Folding Mathematics

Origami and Paper Folding

Course 1 · Chapter 9 · The Mathematics of Folding: Rules Behind the Creases

Origami and Paper Folding 8 treated a crease pattern as a map. This chapter asks which maps are possible. Not every pattern of creases can be folded flat, and a few short rules decide which can. Those same ideas let folders divide paper into exact thirds, reach constructions that a compass and straightedge cannot, and design real-world folding structures such as solar panels and medical stents.

What Flat-Foldable Means

A crease pattern is flat-foldable if the whole sheet can be pressed flat with every crease folded and no paper torn or stretched. The key place to look is a vertex, a point where several creases meet. Every vertex of a flat-foldable pattern must itself be able to fold flat, so we start by testing vertices.

Maekawa's Theorem: Counting Mountains and Valleys

At any flat-foldable vertex, the number of mountain creases and the number of valley creases differ by exactly two:

|M − V| = 2

A vertex with 4 creases is 3 mountains and 1 valley, or 1 mountain and 3 valleys.
A vertex with 6 creases is 4 and 2 either way round.
The number of creases at a flat-foldable vertex is always even, because M and V differ by an even number.

Kawasaki's Theorem: Adding Up the Angles

Maekawa's rule counts the creases. Kawasaki's theorem measures the angles between them. Go around a vertex and label the angles between neighbouring creases in order. The angles in odd positions add up to 180 degrees, and so do the angles in even positions:

a₁ + a₃ + a₅ + ... = 180° and a₂ + a₄ + a₆ + ... = 180°

The simplest case is two creases crossing at right angles: four angles of 90 degrees, and 90 + 90 = 180 in each group. Order matters, because the angles are summed in alternate positions around the vertex.

Rule	Says	Checks	Example
Maekawa	\|M − V\| = 2	Crease types at a vertex	4 creases: 3 M and 1 V
Kawasaki	Alternate angles each sum to 180°	Angles at a vertex	30, 70, 150, 110 degrees
Two-colouring	Regions can be shaded with two colours so neighbours differ	The whole pattern	A chessboard of faces

Necessary, not sufficient Both rules are tests for a single vertex. A pattern whose vertices all pass can still fail, because layers elsewhere may be forced to cross through each other. Deciding flat-foldability for a whole pattern is a hard problem in general, so folders treat these rules as a first check, then fold a test sheet.

Checking the Rules With Code

This small script tests a vertex against both rules:

def maekawa(creases):
    m = creases.count('M')
    v = creases.count('V')
    return abs(m - v) == 2

def kawasaki(angles):
    return sum(angles[0::2]) == 180 and sum(angles[1::2]) == 180

print(maekawa('MMMV'))
print(maekawa('MMVV'))
print(kawasaki([30, 70, 150, 110]))
print(kawasaki([100, 80, 90, 90]))

Output when run:

True
False
True
False

The second vertex fails Maekawa because 2 mountains and 2 valleys differ by 0, and the fourth fails Kawasaki because 100 + 90 is 190, not 180.

Folding Thirds

Halves are easy: fold one edge onto another. Thirds are harder, because no single fold of a strip gives one third. Haga's method finds an exact third with one extra fold:

Start with a square and mark the midpoint of the bottom edge by folding in half and unfolding.
Fold the top-left corner down to that midpoint and crease firmly.
Look at the right edge. The folded-down top edge crosses it at a point exactly 2/3 of the way up from the bottom (so 1/3 from the top).
Pinch at that crossing to mark the third, then unfold.

You can check this with coordinates. On a unit square, the fold sends the corner (0, 1) to (0.5, 0), and the image of the top edge meets the right edge at height 2/3. From one exact third, sixths follow by folding in half, and the other third by folding the bottom edge up to the crease made at the mark.

What Origami Can Do That a Compass Cannot

Straightedge and compass constructions have limits. In 1837 Pierre Wantzel proved that a compass and straightedge cannot trisect an arbitrary angle or double a cube (construct a length of the cube root of 2). Both problems lead to cubic equations, and a compass and straightedge can only reach lengths found from square roots.

Folding is more powerful. The Huzita-Hatori axioms list the seven basic single folds that align given points and lines. One of them, which folds two points onto two lines at the same time, amounts to solving a cubic equation. That gives folders exact methods for:

Trisecting an angle, by a method published by Hisashi Abe.
Doubling the cube, by a construction credited to Peter Messer.

These use real folds, not approximations. They show that paper folding, with this fold allowed, is strictly stronger than the classical compass and straightedge.

Origami in Engineering

Folding solves real packing problems: how to carry something large in a small space, then deploy it reliably.

Miura-ori. A pattern of parallelogram creases, developed by Koryo Miura, lets a large sheet fold down into a compact stack and open with a single pull. It has been used for deployable solar panels on spacecraft.
Stents. Folded tube patterns allow a medical stent graft to be compressed small, inserted through a narrow vessel and expanded in place.
Airbags. Folding patterns have been studied to make airbags pack tightly and inflate evenly.

Do not force a failed collapse If a pattern whose vertices all pass the rules still will not close, do not push. Re-check each mountain and valley against your crease map, because a single reversed crease can stop the whole sheet from folding flat.

Challenges

Each challenge has a worked solution you can open after you have tried it.

Challenge 1

Crease a square along both midlines to make a vertex of four 90 degree angles at the centre. Try to collapse it with 2 mountains and 2 valleys, then with 3 mountains and 1 valley. Which flattens, and how does Maekawa's theorem explain it?

→ Solution

Challenge 2

A vertex has four angles, three of which are 45, 80 and 135 degrees, in that order. Find the missing angle, then use Kawasaki's theorem to show the vertex could fold flat. Draw it with a protractor and test it.

→ Solution

Challenge 3

Use Haga's method to mark exactly one third of a square's edge. Measure it with a ruler to check, then use it to fold the square into three equal strips.

→ Solution

Quick Reference

Maekawa: at a flat-foldable vertex, mountains and valleys differ by 2
Kawasaki: alternate angles around a vertex each sum to 180 degrees
Haga: fold a corner to an edge midpoint to find a third
Beyond the compass: folds can trisect an angle and double a cube
Engineering: Miura-ori for solar panels, folded stents, airbags

Next, in Origami and Paper Folding 10, we put it all together: designing your own simple model, tree theory in brief, famous folders and communities, and where to go from here.