Stage 01 · A10
26 × 37 mm — a SIM card, the corner of a stamp.
A technical digression · A10 → A0
Eleven sheets of paper, one ratio, and a fold that survives itself forever. The sheet on screen never changes size. Everything around it does. Scroll to unfold it.
Stage 01 · A10
26 × 37 mm — a SIM card, the corner of a stamp.
Stage 02 · A9
37 × 52 mm — a matchbook cover, torn free.
Stage 03 · A8
52 × 74 mm — a business card, passed across a table.
Stage 04 · A7
74 × 105 mm — a passport photograph, an index card.
Stage 05 · A6
105 × 148 mm — a postcard, sent from somewhere.
Stage 06 · A5
148 × 210 mm — a paperback page, a field notebook.
Fold most rectangles across their short side and you get a different shape. Fold this one, and you get itself again — smaller, but exactly the same proportions, forever. That isn't an accident of the paper trade. It's the one ratio that survives being halved: 1 : √2. Cut a sheet with these proportions in half along its long edge, and each half keeps precisely the width-to-height relationship of the whole.
No other rectangle can say that. It's why enlarging A4 to A3 on a photocopier reads 141% — √2 × 100 — and not some number somebody had to go and measure. Every A size is simply one square metre divided by a power of two: A0 is a square metre outright; A4 is a sixteenth of one, 2⁻⁴. You can find any sheet's area with nothing but subtraction.
The whole system is one constraint, run out to its logical end. It's a little bit beautiful.
Stage 07 · A4
210 × 297 mm — the printer sheet everyone already knows.
Stage 08 · A3
297 × 420 mm — a folded newspaper.
Stage 09 · A2
420 × 594 mm — a film poster outside the cinema.
Stage 10 · A1
594 × 841 mm — a concert poster, a drafting sheet.
Stage 11 · A0
841 × 1189 mm — one square metre, exactly. The parent of them all.
US Letter — 215.9 × 279.4 mm — has no such property. Halve it and the two halves are a visibly different shape from the whole. Overlay the two here, corners pinned together: the mismatch is small, but it's real, and it never resolves.
Enlarging Letter to its next size up isn't 141%; it's whatever ratio that particular pair of sizes happens to be, worked out case by case, because there's no underlying rule generating them. It works. It just doesn't fold.
The sheet held its shape the entire way. It always does.
The ladder doesn't stop at either end. Keep folding past A10 and you pass the width of a human hair, then further still — the same shape, still halving. Keep unfolding past A0 and you pass a tabletop, a room, a field, the shape of a small country, if paper came in country sizes.