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← The Folded Line

The Folded Line

Four short lessons on paper: a fold is a mirror, what a corner allows, paper that flew, and one straight cut.

Spark & Whetstone · sparkwhetstone.org/learn/the-folded-line · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. A fold is a mirror

Put a dot near each of two corners of a sheet and fold so one dot lands exactly on the other. Unfold it. The crease runs exactly halfway between the dots, and square to the line joining them.

That is all a fold is: a mirror. Every point on one side lands on its reflection on the other side, the same distance from the crease. Folding edge onto edge does the same for lines — the crease splits the angle between them in half, with no protractor.

Mathematicians have listed the ways a single fold can line things up — seven of them. One can bring two points onto two lines at once, and with it a sheet of paper can trisect an angle, which a compass and ruler cannot.

Question. You fold one dot exactly onto another. Where does the crease lie?

  1. Through both dots.
  2. Halfway between them, at right angles to the line joining them.
  3. Wherever your hand happened to press.

Say it back. In your own words: why does folding one dot onto another always put the crease halfway between them?

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Lesson 2. What a corner allows

Unfold any flat model — a paper crane works well — and look at a point where several creases meet. Some creases stand up as ridges (mountains), some sink as valleys.

Count them: around every such point, mountains and valleys differ by exactly two. And if you add every other angle around the point, you get 180°. These two facts (Maekawa’s and Kawasaki’s) decide whether a single meeting point can lie flat.

They are why a road map fights back when you refold it the wrong way, and why a pattern that passes at every point can still jam as a whole: layers from far apart can collide. Deciding that for a whole sheet is, in general, a very hard problem.

Question. Four creases meet at a point with angles 80°, 100°, 120° and 60°, in that order around it. Can it fold flat?

  1. Yes — any four creases can.
  2. No — alternate angles do not add to 180°.
  3. Only if three are mountains.

Say it back. Say it back: what two things must be true at a point where creases meet, for it to fold flat?

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Lesson 3. Paper that flew

Koryo Miura designed a fold for large panels in space: a zig-zag in both directions, at a slight slant. Pull two opposite corners and the whole sheet opens in one motion; push them and it closes along its own creases.

A Miura-folded panel was tested on Japan’s Space Flyer Unit, launched in 1995, and Miura maps are sold today. Other ideas from the same family — folding sunshades for telescopes, folding stents for blood vessels — have been built and tested, but not all of them have flown.

When you read about folding engineering, ask which it is: flown, in everyday use, tested in a laboratory, or still a design. The headline often doesn’t say.

Question. What makes the Miura fold useful for something that must open in space?

  1. It opens in a single motion.
  2. It never needs creases.
  3. It packs smaller than any other fold.

Say it back. In your own words: what is the difference between a design that has flown and one that has been proposed?

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Lesson 4. One straight cut

Fold a square along both diagonals, so its four edges lie on top of each other, and make one straight cut across them. Unfold: a square falls out. A Japanese puzzle book of 1721 already had this kind of trick, and Houdini printed a one-cut star in 1922.

In 1998–99 three mathematicians — Erik Demaine, Martin Demaine and Anna Lubiw — proved something remarkable: any shape with straight sides, however lopsided, can be freed with one straight cut, if you fold the paper the right way first.

The trick is always the same: fold until every edge of the shape lies on a single line. For a symmetric shape, folding along its mirror lines does it. For a lopsided one you need many folds made at once.

Question. What has to be true just before the single cut?

  1. The paper is folded in half exactly once.
  2. Every edge of the shape lies on one straight line.
  3. The shape is symmetric.

Say it back. Say it back: why does a one-cut shape need every edge folded onto the same line?

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At the table: The one-cut table

Players: 2 or more, any ages. You'll need: Paper, scissors, a pencil.

  1. Fold a square along both diagonals, cut once straight across, and open it: a square comes out. A grown-up cuts for young children.
  2. Fold a square in half, then fold it into a fan of five equal wedges, cut once at a slant, and open a five-pointed star.
  3. Each person draws a shape with straight sides. Pass it to the left: that person works out how to fold it so every edge lies on one line.
  4. Cut, open, and compare. Which shapes were easy, and why?

Seat sheets and a run-through: sparkwhetstone.org/fold-together

Answers

  1. Halfway between them, at right angles to the line joining them. — The crease is the perpendicular bisector: every point on it is equally far from the two dots.
  2. No — alternate angles do not add to 180°. — 80 + 120 = 200 and 100 + 60 = 160. Kawasaki’s condition fails, so no choice of mountains and valleys will lie flat.
  3. It opens in a single motion. — The whole pattern has one way to move, so one pull opens it — fewer motors and fewer things to go wrong.
  4. Every edge of the shape lies on one straight line. — One straight cut cuts along one line, so every edge of the shape must be folded onto it. Symmetry helps but is not required.