Origami Mathematics: The Shape of Things to Come

Benjamin Skuse

Though Lexus factories are about as hi-tech as you can get, the Japanese car-maker still employs highly skilled ‘Takumi’ – artisans who oversee and ensure the quality of every aspect of the vehicle manufacturing process. It takes decades to reach the status of Takumi, and one of their final tests is uniquely Japanese. To prove their dexterity, they must fold an origami cat with their non-dominant hand in less than 90 seconds.

This is not the only realm in which the 18th Century Japanese paper-folding art form crops up nowadays. For years, origamists’ efficient and elegant transformation of compact 2D sheets into elaborate 3D shapes has been a source of inspiration for the design of diverse devices and products, not least vehicle airbags, takeaway and mailboxes, and various collapsible everyday items, from chairs to colanders. All these items share one common property: They need to go from being a compact structure to a large one (or vice versa) reliably and with little-to-no human input.

Expanding Miura Horizons in Space

Nowhere is this property more important than in transporting items from Earth into space. Here, every gram of the rocket’s payload is measured and minimised to ensure those involved get the most bang for their buck. For example, the 1995 launch of Japan Aerospace Exploration Agency (JAXA) satellite Space Flyer Unit was the first time solar panels were packed using the Miura fold.

Named after the Japanese astrophysicist Kōryō Miura who devised it in 1985, the solar array was stowed like a long, thick book, and then just required a tug on one corner via a single actuator stroke to expand like an accordion, opening the structure out to reveal a checkerboard of parallelograms.

Also used for decades as tourist maps and more recently in ultra-compact packaging, such as lightweight cardboard boxes, the Miura fold is both aesthetically pleasing and interesting mathematically. It has two types of creases – the mountain crease (red dotted lines) and the valley crease (blue dotted lines). These creases form rows of slanted parallelograms. Where four parallelograms meet is a vertex consisting of exactly four joined creases. Adjacent rows of parallelograms are slanted in opposite directions, in turn causing the adjacent rows of vertices to look like they are pointed in opposite directions.

These creases and vertex angles allow the Miura pattern to fold flat, governed mathematically by two theorems: Kawasaki’s theorem and Maekawa’s theorem. Kawasaki’s theorem states that a vertex can be folded flat if and only if the sum of alternating angles around it is \(180^\circ\). If the four angles meeting cyclically around a Miura fold vertex are represented by \(\theta_1, \theta_2, \theta_3, \theta_4\), then \(\theta_1 + \theta_3 = 180^\circ\) and \(\theta_2 + \theta_4 = 180^\circ\). Because a Miura vertex uses identical parallelograms, its interior angles are arranged as \(a\), \(a\) \((180^\circ – a)\) and \((180^\circ – a)\), satisfying this condition.

However, Kawasaki’s theorem is not a sufficient condition to guarantee flat foldability. This is because it ignores the folding pattern needed to achieve a flat fold. For this, Maekawa’s theorem states that, when a vertex folds flat, the number of mountain creases \(m\) and valley creases \(v\) joined in the vertex always differ by two: \(|m – v| = 2\). The Miura fold only has four creases joined at each vertex: either three mountain creases and one valley crease or three valley creases and one mountain crease.

With flat-foldability constraints defined, the transition from 2D to 3D can be modelled by a set of three equations corresponding to the 3D spatial dimensions \((x, y, z)\). These equations reveal that both \(x\) and \(y\) depend on the shared quantity \(\sin(a)\cos(\frac{\lambda}{2})\), where \(\lambda\) is an angle representing the current 3D folding state (\(\lambda = 0^\circ\) represents the flat expanded state). This means the fold cannot expand in one direction without changing in the other; expansion is bi-directionally coupled.

A Flasher Origami Pattern

More recently, JAXA had a more difficult problem to solve: how to deploy a large antenna from a tiny \(10 \times 10 \times 10\) cm cubesat, named OrigamiSat‑2 (OrigamiSat-1 launched successfully in 2019, but communications were lost days into the mission). For this, they used a different origami structure that has been explored for various purposes, including in skeletal structures for robotic hands.

