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Paper Play

Geometry & modular

The A4 ratio (√2)

A few folds turn a square into a rectangle with exactly the shape of A4 paper — and show why halving it keeps the same shape.

  • DifficultyEasy
  • PaperSquare, 15 × 15 cm
  • Steps6
  • About3 min

Why this level: Just a few easy folds; only the corner-onto-diagonal fold needs care.

You need: 1 sheet of origami paper

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Story

In 1786 the German scientist Georg Christoph Lichtenberg wrote about the advantages of paper whose sides are in the ratio √2 : 1: folding it in half does not change its shape. In 1922 Germany made this the paper standard DIN 476, with the largest size, A0, measuring exactly one square metre. By 1975 so many countries used it that it became an international ISO standard.

Did you know? A4 (210 × 297 mm) is A0 halved four times, so sixteen A4 sheets together make one A0 sheet of one square metre.

Learn more (external sources)

Steps at a glance

  1. 1. Crease a diagonal

    Coloured side up. Fold the bottom-right corner to the top-left corner and unfold.

  2. 2. Bottom edge onto the diagonal

    Fold the bottom edge up to lie along the diagonal, creasing from the bottom-left corner. The bottom-right corner lands on the diagonal.

  3. 3. Fold the top behind

    Mountain-fold the top strip behind along a horizontal line through the folded corner.

  4. 4. Unfold the corner

    Unfold the bottom corner. You have a √2 : 1 rectangle — the shape of A4 paper.

  5. 5. Fold in half

    Fold in half, left edge to right edge. It is smaller but the same √2 : 1 shape — just like A4 halved into A5.

  6. 6. Unfold: done

    Unfold again: one √2 rectangle is made of two smaller √2 rectangles.

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