (exactly). You get 1.5" for free, and can fairly accurately get exactly 1" by rolling the other 3" side into thirds.
If you want to get an exact 1", you can technically get there via 11"-8.5"-1.5", and that gives you the full imperial (fractional) measurement basis, all from folding a (presumably accurate) 8.5x11" piece of paper.
A[n] sizes are useful when enlarging or shrinking documents. Enlarge or shrink by muliples of sqrt(2) and there's always a fitting paper size available. Or you can put two A5s together on an A4, or two A4s on an A3.
As a long time European I never thought I’d come to see the sense of American ways, but having lived here now for a couple of years, it actually is easier for it to just be straight up 8.5 x 11 rather that a ratio that includes a square root.
It's an interesting tiny trade-off.
Everyone makes paper the same hypothetical way, by producing large sheets and cutting them in half, and ANSI E (34"x44" or 864mm x 1118mm) isn't that different than A0 (841mm x 1189mm), but the slight starting difference means that there are two aspect ratios for ANSI (17/22 and 11/17). On the one hand, they're simple fractions and not irrational numbers; on the other, they're different, so you can't just double the size of something printed on ANSI A/letter sized to fill ANSI B/tabloid size, the way you can go from A4 to A3.
Only a small subset of users will ever want to do that (since if you're printing text you probably need to re-typeset it to keep the type a good size for reading), but only a small subset of users actually care about the aspect ratio or exact dimensions of their paper at all, so whether it is 8.5 or 8.11 or 8.314159... inches doesn't really matter.
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I've been working with paper sizes a lot for the last year, and I've rarely thought about the square root of two ratio and when I've, it has been just to amuse myself. However, knowing that to get an A5 piece of paper I just need to cut/fold in half an A4, and that I can get to A3 and A2 by adding A4s, has been really useful. If I were in USA, didn't have that, and instead would have to install yet another new size system in my head[^1], I would despair.
[^1]: This is fun! https://papersizes.io/us/
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What bothers me mostly about American papersizes (I’m also a European immigrant) is that the ratio is not consistent between sizes. So if I design a poster, but want a couple of letter sized printouts for some reason, I have to create a whole new design, rather then just shrink everything down. Otherwise the margins get all wonky.
One nice thing with Letter size is you can fit 80 columns of 10 dpi text with standard LaserJet margins. With A4 you have to squeeze the characters together slightly to make that fit.
> I wonder if there's some way one can take advantage of the special properties of A[n] paper.
Not as a consumer. As a paper producer, you take advantage of it by cutting large sheets of paper in half to produce smaller sheets. Since you never sell any sizes that aren't clean multiples of each other like this, you've minimized the amount of paper you waste. That's the "advantage".
This was once the standard way of making paper; I don't know if it still is.
As a consumer I used to use it all the time, though it matters a lot less these days. Two A4 pages at 50% zoom (A5) fit on one A4. You could cut your printing cost for drafts in half by doing that, back when we had to actually print to check the layout. Same went for posters etc, and since the aspect ratio was preserved it was really handy to preview at home on A4 sheet before taking it to the print shop.
I’m sure you can do that on other size systems, but ISO paper sizing gives you accurate scaling.
Same goes for photocopies, photocopiers can scale copies so two A4 sheets copy to one, if you don’t need the same size.
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Pythagoras would appreciate her usage of his theorem, but I'd have just laid my papers diagonally across the screen to measure it directly without computing any square roots. Since I'm a yank, 11 + 11 + 5.5 would do quite nicely.
I use my phone when I want to measure stuff. Not an app, just the physical phone as a ruler. Almost always the dimensions of whatever phone I've got is published on the internet. It's a quick hack and better than carrying around A4 papers ;)
Ha - I have made so many measurements using an A4 with great accuracy that this monitor-story might as well have been me :-)
I never understood the US paper size system while living there (or since...!), don't get me started with feet and inches and 16'ths etc - ISO, metric and base10 is just so much more logical and easy to use...
Elegant, sure, but... fold a sheet of US Letter (8.5 x 11.0 inches) in half, and you're on your way to a pirate hat. Fold and pull it several more times, and you have a boat.
You can do the same with A4, of course. Metric and imperial pirates will have to battle it out.
At one point on an international project I had to fly a box of UK A4 to the USA in my luggage so the Americans could check their software could cope with the different size. It did, but lugging it around was a pain - paper is heavy!
I wish we could buy A4 paper in North America! I find it surprising it's not available even in specialty stores. The rest of the world uses it!
You may need to look for 8.27" x 11.69" - https://www.staples.com/hammermill-copy-plus-8-27-x-11-69-co...
