The mathematics of origami

Origami is paper folding. And paper wasn’t invented until about 2000 years ago, and the first documents about the art of origami date from the 1700s, so it’s a relatively young art, but one that Adam Murphy was keen to master, with the help of origami artist Robert Lang...

Adam - Okay. So straightforward so far. Corner to corner, fold side to side. And then it... then it's just... some kind of magic apparently that... nope. Aargh!

Adam - Origami. Paper folding. It's a new hobby I'm trying to learn. And when you want to learn something new, you go to an expert. For origami, that's Robert Lang. Robert used to work in laser physics, developing complex mathematical models that would predict how lasers would behave in different situations. And one day he realised the approach you took to physics might also work for his hobby.

Robert - Origami felt like it was amenable to that sort of approach, that there were simple mathematical laws that described what you could and couldn't do in the world of origami. And so if I learn those mathematical laws, if I could put them in the language in mathematics, then I could use the tools of math to accomplish the goals that I wanted to accomplish in the world of origami. And that actually worked out quite well. So first off, the math helped me design origami figures that I really wanted to design and had not been able to figure out a way to design; but also the math that describes origami for the design of artwork, also works to describe things that fold that might have technological applications - deployable structures and the like. And then the math itself is interesting. So sort of all three of those things led to an expansion of my world of origami, expanding the art, and leading to applying folding and origami to technological applications; and also just leading to exploring the mathematics of folding itself.

Adam - It's been around for centuries growing slowly, but that growth has been getting faster and faster. At the time, origami only went so far. Robert was looking to make more detailed animals, things like spiders that have all their legs, or a deer with its complex antlers.

Robert - The goal was to create realistic representations of shapes that had appendages; arms, legs, wings, antennae, and the like. And those were considered very, very challenging to do in origami. In fact, there are books from the 1970s that say, "to do a grasshopper you would need to use two squares of paper." And people just didn't know how to get lots of features from a single sheet of paper. And it turned out that there was some very elegant mathematics that told how to get any number of arms and legs and wings and antennae. You could just follow a mathematical procedure that would give you the fold pattern, that would fold into a shape, that was guaranteed to have exactly the right number of arms and legs and so forth, at exactly the right length and connected in exactly the way they specified.

Adam - So by applying maths to origami, folding paper took a quantum leap forward and a whole range of new things could be made. But maths sometimes feels like it's just equations on a piece of paper. How does it work when you want to fold that paper into a new shape?

Robert - A flap, you know, an arm or a leg, can be represented by a circle. That circle represents the minimum amount of paper needed to produce that flap. And the problem of getting all of your flaps from a single square of paper boils down to fitting all of those circles inside your square in such a way that they don't overlap. That's like stacking a bunch of balls in a square box. You could figure that out. And then we can follow a second set of rules that creates the creases, that involves drawing lines between the centres of touching circles. And by following these step by step processes, we can construct the pattern of all the folds you need to make that shape. It may seem mysterious if you just hear the rules; pack circles, and then construct lines according to these rules that don't seem to have anything to do with folding, and then once you get your finished drawn pattern, you can fold on all those lines and magically it will have the right shape. It may seem mysterious, but math tells us it's not mysterious. I mean, mathematics establishes exactly why those particular rules give you the resulting shape that you're after.