## Origami notes

We had a brilliant lunchtime seminar by a guest speaker at work today. She talked about origami.

I wasn’t sure what to expect – if it’d just be demos of paper folding, or cool models that people had made, or what. There were both of these things, but there was much more. The speaker is doing a Ph.D. in statistics, so has a strong mathematical background. She started by showing some examples of origami with a slide presentation. But then… she went to the whiteboard…

She presented origami as a composition of two functions: a first one-to-one (and thus bijective) mapping from a flat sheet of paper to a 3-dimensional folded version of that sheet of paper – specifically a folded version which produces a number of points that can be used as the basis of extremities of a model (for example, animal limbs, fingers, horns, etc); and a second function mapping that 3-dimensional structure to a tree diagram with nodes at end points and junctions. The second mapping doesn’t preserve point correspondences or distances, but (and she proved this as a theorem) points on the tree diagram corresponding to points on the sheet of paper are separated by a distance less than or equal to their separation on the paper.

The upshot of this is that you can sketch a three-dimensional stick figure of anything you like, with correctly proportioned stick lengths; then map that on to a flat sheet of paper such that you produce a tree with nodes and branches in analogous positions such that the distances between them are equal to or greater than the corresponding distances in the stick figure; then from there you can apply a deterministic algorithm to figure out the arrangement of mountain and valley folds to produce the points at the nodes. There are programs that can do this and solve the fold pattern for any desired stick figure in a few minutes.

The result is a map of exactly how to fold the sheet of paper to produce a 3-D model that resembles your stick figure, with the correct number, arrangement, and lengths of all the extremities. From there it’s a simple matter of subtly modifying the shapes of the points with additional folds to produce an accurate model of your desired shape. She showed an example of a buck deer with four legs, a tail, a head with a nose, and two antlers, each with 4 distinct points. The resulting tree has 14 leaf nodes, and produces a complex fold map, but it’s perfectly doable, and the resulting origami model looked amazing.

I was utterly blown away. I always figured origami artists created new models by trial and error and months of hard work. But apparently you can do it with a stick figure sketch and a simple piece of mathematical computer code, in a few minutes. I should stress this only produces the basic “stick figure” form, and the subtle shaping of a 14-pointed piece of folded paper into a realistic deer shape still takes artistic talent, but still, I was amazed that the hard structural part of the work was so tractable and solvable by a beautiful application of mathematics. It shows how mathematics can be applied to surprising fields and come up with usable models and solvable answers for problems that seem unapproachable any other way. Very nice stuff.