We all know the A4 format as a rectangle of 21x29.7 cm. But who knows that it is a mathematical construction that implies an A0 format and subsequently A1, A2 and A3?
The basis is the A0 format, a rectangle with a 1 square meter area, with particular proportions. We shall find the ration of the length L0 over width l0 when we construct the A1 format.
The A1 format is half a A0 format. More precisely, its length L1 is the width l0 of the A0 format, and its width l1 is half the length L0 of the A0 sheet.
The characteristic property of the A0 to A1 construction is that the ration of the length to the width is the same for both formats: L1/l1=L0/l0.
But L1=l0 and l1=L0/2, so that L1/l1=l0/(L0/2)=2x(l0/L0)=2/(L0/l0). As this is equal to L0/l0, if we multiply both sides by L0/l0, we get the square of L0/l0 is equal to 2.
So that L0/l0 is the square root of 2. L0 is approximately equal to 1.41xl0.
But the suface of the A0 format, l0xL0, is equal to 1 square meter, or 100x100=10,000 square centimeters.
So that the square root of 2 multiplied by the square of l0 in centimeters is equal to 10,000. Solving that leads to l0 is the square root of 10,000 divided by the square root of 2. And L0 is l0 multiplied by the square root of 2.
This gives l0 approximatlely equal to 84.1 cm and L0 is approximatly equal to 118.9 cm.
Thus the A0 format is 84.1x118.9 cm.
For the A1 format, we exchange the length and the width and divide the old length by 2 to obtain the new width: A1 is 84.1x59.5 cm, with the same proportions as A0 format, as we constructed the A0 format so.
Then we continue: the A2 format has length L2=l1 and width l2=L1/2.
The proportions of A2 format are stil the same: L2/l2=2/(L1/l1), that is 2 divided by the square root of 2, that is the square root of 2.
A2 format is 42x59.5 cm.
Then A3 format is constructed by dividing the A2 rectangle in 2. The ratio L3/l3 is also the square root of 2, as it may be shown the same way as for A2 construction from A1 format.
A3 format is 29.7x42 cm, as it is known.
And now, the famous A4 format, dividing the length of the A3 sheet in 2. The ratio L4/l4 is also the square root of 2.
And the A4 so constructed is 21x29.7 cm, as we already knew!
But now we know where it comes from. And we know as well why half an A4 is A5 and half a A5 is A6, and so on.
Amazing, isn't it?