Orizuru (paper crane) · verified flat-foldable
The Halved Diagonal
This family is not rigid-foldable — the paper has to bend on the way, so there is no continuous 3D motion to show. Step diagrams are the honest way, and the way every origami book does it.
Open the diagram sheet →How to fold it
- Fold the square in half vertically.
- Fold the folded shape in half horizontally — front base ready.
- Bring the horizontal raw edge to the center line and unfold (bottom-left).
- Bring the vertical raw edge to the center line and unfold (bottom-left).
- Lower the top triangle and unfold — petal hinge (bottom-left).
- Bring the horizontal raw edge to the center line and unfold (bottom-right).
- Bring the vertical raw edge to the center line and unfold (bottom-right).
- Lower the top triangle and unfold — petal hinge (bottom-right).
- Bring the horizontal raw edge to the center line and unfold (top-right).
- Bring the vertical raw edge to the center line and unfold (top-right).
- Lower the top triangle and unfold — petal hinge (top-right).
- Bring the horizontal raw edge to the center line and unfold (top-left).
- Bring the vertical raw edge to the center line and unfold (top-left).
- Lower the top triangle and unfold — petal hinge (top-left).
- Make the petal fold: lift the bottom point upward, letting the side edges come in on their own; turn over and repeat on the back — bird base complete.
- Collapse the bird base along all the existing creases.
- Inside reverse fold one pointed end between the two layers — neck.
- Inside reverse fold the other pointed end the same way — tail.
- Make one more small inside reverse fold at the tip of the neck for the head, then fold the wings down — crane complete.
Pre-creasing every line before you collapse is the whole trick. Paper remembers.
The story below, read word for word by a synthetic voice over this fold's own music.
At the lost property counter of a station, in the shallow tray where they keep the things nobody comes back for, there was a paper crane. Bus ticket paper, folded soft at the wings, one point slightly crushed as though it had ridden in a pocket for a while. The clerk had set it upright instead of flat. That small decision — upright, facing the queue — is most of what I want to say about this fold.
Nobody knows who invented The Halved Diagonal. It is the orizuru, the crane, and it is traditional: no name attaches to it, no estate, no signature. The oldest printed trace we can point to is Hiden Senbazuru Orikata, from 1797, which already treats the shape as common knowledge and goes on to describe cranes joined at the wings, cut from a single sheet. By then the fold was old furniture in a house whose builder had been forgotten. What we made here is narrower: the measurements on this page, the step diagrams, the writing, the music, the recorded conversation. The pattern itself was never ours to hold.
The crane refuses to be a machine
There is a class of folds you can build out of stiff panels and hinges — cardboard, sheet metal, solar arrays that open in orbit. The crane is not in that class. Somewhere between the bird base and the finished neck, the paper has to flex across its faces; it curves, briefly, and then relaxes. If you replaced the sheet with rigid plates, the motion would jam.
This is why the page carries drawn steps rather than a smooth three-dimensional animation. An animation of a fold like this either lies about the paper or stalls. A diagram in the Yoshizawa–Randlett convention — dashes, dots, arrows — does not pretend to show the in-between. It shows the before and the after and trusts your hands with the middle, which is what your hands are for.
The mathematics we can state cleanly is the flat state. Kawasaki's condition governs the alternating angles around each interior vertex; Maekawa's counts the mountains against the valleys. Both are checks on the finished, pressed shape, not on the journey.
Counted, from the sheet up
The pattern has 48 vertices and 109 creases. Of those vertices, 38 sit in the interior of the square, and all 38 satisfy flat-foldability. The crease assignment comes to 41 mountains, 58 valleys, and 10 boundary edges.
The angles are the quiet surprise. They cluster on the familiar eighths — 45°, 67.5°, 90°, 112.5°, 135° — and then drift into values that no one designed on purpose: 33.7°, 33.8°, 56.2°, 56.3°, 78.7°, 78.8°, 101.2°, 101.3°, 123.7°, 123.8°, 146.2°, 146.3°. Those near-twins are the diagonal halving working against the square's own symmetry, landing a tenth of a degree apart. The whole spread runs from 33.7° to 146.3°, comfortably inside the range we accept. All five of our checks passed; that is the least interesting sentence on this page.
Upright, facing the queue, folded by someone who never came back for it.