How You Really Use Mathematics To Define Paper Size

Have you ever wondered how paper size is defined? The most common paper size you and I probably use is the A4. If you work in or have exposure to technical disciplines, you would know about other paper sizes such as A0, A1, A2, etc. ISO 216:2007 defines most of the international paper sizes, whereas other ISO standards such as ISO 269 and ISO 217 cover the rest.

But where did these norms come from? 

What existed before these norms and definitions? 

Who came up with these specific numbers? 

And why?

Well, that’s an interesting set of questions. It turns out that there is a rich and mysterious mathematical history behind the invention of paper size standards.

The mathematics as such is subtle, yet deep. In this essay, I start with the history and proceed to explore the mathematics behind the paper size standards. Finally, I touch upon the beautiful human aspect that is also subtly embedded in the expression of this technology. So, sit back, relax, and enjoy the pleasant ride!

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The History

Georg Christoph Lichtenberg was a well-known German physicist and satirist in the 18-th century. Honoured as an extraordinary professor of physics at the University of Göttingen, he was known to be one of the first scientists to introduce experiments with apparatus in their lectures.

He also maintained relations with other great German figures of the era such as Johann Wolfgang von Goethe and Immanuel Kant. Legendary mathematician Karl Friedrich Gauss is known to have sat on Lichtenberg’s lectures. He is also well known for his discovery of tree-like electrical discharge patterns which came to be known as Lichtenberg figures.

Among the array of scientific problems that he was solving at the time, Lichtenberg took a specific interest in standardizing paper size. He wanted to derive a standard that would enable the perfect scaling of content size on paper. Perfect scaling here would mean neither waste nor want of space on paper across a range of potentially standardized magnification ratios.

He posed this as an exercise problem to one of his British algebra students. The student came up with a specific ratio that has the preferred property (more on the ratio later). When Lichtenberg proceeded to try and apply this ratio practically to a sheet of paper, he was pleasantly surprised that the paper of the time in Germany already featured this ratio.

In a letter to Johann Beckmann in 1786, he expressed uncertainty as to whether this ratio had occurred due to tradition (naturally) or if it came from precise mathematical calculations (for more details, the letter is linked in the references at the end of this article). Either way, this story is the first recorded occurrence of the mathematics behind the paper size standards that are followed until this day.

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The Ratio of √2

√2 is a very interesting number in mathematics. When you apply the Pythagorean theorem to a right-angled triangle that has unit length and unit height, the length of the hypotenuse turns out to be √2. 

The diagonal of a unit square would hence be √2 as well. √2 is an irrational number with an infinite decimal value of 1.4142135623730950488016887…

What Lichtenberg (and his student) figured out is that when a sheet of paper is treated as a rectangle whose longer side is √2 times the shorter side, the paper size scales with neither waste nor want along with magnification.

We will see how this works mathematically in a bit. But first, it perhaps helps to understand this concept geometrically.

Consider an A0 sheet of paper. It is rectangular with dimensions of 841 mm x 1189 mm. If you fold it in half along its longer side, each folded side would represent one A1 sheet each. If you fold the two A1 sheets along their respective longer sides, you will end up with four A2 sheets. Repeating the process would yield eight A3 sheets, sixteen A4 sheets, thirty-two A5 sheets, sixty-four A6 sheets, 128 A7 sheets, and 256 A8 sheets respectively each time around.

This is the reason why modern printers can quickly scale printable content to different paper sizes. For instance, if one wishes to save paper, one could shrink the pages of a book (digitally) to A6 scale first and then print the book on an A4 sheet of paper. Each A4 sheet would then contain 4 pages of the book (on each side), thereby compressing the information-to-space ratio.

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What Is Special about √2?

Remember when I told you that we will cover the mathematics behind the ratio? Well, this is it. It turns out that the mathematics behind this ratio is quite clean and straightforward.

Let us consider a sheet of paper whose longer side is ‘a’ units long and shorter side is ‘b’ units long. If we fold this paper along its longer side, we would end up with two sheets of paper, each with a longer side that is b units long and a shorter side that is ‘a/2’ units long.

Now, replicating Lichtenberg’s exercise problem to his algebra student, we require that the ratio between the long side and the shorter side be preserved in the bigger sheet of paper as well as the two smaller sheets of paper (after the fold). Then, this becomes a simple math problem that can be solved as follows:

When we solve the problem mathematically, it becomes clear as to why the ratio cannot be anything other than √2 when we consider positive lengths only.

The Rest is History

Following Lichtenberg, the French published a law on the taxation of paper in 1798, which turned out to be direct ancestors of the present ISO norms (link to reference at the end of the article). 

W. Porstmann argued in a 1918 article that the paper size standards need to incorporate the surface area involved as well. He also argued that envelopes of said paper sizes should be 10% larger. Owing to his influence, the Standardisation Committee of German Industry (Deutsches Institut für Normung — DIN) published the DIN 476 for four series of papers, each featuring an aspect ratio of √2.

The A0 size was defined to have a surface area of 1 square metre (841 mm x 1189 mm) when rounded to the nearest millimetre. A4 was recommended as the standard paper size for business and administrative activities. A6 was recommended for postcards. The series B sheets were based on B0 with a width of 1 metre. The series C sheets were developed as the basis for envelope formats.

Fast forward to today, these standards have been adopted by almost every country except a few such as North America, Peru, Colombia, etc.

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Further Mathematical Significance

The ratio of √2 features some counter-intuitive properties.

Portrait and Landscape

We have seen so far that we can fit two A5 sheets of paper in one A4 sheet. Let us say that we are interested in printing in landscape rather than portrait. By how much do we need to shrink the original A4 (portrait) content in order to fit two pages on a single A4 sheet in the landscape orientation? Intuition says 50%.

However, since the aspect ratio is √2, we need to just shrink the content by 70% and not 50%. This is because (1/√2) = 0.7071…, which is approximately equal to 70% (0.70).

Geometric Mean

It turns out that the concept of the geometric mean is very useful when it comes to packaging different sizes of paper. For instance, the envelope size C2 is the geometric mean between A2 and B2. Similarly, the entire C series formats are geometric means between the corresponding A series and B series numbers.

Final Thoughts

When I started the investigation and research into this topic, I had a ruler and measuring tape constantly by my side. I found myself measuring the aspect ratio of any rectangular shape that pleased me aesthetically.

Among objects that I measured were the following: my writing table, monitor, mouse pad, tablet, a mint-confectionary box, a rectangular plate, physical books, etc. The rectangular shapes that pleased me the most had an aspect ratio between 1.31 and 1.64 (√2 = 1.4142..).

Firstly, I was taken aback by just how many objects around me featured the rectangular shape. I did not pay as much attention to their shape as I did after I started work on this topic. Secondly, I was really surprised by how much my aesthetic taste was aligned towards the ratio of √2.

Whether this is something that I learned through cultural conditioning or if this is a phenomenon that occurs in nature is something that I am unsure of. For now, all I can do is appreciate the fact that there is a beautiful human aspect to this besides the clean mathematics.

Sometimes, the deepest of realisations come from the simplest of observations!

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References: Lichtenberg (letter to Johann Beckmann) and Loi sur le timbre — no. 2136 (French law on the taxation of paper).

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Further reading that might interest you: Why Do You See Mirrors Flipping Words? and How To Tell A Real Law From A Fake One?