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linuxdoctor
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Joined: 23 Apr 2005
Posts: 1307
Location: Ottawa, Canada
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 Posted: Sun Feb 10, 2008 3:05 pm Post subject: ISO 216 Paper Sizes |
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One of the things that always fascinates me is mathematical concepts that turn into real life products and inventions. One of these inventions is the ISO 216 paper sizes which are derived from the German DIN 476 standard. The most familiar aspect of these standards are the well know A-series of paper sizes of which A4 is used as the standard European 'Letter' size.
The concept behind the DIN 476/ISO 216 standard sizes is based on a simple mathematical principle first noted in the 18th century by the German scientist Georg Lichtenberg. What he notes was that if a sheet of paper were cut in such a way that the aspect ratio, that is to say the length of the sheet divided by its width, is √2 then each time you divided the sheet of paper into two equal halves along its width (making it half the length) the aspect ratio would again be √2.
This can be shown mathematically as follows:
Let the length of the sheet be l and the width of the sheet be w such that l / w is √2. If we now divide the sheet in half along it's width l / 2 becomes the width and w becomes the length of that new sheet with aspect ratio w / ( l / 2 ). This is the same as 2w / l. We aleady know that l / w is √2 so 2w / l is 2 / √2 which is again √2. QED.
It doesn't matter how many time you divide the sheet in half in this way, the aspect ratio will aways be the same. For me this is fascinating stuff and this fascinating mathematical principle has it's application in the A-series of paper sizes. This series begins with A0 which is a sheet of paper that has an area that is exactly one square metre is size. Each succeeding size A1, A2, etc. has an area exactly one half previous size so A1 is 1/2 m², A2 is 1/4 m² and so on. The standard A4 'Letter' size would have an area of 1/16 m². The dimensions of each size could then be calculated based on its proportion of the A0 sheet.
What are the dimensions of the A0 sheet? That's easily calculated. We know that the A0 sheet is 1 m² it's aspect ratio is √2. This means that product of it's length and width is 1 while the quotient of dividing the length by the width will be √2. Calculating we get:
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(1) l · w = 1 square meter
(2) l / w = √2.
This means that
(3) l = w √2.
Replacing l in equation (1)
(4) (w √2) · w = 1, and
(5) w² √2 = 1 meaning
(6) w² = 1 / √2 or
(7) w = 2^(-¼) = 0.840896 metres.
We know from (1) that the length times the width must be 1, so
(8) l = 1 / w = 1.189207 metres.
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For each succeeding paper size, the area is one half the previous one and so the dimensions are calculated in the same way replacing 1 square meter appropriately. For the standard A4 'Letter" size we get:
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(1) l · w = 1/16 square meter
(2) l / w = √2.
This means that
(3) l = w √2.
Replacing l in equation (1)
(4) (w √2) · w = 1/16, and
(5) w² √2 = 1/16 meaning
(6) w² = (1/16) / √2 or
(7) w = (1/4) · 2^(-¼) = 0.210224 metres or 210.224 millimetres..
We know from (1) that the length times the width must be 1/16, so
(8) l = (1/16) / w = 0.297302 metres = 297.302 millimetres.
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The standard rounds these numbers out the nearest centimeter so the dimensions of the A0 sheet is 1189 by 841 millimeters and the A4 sheet is 297 by 210.
Neat stuff, in my view.
_________________
Misanthrope: someone who realizes that humans really are as stupid as they appear.
If you think I'm 'politically' incorrect you have the wrong politics.
Last edited by linuxdoctor on Sun Feb 10, 2008 3:46 pm; edited 1 time in total |
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Pie32
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LP-SolidRaven
Dictator of the Dump
Joined: 06 Jun 2004
Posts: 7359
Location: The cheese is made out of moon
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| Posted: Sun Feb 10, 2008 3:39 pm Post subject: |
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297cm would be 2.97m. It's 297mm and 210mm 
_________________
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<bart416> I just realized something
<bart416> we celebrate the fact that this piece of rock made one rotation around a glowing ball of plasma that is kept together due to its own gravity well
<njsg> HAPPY NEW YEAR
<Easter> ^^
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linuxdoctor
Infallible Persona
Joined: 23 Apr 2005
Posts: 1307
Location: Ottawa, Canada
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| Posted: Sun Feb 10, 2008 3:44 pm Post subject: |
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| SolidRaven wrote: | | 297cm would be 2.97m. It's 297mm and 210mm |
Oops. My infallibility is slipping again. Right, that's millimetres. 
I went back and changed the text so now it reads correctly.
Or was it ever wrong in the first place??? Right, it was your imagination guys. My infallibility remains intact. 
_________________
Misanthrope: someone who realizes that humans really are as stupid as they appear.
If you think I'm 'politically' incorrect you have the wrong politics. |
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LP-SolidRaven
Dictator of the Dump
Joined: 06 Jun 2004
Posts: 7359
Location: The cheese is made out of moon
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| Posted: Sun Feb 10, 2008 4:02 pm Post subject: |
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| linuxdoctor wrote: | | SolidRaven wrote: | | 297cm would be 2.97m. It's 297mm and 210mm |
Oops. My infallibility is slipping again. Right, that's millimetres.
I went back and changed the text so now it reads correctly.
Or was it ever wrong in the first place??? Right, it was your imagination guys. My infallibility remains intact. |
Didn't see a thing  Also, there are variants on this sizes though.
(I'm typing these over from a little pocket book with most common standards)
The most common ones are:
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A0 841x1189
A1 594 x841
A2 420x594
A3 297x420
A4 210x297
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Long formats:
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A3x3 420x891
A3x4 420x1189
x4x3 297x630
A4x4 297x841
A4x5 297x1051
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Special formats:
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A0x2 (2A0) 1189x1682
A0x3 1189x2523 *
A1x3 841x1783
A1x4 841x2378 *
A2x3 594x1261
A2x4 594x1682
A2x5 594x2102
A3x5 420x1486
A3x6 420x1783
A3x7 420x2080
A4x6 297x1261
A4x7 297x1471
A4x8 297x1682
A4x9 297x1892
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* next to one means that you can't put it on a single microfilm.
And I'm now going to continue to play on my guitar or go to sleep (hard choice).
_________________
| Quote: |
<bart416> I just realized something
<bart416> we celebrate the fact that this piece of rock made one rotation around a glowing ball of plasma that is kept together due to its own gravity well
<njsg> HAPPY NEW YEAR
<Easter> ^^
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