Known as the flasher origami pattern, it features a central rigid polygon surrounded by a spiral ring of intricate folds. When being unfurled, it unrolls and untwists from a central point outward in a 360-degree circle. This should maximise the OrigamiSat-2 array’s surface area in every direction at once to measure \(~50 \times 50\) cm; an impressive 25 times its stowed area. Launched aboard Kakushin Rising in April 2026 from New Zealand, OrigamiSat‑2 is currently undergoing health and status checks before undertaking its core mission to deploy the reflectarray antenna.

Like the Miura fold, the flasher fold’s local flat-foldability relies on Kawasaki’s and Maekawa’s theorems. Because the flasher is built on a grid of concentric rings surrounding a central polygon (usually a square or hexagon), Kawasaki’s theorem dictates that at each internal vertex, the alternating angles around it must sum to exactly \(180^\circ\). For a standard square-hub flasher, the interior vertices feature four right angles (\(90^\circ+90^\circ=180^\circ\)).

Additionally, since the basic flasher geometry utilises degree-4 vertices throughout its spiralling concentric tracks, each internal junction must maintain a strict assignment of three mountain creases and one valley crease (or vice versa) to compress, satisfying Maekawa’s theorem.

The flasher’s current physical state is represented by folding angle \(\theta\) (where \(\theta = 180^\circ\) corresponds to the flat 2D state). And each vertex point is represented by cylindrical coordinates \((r, \phi, h)\), where \(r\) is the radius from the centre, \(\phi\) is the angular wrap and \(h\) is the vertical height.

The equations for \((r, \phi, h)\) involve a number of other parameters and so will not be written here. But what can be said is that they are each bound to the folding state parameter \(\sin(\frac{\theta}{2})\). This makes the flasher fold a highly coupled, single-degree-of-freedom system capable of being completely deployed or retracted using a single tug at one point, just like the Miura fold.

However, the flasher pattern suffers from a geometric conflict that means that, unlike the Miura fold, it is not rigidly foldable, i.e. the panels have to bend during transition, otherwise the mechanism will jam. To get around this, engineers add a secondary crease along the panel diagonals, cut away material at the vertices or use rolling hinges; all with the intention of introducing a little more flexibility than the equations allow.

A similar unfurling mechanism based on the flasher pattern is being explored at NASA’s Jet Propulsion Laboratory to allow a huge flower-shaped shield to fly in space. Such a starshade would be key to potential future missions observing exoplanets for signs of life. In particular, the Habitable Worlds Observatory, set for launch in the 2040s, could employ a 60-metre starshade to help block the light of nearby stars in order to observe their orbiting exoplanets in unprecedented detail.

The Elusive Yoshimura Pattern

Another origami fold – the Yoshimura pattern – is even more intriguing. This pattern consists of interlocking diamond and triangular facets. Crushed flat, it resembles a circle. Fully extended, it forms a rigid, highly stable cylinder. This makes it ideal as structural tubes in the crumple zones of cars and as self-expanding stents in the human body. It is even being explored as a structural element of space habitats for future trips to the Moon and Mars.

Discovered in 1951 by Japanese aerospace scientist Yoshimaru Yoshimura, his eponymous fold was the output of investigations into how thin-walled metal cylinders buckle and crumple under extreme vertical compression. As he discovered, this pattern packs down tightly into a short stack and expands to a cylinder with exceptional stiffness once fully extended.

However, apart from a few notable examples, translating these benefits into applications has proved challenging. One reason is that, whereas the Miura fold and flasher pattern are mathematically well understood, modelling the transition between the Yoshimura pattern’s 2D and 3D states remains difficult. This is because, during the mid-fold transition, the triangular facets must bulge, flex or twist, inserting a degree of nonlinearity into the equations that is not so easily overcome through engineering trickery.

Another reason is that the Yoshimura pattern is a complex cluster of degree-6 vertices, none of which have a unique folding path. Unlike the Miura and flasher folds, this introduces multiple degrees of freedom, requiring complex matrix algebra to track the many different potential folding paths. Yoshimura structures are therefore unpredictable and difficult to model and control.

Yet, just as the Takumi on the Lexus factory floor must master the manual manipulation of paper to prove their dexterity, modern mathematicians view the complexity implicit in the Yoshimura pattern and other origami folds not as a dead-end, but as a challenge they must overcome. The result will likely unlock the full benefits of origami mathematics, and a host of structures and devices inspired by the ancient art of paper folding.

The post Origami Mathematics: The Shape of Things to Come originally appeared on the HLFF SciLogs blog.