Or you can buy a ream of legal-size and have a printshop slice it down (which is how I got ahold of B4 or B5 IIRC).
I don’t know what specialty stores you’re talking about but A4 is readily available at most stationary stores, or anything related to letter writing, pens, or paper. I got the A5 notebook I’m currently using at Barnes and Noble, they also have A4.
Heck, I’m pretty sure you could get a sheaf of it at any number of office supply stores right now if you wanted.
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Kodak used to have an industrial printer partnership with Heidelberg. They would test their printers with pallet loads of A4. Most I've ever seen in the US.
Yo do know all the jokes about how the US would anything as a measuring system except the metric system? Same with paper.
For printers?
I buy A4 notebooks all the time. I use fountain pens, so many of the notebooks and even loose paper with the proper sizing (coating, that is) usually come in EU sizes. Tomoe River... Clairfontaine... etc.
I had the reverse, we had to get a ream of US Letter and corresponding envelopes sent over so we could ensure the layouts printed properly. Also some chequ… “checks” which were fascinating.
Couldn't they... just cut it according to A4 dimensions?
An interesting question, but I think it would be very hard to do it accurately. Also, some of the reports the needed to print during testing were looong.
Hint: The aspect ratio sqrt(2) should be quite convenient for foldable phones. Current ones seem to be more or less square if unfolded - what use case is that good for?
That would make the folded phone too wide though. More like a mini tablet.
ref: Huawei Pura x
Yeah, I’ve used a sheet of paper as a ruler too...
As regards metric/A* paper sizes, it seems like just a coincidence that this scheme resulted in a standard size that is useful for everyday documents, since it only works for powers of 2 and starts with the definition of 1 square meter. If a meter were 1.5x smaller or larger, then I don’t think there would be a standard size that works so well.
EDIT: Being curious about this, I did some more reading, and discovered there is a “B” series of paper sizes that maintain the same ratio relationship, but are exactly in between all the A sizes! That’s useful.
The creators of metric weren't above buggering it to fit human scale needs.
Take the length/weight relationship.
Definitionally, it'd be way more elegant for the unit of mass to be based on the unit of length directly, a cubic meter of something, but having your base unit of mass be a ton wasn't going to fly.
So they instead tried for 1/100th of the meter and landed on the gram, but it turns out they misjudged and now your standard unit of weight is the prefixed kilogram instead, because everyone used kilograms instead.
Which is to say, if you didn't get a pretty good paper size out of the definition used for A0, someone would have found a different definition which did produce a pretty good paper size, and then declare it was the only natural one.
I don't think anybody loses sleep over the kilogram issue because, well, it's metric after all. A kilogram is exactly 1000 grams, so the gram is just as perfectly well defined. Nothing would really change if they were to promote the gram to be the standard unit of mass (not weight!) someday.
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There is also a C series of sizes which are slightly larger than the A sizes and therefore useful as envelope sizes.
I recently went down the A4 paper sizes rabbit hole and found out that there is an ISO standard for this 'ISO 216'
https://en.wikipedia.org/wiki/International_standard_paper_s...
Edit: while going through the article again, I noticed that's a brief mention of the ISO standard, which I missed in the first round.
Same, last year. I like A4 for large notebooks, and B7nos the perfect size for a pocket journal but it is hard to find.
A0 paper is for drafting. That's what people used for engineering drawings before we had AutoCAD with zoom.
I have heard German toilet paper is A6, but I never got to verify that. If true, it's one of the more German things ever.
Almost. In Germany, standard tp is 100 × 140 mm, a bit smaller than DIN A6 (105 × 148).
I don't think it is, the aspect ratio seems off, but maybe I'm misremembering (not in Germany right now)
21cm is close to 20 which means A5 has a side close to 10.
29.7 is close to 30. So why not use A4 sheets to install kitchen cabinets? A friend of mine advised me to buy a laser level. A tiny level was quite enough. A laser meter is nice though although I don't understand why it's inaccurate sometimes. Maybe it depends on the surface matter/paint.
I always liked the Japanese idea of measuring room size by the number of tatami mats.
Fun fact: tatami mats are slightly smaller in Tokyo than in other parts of Japan.
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Sqrt(2) being halfway between 1 and 2 multiplicatively, leads to interesting stuff. For example consider two integers A and B. They have a dual in the pair A+B and A-B. Well, not quite. You need to scale them down by 1/sqrt(2). If you do that to the duals, you get the originals again.
