The Other Love Story

Paper sizes in formats A0 to A8 exhibited at the science museum CosmoCaixa Barcelona. Source Wikimediacommons.org (Creative Commons License)

Many in the world have caught up in the frenzy of Valentine’s Day (February 14th), which started as a Christian feast-day honoring Saint Valentine, one of two early Christian martyrs. Valentine’s Day is now a global event celebrated by Christians and non-Christians alike and driven by aggressive marketing. In the US, for example, Valentine’s Day spending is expected to reach $23.9 billion, the second-highest spending ever, and representing an increase of almost 10 percent on sales in 2020.

I argue that the Valentine’s Day frenzy is a good opportunity to talk about the other love affair, between root two, and paper. “Root two” is shorthand for the square root of 2 (i.e., √2). For those who suffer from math phobia, it should encourage you to note that all you have to remember is that a root (generally taken to be the square root, unless otherwise stated) of a number (say, “A”) is that number (say, “B”) which multiplied by itself will equal A. For example, the square root of 4 is 2 because 2 times 2 is 4. Similarly, the square root of 100 is 10 because 10 times 10 is 100. On the other hand, the cube root of 1,000 (∛1,000) is 10 because 10 times 10 is 100 which when multiplied by 10 gives 1,000.

If punch the number 2 on your calculator and hit the square root (√) button, you will get the answer 1.414 (to three decimal places). To make sure, multiply 1.414 by 1414. You will get 1.999, which can be rounded up to 2. The result you got is not exactly 2 because we chose only three decimal places of root two to square. The more decimal places you choose, the closer its square will be to 2.

So why the fuss about root 2?

Well, I found this out by accident a month or two ago when I searched for the standard for paper sizes around the world. Specifically, I wanted to know which countries in the world use the A4 paper, and which countries use Letter size paper.

What I found is what I call a love story between root two and paper. The love story starts in 1786, when a German physics professor Georg Christoph Lichtenberg found that any sheet of paper whose length is √2 times its width will, if the long side is folded in half, yield two identical two sheets, each the same length-width (aspect) ratio as the original.

No wonder then, that when Germany standardized paper sizes in 1921, they used the 1:√2 aspect ratio for four series of paper formats. In addition, the Germans linked what became their DIN 476 system to the metric system by using a width of 1 cm for the base format. The A series is the preferred format and form the basis for the other series. The A0 size, which is 841 mm wide by 1,189 mm long, is the largest in the A series, and has a nominal surface area of 1 square metre. However, its actual area is 0.999949 square metres, because of rounding errors.

Many countries contributed to the development of the DIN 476 standard for paper sizes, and many adopted it as their national standard. Among these countries are Belgium (which adopted it in 1924), the Soviet Union (1934), Argentina (1943), Japan (1951), Venezuela (1962), Greece (1970), and Kuwait (1975). The DIN 476 standard became international standard (ISO 216) and a UN document format in 1975.

ISO 216 defines the “A,” “B,” and “C” series of paper sizes, including the A4. ISO also defines two supplementary standards, ISO 217, and ISO 269 to define related paper sizes, with ISO 269 “C” series sizes used for envelopes, and often listed with A and B sizes. All ISO standards (216, 217, and 269), except for some envelopes, have the same 1:√2 width to length aspect ratio.

This unique property of the ISO 216 and related standards means that starting from the largest size A0 (1189 mm x 841 mm), each successive smaller size can be produced by folding the preceding sheet into equal halves along its width. Thus, A1 size is produced by folding the A0 paper widthwise into two (841 mm x 594 mm) halves. The A2 size is produced by folding the A1 size widthwise into two 594 mm x 420 mm halves, and so on.

When we fold the A0 sheet four times, we will get 16 A4 sheets, each 297 mm long, and 210 mm wide. The aspect ratio of the A4 paper is 297 mm divided by 210 mm which yields (to three decimal places) 1.414, or (√2). The fun doesn’t stop there, because if you fold an A4 paper parallel to its width into two equal halves, what was the width of the A4 paper now becomes the length of the two halves, i.e. 297 divided by 2, which is 148 mm (after rounding the result down). The aspect of the two new halves (each size A5) then become 210 mm divided by 148 which is 1.419, or roughly √2!

The A4 paper size is the standard worldwide, except for North America (US and Canada), a few countries in Latin America, and the Philippines, which use the Letter paper size (216 x 279 mm or 8.5 x 11 inches). Although the Letter size format was introduced in the US in 1921, it was only declared the official US government paper size by President Ronald Reagan in the early 1980s.

Since its invention about 2,000 years ago in China, paper has become an integral part of our lives. The industry also has a huge impact on the environment, given that 20 million trees are cut down to make paper for printing books, and that 40 percent of the 4 billion trees that are cut each year is used to make paper. Despite this, the love affair between √2 and paper will see many more Valentine’s Days to come because the global pulp and paper market is expected to grow from $351.51 billion in 2021 to $370.12 billion in 2028. I guess you can say that again, love conquers!

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