Ever wondered why a piece of A4 is the size it is? No? OK, let's talk about aliens and we'll come back to it.
How would you communicate with an alien? Without a common language, talking or writing would not work. There's no guarantee they'd have arms or eyes, so sign language would be out.
This problem puzzled Nasa when the Pioneer 10 probe was launched in 1972.
This was to be the first man-made object to leave the Solar System, and the scientists wondered how it would be received if intercepted by aliens. They fitted a plaque to the side of the probe, with directions to Earth and crude representations of human beings.
Because they knew that aliens would not understand metres, they showed the size of people in terms of the wavelength of a universal vibration in the hydrogen molecule, represented in binary arithmetic. The whole exercise was a bit of a gesture, because it is estimated that the time taken for the probe to come within detectable distance of another star system would be comparable with the current age of the galaxy. But it did get people thinking about what is truly universal.
This is where mathematicians wax lyrical on their subject. English literature is relevant in English-speaking countries and history is relevant on Earth - but maths applies everywhere.
Now return to the sheet of paper. Suppose you give your class a sheet of A4 each, and the following scenario: aliens have landed on Earth and you have to deliver a message of peace demonstrating that we are a civilised race.
All you have is this piece of paper. Promise them you will do the same exercise and compare results in the next lesson. Imaginative pupils will come up with some intriguing ideas. You might even get a paper aeroplane.
But when you present your offering, it will be completely blank.
This is only natural, because aliens aren't going to understand written English. It took Egyptologists years to decipher the ancient scripts of the Nile, and they were written by fellow Earthlings with the same concerns of harvests, love and wars. Aliens may be completely different, but they will understand maths.
The first thing they would notice is that the paper is right-angled. That is unlikely to have occurred naturally, but not impossible. Some crystals have right-angled structures. Next, they would measure the sides of the paper.
You and your class should do the same. Do you get 297 x 210 mm? Good - the aliens would get the same. Only they wouldn't because they would get 1,692 x 1,196 zlhurgs. None of these numbers is particularly special until you divide the length of the long side by the length of the short side.
Whichever units you use - millimetres, inches or zlhurgs, the ratio will be the same - about 1.414.
Your class might not recognise that number as anything special either, but ask them to multiply it by itself. They should get a number staggeringly close to 2.000.
Coincidence? Of course not. The people who designed the sizes of paper, from A0 down to A6, knew it would be handy if each sheet of paper had the same proportions. That way, the same image or text could be reproduced at different sizes and still fit the shape of the paper. We couldn't use photocopiers otherwise.
They also wanted each paper to be the same size as its predecessor, folded.
Maths determines that the ratio has to be the square root of 2. So the aliens would conclude, just from a blank sheet of paper, that we were sufficiently advanced to know the square root of 2 and that it is a useful constant in stationery.
Although this explains the paper proportions, it doesn't answer the original question about the size. For this, you need to know, or to work out, the size called A0, the largest of the paper family. Remember, a sheet of A3 has a short side the same as the long side of A4, and so on.
So, how many square millimetres is a sheet of A0? Thanks a million for working it out!