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Geometry & modular

Sevenths (Fujimoto’s method)

Seven can’t be reached by folding in half. But repeat the same three folds and the creases creep closer and closer to 1/7.

  • DifficultyIntermediate
  • PaperSquare, 15 × 15 cm
  • Steps6
  • About8 min

Why this level: Only easy edge-to-crease folds, but many of them, repeated carefully.

You need: 1 sheet of origami paper

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Story

Folding in half is easy, so halves, quarters and eighths are quick to make. Sevenths are harder, because you never reach 7 by doubling. Origami designers still need them sometimes, and this trick is known as Fujimoto’s approximation, after the Japanese mathematician Shuzo Fujimoto. Start from a rough guess and keep folding an edge to the newest crease: the creases creep closer and closer to the right places. The same idea gives fifths, ninths and other odd divisions.

Did you know? With this kind of repeated halving you can find any position along an edge to within one part in a thousand using no more than 9 creases.

Learn more (external sources)

Steps at a glance

  1. 1. A first guess: 1/8

    White side up; we will divide the bottom edge into seven. Fold the right edge to the left edge and unfold (1/2). Fold the left edge to that crease and unfold (1/4). Fold the left edge to the newest crease and unfold (1/8).

  2. 2. Another round

    Fold the right edge to the 1/8 crease and unfold (close to 4/7). Fold the left edge to that new crease and unfold (close to 2/7). Fold the left edge to the newest crease and unfold (close to 1/7).

  3. 3. The last round

    Repeat the same three folds once more: right edge to the newest crease, left edge to the new one, left edge to the newest again, unfolding each time. Now you have creases at 1/7, 2/7 and 4/7.

  4. 4. Make 3/7

    Fold the left part over so that the 2/7 crease lies exactly on the 4/7 crease, then unfold. The new crease is 3/7.

  5. 5. Make 5/7 and 6/7

    Fold the right edge to the 3/7 crease and unfold (5/7). Fold the right edge to the 5/7 crease and unfold (6/7). All six creases that split the edge into sevenths are made.

  6. 6. Stand it up like a folding screen

    Fold the six creases from 1/7 to 6/7 into valley and mountain folds in turn, and stand the paper up like a folding screen: seven panels of the same width.

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