Folding Physics: The Mathematics of Origami

Origami Spaceship.png

Space origami. Sounds cool, right? It’d be kind of fun if…what? That’s already a thing?

I don’t mean anything like my origami spaceship, of course. I’m talking about solar cell panels, which are useful in spacefaring endeavors, since they can provide power to satellites. These panels are very thin, but since both mass and space are at a premium in launches, the ideal situation is to be able to keep them compressed until they reach orbit, when they can be expanded into a much larger array.

Thanks to astrophysicist Koryo Miura, we have a solution in the form of the Miura fold, which falls in the category of rigid origami techniques. This means that it can be used on a sheet of linked pieces that must all remain flat at all times, making it perfect for solar cell panels. Using Miura’s invention, the Japanese space program has successfully launched panel arrays that remain folded during launch and are unfolded once they reach orbit.

Miura Fold

The Miura fold uses parallelograms instead of squares or rectangles. Also useful for folding maps!

Origami isn’t just for large-scale applications—scroll through this page and you’ll find an image of a microscopic (literally, less than a millimeter long) crane made from a self-folding polymer gel sheet. This is insanely cool on its own, but the researchers who create these self-folding microscopic sheets are hoping to develop practical applications as well. Something that can fold up tightly and then unfold easily at the right moment could be extremely handy for tasks like delivering drugs to particular cell populations or regions without touching the healthy cells.

These micro- and macro-scale applications are innovative, but it’s not too much of a surprise that origami can provide elegant solutions to spatial problems. The incredibly complex and beautiful shapes that can be made with origami are clearly a combination of art and rigorous logic. It’s also a natural step for mathematicians, physicists, and engineers to adapt origami techniques to real-world objects, since the folds are developed and used on actual pieces of paper.

However, we can also use origami to deal with some more fundamental mathematics in very imaginative ways. In fact, paper folding can accomplish things that are impossible with the tools of Euclidean geometry, which is defined by axioms that limit what can be proven. These axioms allow us to bisect an angle, or find the line that divides the angle precisely in half. As proven by nineteenth-century mathematician Évariste Galois, though, Euclid doesn’t get us as far as being able to divide that angle in three, or trisect it.

Luckily, Euclid doesn’t have the last word—check out this article, which illustrates how to use origami to trisect an angle with just a few folds. Better yet, grab a square piece of paper and follow my instructions to do it yourself!

Origami Trisect Angle.png

This isn’t even remotely the coolest thing you can do mathematically with origami. For some more ideas, check out this gorgeous collection of mathematical solids and this guide to some mathematical tricks, including the one you just did. If you want to get technical, check out this 2009 paper, or this talk on the history of origami mathematics, or just get some basics on origami mathematics.

Don’t forget to appreciate origami for the aesthetics and skill, too! Here are some beginning origami instructions, and some more challenging ideas for further practice. Enjoy creating beautiful objects and challenging the limitations of conventional geometry!

Origami Monica

 

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