The paper we usually use has a standar size. In lots of countries in the world (but not in North America) we use paper size standars based in ISO 216 and we use world like DIN A0, DIN A1, DIN A2, DIN A3, DIN A4 an so on.

The base DIN A0 size of paper is defined to have an area of one square meter, and successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension. The objective is that this parts again have the same aspect ratio.

We can calculate this aspect ratio:

The aspect ratio verifies (these two rectangles are similar):

Then

Or

Then the larger side is equal to the diagonal of a square of size the shorter side:

DIN A0 size has one square meter. We can calculate his dimensions (rounded to milimeters)

In a photocopier, when we want to reduce from A3 to A4 the display shows a ratio of 71%. ¿Why?

I have used this proportion in the animation about the sum of the geometric series of ratio 1/2.

The doors of this piece of furniture are in the same proportion. It has been designed and made by Roberto Cardil using pine and oak wood. You can see another furniture with the golden spiral.

This proportion is different than the golden proportion.

LINKS

	The golden ratio From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio.
	Sum of a geometric series of ratio 1/2 The geometric series of ratio 1/2 is convergent. We can represent this series using a rectangle and cut it in half successively. Here we use a rectangle such us all rectangles are similar.
	The golden rectangle A golden rectangle is made of an square and another golden rectangle.
	The golden rectangle and the dilative rotation A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.