A letter from 1786
The earliest written trace of the idea comes from the German physicist Georg Christoph Lichtenberg, in a letter to his friend Johann Beckmann dated 25 October 1786. Lichtenberg had noticed a property that most rectangles lack: if you cut a rectangle exactly across the middle of its long side, the two resulting halves keep the same proportion as the original only when the ratio between the sides is 1 to the square root of two. Any other ratio produces a shape that looks visibly different from the one you started with.
The practical consequence arrived 136 years later. In 1922, the German engineer Walter Porstmann turned Lichtenberg's observation into a full industrial standard, DIN 476, adding a second rule: the base size, called A0, has an area of exactly one square metre.
Those two equations fix the dimensions of A0: 841 × 1,189 mm. Every following size comes from cutting the previous one exactly at the midpoint of its long side — A1 is half of A0, A2 is half of A1, and so on down to A10, a 26 × 37 mm sheet about the size of a sugar cube.
Romania, 1949
The German format spread across Europe country by country, over decades rather than years. Romania adopted it as a national standard in 1949, 26 years before the same format became an international standard through ISO 216, in 1975, and well before the United Kingdom or France made the same move.
| Country | Year of national adoption |
|---|---|
| Germany | 1922 |
| Switzerland | 1929 |
| Soviet Union | 1934 |
| Hungary | 1938 |
| Spain | 1947 |
| Austria | 1948 |
| Romania | 1949 |
| Japan | 1951 |
| United Kingdom | 1959 |
| France | 1967 |
| Australia | 1974 |
| International standard (ISO 216) | 1975 |
| United States | never adopted |
The United States kept the Letter format (216 × 279 mm), inherited from eighteenth-century printing sheets and fixed only in 1995 as an ANSI standard. The ratio between Letter's sides is 1.2917, not 1.41421. A document sent from the US for printing in Europe never scales cleanly onto A4: either a white margin stays empty, or a strip of text runs off the edge.
The button on the copier
Because the ratio stays identical at every step, enlarging from any A size to the next larger one always uses the same value: the square root of two. Copiers everywhere have this number fixed on a button, rounded to a whole percentage.
Jumping two sizes at once, from A4 straight to A2 for instance, is simply 141% applied twice, or 200%. The calculator above computes the exact percentage for any pair of sizes.
The physical limit of halving
Mathematically, the A series descends without limit past A10. Physically, a real sheet cannot keep up: with every fold the paper's thickness doubles, so the number of layers grows exponentially rather than linearly. The idea that no sheet can be folded more than seven or eight times circulated for years as settled fact, until the American high-school student Britney Gallivan derived, in 2002, an exact formula linking paper thickness, required length, and the number of possible folds, then demonstrated 12 folds on a specially chosen sheet far larger than an ordinary A4. For a standard 80 g/m² A4 sheet, the limit still sits at roughly seven or eight folds.