Paper Sizes

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Rectangular paper with an aspect ratio of √2 has the unique property that, when cut or folded in half midway between its shorter sides, each half has the same √2 aspect ratio and half the area of the whole sheet before it was divided. Equivalently, if one lays two same-sized sheets paper with an aspect ratio of √2 side-by-side along their longer side, they form a larger rectangle with the aspect ratio of √2 and double the area of each individual sheet.

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North America, including the US, Canada and parts of Mexico, is the only area of the first world that doesn't use the ISO 216 standard paper sizes, instead they use Letter, Legal, Executive and Ledger/Tabloid paper sizes

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ANSI (American National Standards Institute) defined a regular series of paper sizes based around the Letter (8.5" x 11") format, with this becoming the A sizes and larger sizes being B,C,D & E. Surprisingly these ANSI standard sizes were defined in 1995 well after the ISO standard sizes. ANSI A sized paper is commonly referred to as Letter and ANSI B as Ledger or Tabloid.

Unlike the ISO standard sizes which have the single aspect ratio of 1:root2, ANSI standard sizes have two aspect ratios 1:1.2941 and 1:1.5455 which means that enlarging and reducing between the sizes is not as easy as with the ISO sizes and leaves wider margins on the enlarged/reduced document.

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The ISO 216 paper sizes have some useful properties for reducing and enlarging in print or photocopying such that the scale factor needed to convert from A3 to A4, at 71% is the same as the scale factor to convert between A4 and A5, A5 and A6 and so on.

Amos Bannister wrote:

Maybe I am biased, living in a metric country, but I love the ISO paper sizes. I can print my documents 2 or 4 to a page with no distortion and no waste of paper. Being able to accurately copy two pages onto one with no distortion or waste is also a boon when using a photo copier. Also, when I was producing a newsletter for a computer user group years ago, I could print the newsletter on A4 paper, then simply copy the pages onto A3 and fold the A3 pages - no waste, no fuss.

Like the metric system, it is really simple to use and get used to, but people who have grown up with imperial measurement systems sometimes take a lot of convincing that "the way we always do it" is not always "the best way".

-- Amos Bannister

That's interesting using the square root of two as the ratio for the ISO paper sizes. Do you know if that is the only ratio which has those properties? Wouldn't any whole-integer ratio have them?

Unlike the metric system, I can't see any meaningful advantage to using the ISO paper sizing.

If you cut a 4x8 sheet of plywood in half, you get two 4x4s. If you lay two side by side you get 8x8. So the ratio works EVERY OTHER iteration, rather than every halving/doubling.

_Howard wrote:

Wouldn't any whole-integer ratio have them?

I don't know why I wrote that. It should have been "Would any non-whole-integer ratio have them?" (I guess I could have just written "fractional".)

A few quick calculations show that fractional ratios always alternate with doubling in size (at least the ones I tested). And as the ratio approaches the square root of two, the difference in the alternate ratios diminishes, indicating that it would be zero at the square root of two. So I guess that is the only ratio which shows the described properties.

Yup, 71% is the inverse of 1.414213562373095.  Now, does it strike you as odd that reducing an image to 71% results in a duplicate exactly half the original area?

No, it doesn't strike me as odd. 0.7071... squared is 0.5. Conversely, 1.414...squared is 2.
You are not reducing the area of the image to 71%; you are reducing the sides of the image.

But those numbers do support the idea that the square root of two is the only ratio that will support the properties described in your first post.

_Howard wrote:

But those numbers do support the idea that the square root of two is the only ratio that will support the properties described in your first post.

No, that statement is wrong. Any rectangle whose sides are multiplied by the square root of two will have twice the area of the original, while multiplying the sides by 0.7.0... will have half the area. The ratio of the sides does not matter in this case.

So, only the 90º rotation is dependent on √2?

Retaining the aspect ratio when doubling the length of one side seems to be dependent on √2 as the aspect ratio.

I went searching and found a mathematical explanation of all this. Geeky Article (pdf)

Oh that wasn't too geeky. Found it pretty clear actually. 10Q

Geeky doesn't mean it can't be clear.

What I would like to find is a mathematical proof that no other ratio will work. Before finding and reading the article, I had deduced much of what it said, but I am not sure that no other non-terminating ratio will work.

Isn't the proof that 1.414... is the square root of 2?  IOW, if you want to double the area, multiply the height & width by 1.414...

Makes sense to me(?)

And the only shape where this works at 90º is where the height is 1.414... times the width.

NoCoPilot wrote:

Isn't the proof that 1.414... is the square root of 2?  IOW, if you want to double the area, multiply the height & width by 1.414...

No, if you want to double the area, multiply the height or the width by two. For example, take an 8x10 rectangle. Multiply both sides by √2 and you get an area of 128, instead of 160.

For some reason, I suspect that √2 works because it is an irrational number.

What I wonder is if other irrational numbers used as the ratio will give similar results. They won't double the area, of course, but is there some other constant area multiplier that would result while retaining the aspect ratio.

I want to see a formal mathematical proof that rejects all ratios other than √2.