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Geometry & modular

Doubling the cube (∛2)

A puzzle ruler and compass could never solve: find ∛2 with a single clever fold.

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

Why this level: You need accurate thirds, then one fold that brings two points onto two lines at the same time.

You need: 1 sheet of origami paper

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Story

An old Greek legend says the people of Delos, suffering from a plague, asked an oracle for help and were told to double the size of Apollo’s cube-shaped altar. Doubling a cube’s volume means making its edge ∛2 times longer, and in 1837 Pierre Wantzel proved that ruler and compass can never construct that length. Folding can: Peter Messer reported this one-fold method in 1986.

Did you know? In 1936 the Italian mathematician Margherita Beloch showed that a fold putting two points onto two lines at once can solve cubic equations — that is why folding can do more than ruler and compass.

Learn more (external sources)

Steps at a glance

  1. 1. Crease the centre line

    White side up. Fold the left edge to the right edge and unfold to find the middle of the top edge.

  2. 2. Find a third (Haga’s theorem)

    Fold the bottom-right corner to the middle of the top edge. Where the folded bottom edge crosses the left edge is one third of the way up.

  3. 3. Mark and unfold

    Pinch a small mark at that point on the left edge, then unfold.

  4. 4. Lower third crease

    Fold the bottom part up along a horizontal line through the mark, then unfold.

  5. 5. Upper third crease

    Fold the top edge down to the lower third crease and unfold. The paper is now creased into three equal strips.

  6. 6. Two points onto two lines (Messer’s fold)

    Fold the bottom-left corner over onto the right edge so that, at the same time, the left end of the lower third crease lands on the upper third crease. The point where the corner touches marks ∛2.

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