# One Hundred and Forty-One Percent

> The enlarge button on any photocopier, from A4 to A3, reads 141%. That figure is the square root of two, written as a percentage, and it is why paper sizes A0 through A10 can keep halving forever without ever changing proportion.

- **Canonical URL**: https://mariuscomper.uk/o-suta-patruzeci-si-unu/en/
- **Romanian edition**: https://mariuscomper.uk/o-suta-patruzeci-si-unu/
- **Author**: Marius Comper (2026)

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## Key figures

- **141%**: The exact enlargement factor from A4 to A3, on any copier in the world — the square root of two, written as a percentage.
- **4.99 g**: The real weight of one A4 sheet at 80 g/m², computed directly from area and grammage.
- **0.0051%**: How far the real A4 ratio (297/210) deviates from the exact value of √2.
- **163 years**: Between Lichtenberg's letter in 1786 and Romania's national adoption of the format, in 1949.

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## 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.

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, A0, has an area of exactly one square metre.

$$l \times h = 1\text{ m}^2 \quad \Longrightarrow \quad h / l = \sqrt{2}$$

Those two equations fix A0's dimensions: 841 × 1,189 mm. Every following size comes from cutting the previous one at the midpoint of its long side — A1 is half of A0, A2 is half of A1, down to A10, a 26 × 37 mm sheet about the size of a sugar cube.

## Romania, 1949

Romania adopted the German format 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 (1959) or France (1967) made the same move. The United States never adopted it; it kept the Letter format (216 × 279 mm), with a ratio of 1.2917, not 1.41421.

## 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. √2 ≈ 141% (A4 → A3); 1/√2 ≈ 71% (A3 → A4).

## The physical limit of halving

Mathematically, the A series descends without limit past A10. Physically, a real sheet cannot keep up: paper thickness doubles with every fold. Britney Gallivan derived, in 2002, an exact formula linking paper thickness, required length, and the number of possible folds, and demonstrated 12 folds on a specially chosen sheet. For an ordinary A4 sheet, the limit still sits at roughly seven or eight folds.

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## Method note

The official A-series dimensions come from ISO 216:2007. Lichtenberg's letter is cited from the transcription kept by Markus Kuhn, University of Cambridge. Romania's adoption year (1949) comes from the "ISO 216" article and was cross-checked against Printcenter.ro and Papetti.ro. Gallivan's formula comes from her 2002 paper, reported by Science News (2004).