As the ratio used is a rational approximation of an irrational, I would guess that the ratio-preserving feature breaks down well before you get to atomic sizes, though I have neither proved that to be so nor figured out how to calculate the divergence.
If you have a Pro or Pro Max model of iPhone from the last several years, it has a LiDAR that allows the pre-installed Measure app to measure lengths/heights, etc., using the camera. Several higher end Android phones may also have the same.
> "Hold on. I think I hear another heckle. What is that? There are mobile phone apps that can measure things now? Really? Right. Security. Where's security?"
Just quoting the author here, haha.
Adds gsm issue to https://github.com/librasteve/raku-Physics-Unit/issues/32
There's also the fact that if you don't have a convenient straight-edge around -- fold a sheet of paper, not too rough.
It's a good exercise in thinking, dhy that is so.
>Like most sensible people with a reasonable sense of priorities, I do not carry a ruler with me wherever I go.
Let me introduce to you: the IKEA paper tape measure, folded neatly in your wallet.
Or just use your phones measuring App. So far the iPhone one has been precise enough for me when on the go.
Security!
(read the article)
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There's also the fact that if you don't have a convenient straight-edge around -- fold a sheet of paper, not too rough.
What ... not everybody carries a small, 6' tape measure with them??? Surely I'm not THAT abnormal.
Most of the world carries a 2 m tape measure with them. You're only normal in the USA, Myanmar, and Liberia.
While this is nice, it is not inherently related to an A4 paper.
You could have memorized the length of a cheapo Bic pen if that is common in your area; Or a Parker or a Monte-Blanc if you carry one of those.
All recent iPhones (regular models since `03) have a width of 71.5mm. Remember that, and as long as someone near you has an iPhone, you are good to go using it as a ruler. (And people will definitely be, um, impressed).
We have in my kitchen several brands of small forks, all are 19.2mm (just checked. The large forks have a range of sizes). Next time I need to measure something I could just request a fork...
You can use objects of known length to measure other objects of unknown length. Am I a hacker, now?
Obligatory link to Markus Kuhn's excellent page on international/metric paper sizes: https://www.cl.cam.ac.uk/~mgk25/iso-paper.html
American: "Wow -- yet another example of practical math linked to everyday sources."
European: "Yes, but not your everyday sources."
1190mm (A0 height) / 841mm (A0 width) ⩰ √2
11" (US letter height) / 8.5" (US letter width) = 1.29411 ...
Countries using the Metric system: 192
Countries using the Imperial system: 3 (U.S., Liberia, Myanmar)
The U.S. rejection of the Metric system can be traced to Thomas Jefferson, who declared it "Too French."
> We drink our tea from doughnuts.
Lol
Great video about human topology: https://www.youtube.com/watch?v=egEraZP9yXQ
Trying not to be too negative but this is quite a long blog post for what effectively just comes down to using a piece of paper to measure something, which I imagine lots of us have done in a pinch. Also, nit: one of the calculations on paper size seems duplicated due to a typo.
Personally I make sure to always carry around a string and some scissors, so any time I need an arbitrary measurement I can just cut the string to the length of the thing I want to measure (making sure to label it in case of multiple required measurements), then measure the cut string later. Simple.
Although I still haven't figured out the best way to do that in reverse (when someone wants a specific measurement and I cut the string from that number), though I was considering a scheme where I start with strings of known length up-front then repeatedly cut successive halves until I converge on the desired number, accounting for cut accuracy and require precision.
Maybe you could start off with a long string and then mark off its midpoint, the midpoints of the halves, and so on. Then you wouldn't have to cut a new string every time you want to measure something.
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The span from my thumb to my pinky in a “measuring position” is 20 cm (and is easily repeated by moving the thumb to the pinky and then stretching out the pinky again). The length of my “thumbs up” hand is 16 cm. The width of my fist is 10 cm. The length of my pinky is 6 cm. The width of my thumb is 2 cm. This allows me to estimate distances between ~2 m and 2 cm pretty well. Knowing your foot/shoe length also comes handy sometimes.
I do the same, but my piece of string has little lines and numbers telling me how long a thing is.
Reverse: perhaps carrying a measuring tape [0] ?
[0] https://www.amazon.com/Measure-iBayam-Measuring-Measurement-...
How long is a piece of string eh?
That works but with a piece of paper you can tell the time too, so long as you get a nice man to write it down for you[1]. ;)
1. And now the confusion of having to explain the Goon Show: https://www.youtube.com/watch?v=rLQhQSiDR-k
So besides the string and the scissors, you also need a marker that can write really, really tiny?
But this is also quite cool! These kind of articles are what makes HN interesting and fun.