The discussion of A4 paper led me to consider other possibilities. I AM NOT PROPOSING ANY OF THESE. I'm just "thinking out loud"* and you should quit reading now and discard this message if you don't want just to play along.
*(Is there a better phrase we could introduce, like "thinking out loud", but meaning "thinking by composing and sending email or other electronic messages"?) ======================================================== One of the perceived problems with A4 paper (and the rest of the A series) is that the length and width are not, and cannot be made to be, simple numbers. This restriction is caused by the necessity of using the square root of two as a factor relating length (L) and width (W). So, would it be mathematically possible to accomplish the same thing with a different factor, perhaps one that is the square root of a perfect square (which would result in a whole number)? Since the factor or 2 came from the desire to cut a sheet of one size in half and get 2 sheets of a smaller size with the same shape (aspect ratio), we could accomplish our goal by choosing 4 instead of 2; that is, design our paper so that we could cut it into FOUR pieces and get 4 sheets of a smaller size but still the same aspect ratio. Just as the A series gets a L/W ratio of the square root of 2 because the paper is cut into 2 parts, this new series would get a ratio of the square root of 4 because the paper is cut into 4 parts. (The details of the mathematical proof are omitted here because most of you either already know it or don't care.) Of course, the square root of 4 is just 2, so this new paper would series would need to have the length be exactly twice the width (L=2 x W). When cut into four parts (by making three equally spaced cuts parallel to the short side), the new sized sheets (4 of them) would have a length equal to the width of the original, and a width equal to 1/4 the length of the original: new L = old W new W = old L /4 What would such a sheet of paper look like. Let's consider one that would be about the size of an A4 or an 8.5 x 11 inch sheet; standard writing paper sizes. We will figure out later how big the other sheets of the series would be. The length of A4 is about 295 mm (or 11.6 inches) while the length of 8.5 x 11 inch is about 280 mm (or 11.0 inches). So let's pick a nice round number in that vicinity. We could pick 10 inches if we wanted to work in inches; it could work just as well as a metric version. But since we are all metric proponents, let's pick a neat number of millimetres, instead. I'm going to use 300 mm for this discussion (but 256 mm might be a better choice, see later). If we choose 300 mm for the length then the width must be 150 mm. Try sketching or cutting and pasting a sheet of paper that size. It is not very much longer than A4 or 8.5 x 11 in. (300 mm vs. 295 for A4 and 280 for the other one, you know: 8.5 x 11 in.). But it is considerably narrower; the width of 150 mm is less than three quarters the width of either A4 or the other one. This narrowness might be awkward, but I think we could get used to it. The lengths of the smaller sizes would go (as we cut them in two) from 300 mm for our letter size, down to 150 mm for the next smaller size, then to 75 mm, then to 37.5 mm. (The next step down is too small to be of any practical use except maybe for small postage stamps.) Going up the scale to larger sheets, the lengths would be 600 mm, 1200 mm and 2400 mm (anything much larger being impractical for most purposes). Remembering that the width of these sheets would each be half the length, the sizes would be (with examples of things about that size): 37.5 mm x 18.75 mm (big postage stamp) 75 mm x 37.5 mm (Post-It note) 150 mm x 75 mm (index card) 300 mm x 150 mm (letter size) 600 mm x 300 mm (newspaper page) 1200 mm x 600 mm (poster) 2400 mm x 1200 mm (big sign board) We do lose some convenient intermediate sizes with this series. Each sheet size is only one quarter as large as the next larger one. So, if we wish to reduce the size of some printed material we would have to go (for example) all the way from 300 mm x 150 mm (about A4 size) down to 150 mm x 75 mm (index card size). There's nothing in between. But there is a rather simple solution to this dilemma. Just put something in between. Half way between lengths of 300 mm and 150 mm would be 225 mm. If a sheet of this length would be given a width of 112.5 mm, it would still have the same aspect ratio (2:1) as our main series. However, it could not be easily produced by cutting up one the larger sheets in our main series. But we could make this new sheet part of a NEW series. The new series would also have the length equal to 2 times then width and would be formed by cutting each larger sheet into four smaller sheets, just like the original series. This new series would have sizes as follows: 56.25 mm x 28.125 mm 112.5 mm x 56.25 mm 225 mm x 112.5 mm 450 mm x 225 mm 900 mm x 450 mm 1800 mm x 900 mm ALL of the sizes of the new series would fit nicely in between all the other sizes in our main series. We now have a double series, members of one series being interspersed uniformly between the members of the other series. All of them have the same aspect ratio and therefore enlargement and reduction could be easily made to the next size in the OTHER series for a moderate size change, or to the next one in the same series for a larger change, etc. Although ALL the paper sizes could not be produced by subsequent cuttings of the largest ONE, they COULD all be formed by cutting of just TWO larger ones. If the fractional values we encountered in this example are too inconvenient, one could always start with 256 mm instead of 300 mm. (288 mm also works well). Starting with 256 mm and proceding as above, the main series would be: 32 mm x 16 mm (big postage stamp) 64 mm x 32 mm (Post-It note) 128 mm x 64 mm (index card) 256 mm x 128 mm (letter size) 512 mm x 256 mm (newspaper page) 1024 mm x 512 mm (poster) 2048 mm x 1024 mm (big sign board) The secondary series would be: 48 mm x 24 mm 96 mm x 48 mm 192 mm x 96 mm 384 mm x 192 mm 768 mm x 384 mm 1536 mm x 768 mm The numbers are not so simple, but at least they are exact whole numbers with no fractional parts. Which one is nicer is a matter of opinion. The one thing lost in this is the abilitty to calculate the weight of paper easily because the large size from which all the others are cut is not a nice round 1 square metre. (It is 2.88 m^2 for the first main series above and 2.097 m^2 for the second main series, The largest sheets of each of the secondary series have areas of 1.62 m^2 and 1.18 m^2 respectively.) Having labored through all this (I did not know where it was going to lead when I began writing this!), I now wonder if I should retract my statement that "I AM NOT PROPOSING ANY OF THESE." It is beginning to look workable. i do NOT think it is BETTER than the A series, but I think it may be as good, and it soes have the advantage of fairly simple sizes for all lengths and widths IF THAT IS IMPORTANT and I'm not sure it is. (It may be as important as having the largest sheet be exactly one square metre in area.) So there it is, guys and gals. Tear it apart, but please remember, this was just for fun, I'm not suggesting it or promoting it's adoption; just discussing what is possible or might have been. Have fun! Regards, Bill Hooper college physics teacher (retired), USA (Florida) +-+-+-+-+-+-+-+-+-+-+-+ Do It Easy, Do It Metric! +-+-+-+-+-+-+-+-+-+-+-+
