Origami and Mathematics

Fold point P onto point Q. Every point on the crease is equidistant from P and Q. The crease is therefore the perpendicular bisector of segment PQ.

One side of the paper rotates about the crease. When folded flat, each moved point is the reflection of its original position across the crease. Distances along the paper stay the same.

Folding one edge onto another bisects the angle between them. Bringing points or lines together is the basis of origami constructions.

Take a square with side length 1. Choose point P on the top edge, a distance a from the top right corner. Fold the bottom right corner onto P.

Let Q be the intersection of the folded bottom edge with the left edge. The distance from the top left corner to Q depends only on a.

Setting a to 1/2 gives a distance of 2/3. Folding an edge in half marks the point needed to construct this distance without a ruler. It can then be used to divide an edge into thirds.

A fold can pass through a point, bring a point onto a line, or bring one line onto another. Combining these conditions gives seven operations that describe simple origami folds.

The conditions do not always determine exactly one crease. Operation 5 has zero to two solutions, and Operation 6 has zero to three. Blue dotted lines show all creases that satisfy the conditions. Each is drawn as a valley fold.

Operation 6 brings two points onto separate lines at the same time. The next section uses this operation to solve cubic equations.

Ruler-and-compass constructions can solve linear and quadratic equations. These tools alone cannot trisect an arbitrary angle or construct a cube with twice a given volume.

Origami's Operation 6 constructs a crease that meets two conditions at once. Hisashi Abe's method trisects an angle. Peter Messer's method constructs the cube root of 2, the ratio of side lengths needed to double a cube's volume.

Abe's method simultaneously brings the angle's vertex O onto a horizontal line and point B on the left edge onto the angle's sloping side. The line joining O to its destination O′ trisects the angle.

Creases meeting at a single point divide the surrounding paper into angles. Add the alternating angles to form two sums. Kawasaki's theorem states that an even number of creases can fold flat with suitable mountain and valley assignments if both sums are 180°.

Maekawa's theorem states that the numbers of mountain and valley folds at a vertex folded flat differ by exactly two.

When mountain and valley folds are assigned, their order also matters. The creases on either side of an angle smaller than both its neighbors must have opposite assignments. The diagram checks the assigned folds as well as the angles and counts.

A crease pattern may fold around each vertex yet fail to fold flat as a whole because distant parts of the paper intersect. A global test must also account for the order of overlapping faces.

For a narrow strip, foldability can be tested by repeatedly applying crimps and end folds. A crimp folds a short interval between neighboring mountain and valley creases. It is allowed when that interval is no longer than either neighbor and its bounding creases have opposite assignments.

Deciding whether a general crease pattern folds flat is NP-hard. This means that a broad class of difficult computational problems can be reduced to this decision problem.

For m×n map folding with assigned mountains and valleys, it remains open whether the problem is NP-hard or solvable in polynomial time. Narrow strips and 2×n maps can be tested in polynomial time.

Mathematical paper may have no thickness, but real paper does. When folded in half repeatedly in the same direction, the outer paper must wrap around the inner layers. The required length grows rapidly with the number of folds. For a fixed number of folds, it is proportional to the paper's thickness.

Britney Gallivan derived this formula as a high school student. In 2002, she folded 1,219 m of toilet paper in half 12 times in the same direction.

As the fold count rises, the 2ⁿ terms make the required length grow. Using thinner paper reduces the required length in proportion to its thickness.

A leg or horn needs enough paper for its length. Associate a flap of length r with a circle of radius r. The design then becomes a circle placement problem.

Represent the intended shape as a tree of branching lines. When placing its tips on the paper, keep each pair at least as far apart as their distance along the tree.

The diagram updates positions gradually to reduce circle overlaps and keep the circles inside the paper. Longer branches need larger circles, making it harder to fit them on the same sheet.

Any shape made of line segments can be cut out by folding the paper flat and making one straight cut. This is the fold-and-cut theorem.

This example uses fivefold symmetry. Fold a circular sheet in half. Further folds stack 10 sectors together. A straight cut through the stack produces an outline made by reflecting the cut across the 10 sectors as they unfold.

Changing the cut angle produces a star. A cut perpendicular to an axis of fivefold symmetry makes neighboring segments line up to form a regular pentagon.

Fold-and-cut problems appear in Wakoku Chiyekurabe from 1721. The story that Betsy Ross cut a five-pointed star with a single cut is a legend.

The Miura fold uses a repeating pattern of parallelograms. Pulling opposite ends opens the whole sheet at once. Every face stays flat as it moves.

Four creases meet at each vertex. There are three mountain folds and one valley fold, or the reverse. Each sum of alternating angles is 180°. This satisfies Maekawa's and Kawasaki's conditions from Section 5.

This structure, developed by Koryo Miura, was deployed in space aboard a satellite. It was used in an experiment on the Space Flyer Unit, launched in 1995.