Presenter: Robert Lang, origami expert
Origami has been around for 100s of years. It didn't change until 1970s when it experienced a Cambrian explosion in variety and techniques. It got richer and more interesting because people started applying math.
The secret to origami, and so many other things, is to let dead people do your work for you, like looking at the geometry of disk packing.
Four simple laws can give rise to very rich complexity in origami. They have to do with properties of crease patterns, angles around a vertex, layer orders, and valleys and ridges. If you obey these laws you can make anything. He has a program on his website that will show you the fold patterns needed to make anything. (You give it a stick figure, it shows you the folds.)
He shows how he uses these mathematical ideas to fold a square sheet of paper into anything.
Origami has applications in other areas, like a solar array that flew in a Japanese satellite telescope, umbrella telescope, solar sail, airbag, heart stent (origami may save a life).