Like so many other videos, I happened upon this via another video.
In Japan, the art of origami (which literally means "fold paper") is something that all Japanese children are introduced to from an early age. It's an integral part of the culture. Although it's purportedly quite old, the mathematical theory of origami is only a few decades old. Before the discovery of its mathematical theory, it wasn't very easy (or possible in many cases) for people to derive new forms from the paper. (There may also have been some level of social conservativeness that slowed adoption of novel forms, but that's another topic.) However, certain forms became iconic (and still remain so), the crane being perhaps the most famous.
I like making cranes for the kids of my friends, using bits of paper I find around the place. Most are sufficiently surprised and tickled with my quite limited ability of making this form. My brother -- a far greater origami worker -- achieved official ranking in Japan while he was still in high school, and can make far more interesting and intricate forms than me. However, the traditional crane has been (and still remains) the icon of origami (even though more realistic ones are possible).
However, with the discovery of the mathematical theory of origami, completely new shapes are able to be produced, sometimes with exquisite intricateness. These forms have exploded onto the scene in a Japan that is now far more obsessed with the novel (and new uses for traditional things) than with the conservation of tradition for traditions sake alone. Thus, we can get things like this:
The following video describes the secret of the mathematics behind origami: