Over the first weekend in November the weather in Ithaca has changed to warnings of winter. Monday, November 4th afternoon is still bright and sunny but with significant chill in the air. Good reason to come to Rockefeller Hall Schwartz Auditorium to listen to Robert Lang who is visiting Ithaca.
He is introduced by Itai Cohen who ends his intro by saying: " And now Robert Lang will show you what can you creatively do with physics degree."
Nobody really knows when origami started - it was (and still is) a popular past time activity for kids - everybody has made a boat, flapping bird or jumping frog at school. Earliest pictures showing folded butterflies are from 1680, pictures in 1734 Japanese manuscript let conclude from the complexity of origami that it was already well developed art. Well known around the world is Senbazuru origami based on ancient Japanese legend about the thousand folded paper cranes. The most significant in the art of origami happened in 20th century when Akira Yoshizawa developed the language of origami.
The next big step in origami happened when after 80 publications in physics and 50 patents in optoelectronics Robert Lang decided to turn to origami full time. The reason? It was physicists "instinct" that told him - origami must have underlying mathematical properties and it is time to find them. And he did.
There are four rules for creating flat origami:
1. Two colorability - which means if we map necessary creases, we should be able to color all regions in two colors so that no two adjacent regions are in the same color.
2. At each interior vertex Mountains - Valleys = +/- 2.
3. Odd and even numbered angles around the vertex has to be the same amount and add up to 360 degrees.
4. Paper cannot intersect itself.
First 3 rules are easy to create, the fourth one is hard.
Whenever something is hard we want to know how hard. Pleats in origami are not all equal - they can be denoted true/false. That makes digital circuits of the folds, which turns into problem in mathematical logic - "not all equal". It is proved that crease assignment complexity (NAE-3-SAT) in origami is NP-complete.
If the flat origami can be so hard, then what about three-dimensional one? Robert Lang shared with the audience how he develops ideas of making 3-D objects from one sheet of paper - how the amount of paper for flaps is calculated, how the various regions are allocated on the paper, how to get from string figure which represents the object you want to fold to the number of flaps needed. These problems lead to discussions about intelligent design in origami which is connected with nonlinear constrained optimization.
For my delight Robert Lang was mentioning folding of hyperbolic tiling and even 3-D printed hyperbolic paper which he used to fold "hyperbolic crane". Well, that crane came out with two heads...
Once origami even led to "bug wars" - competition among the origami enthusiasts who can fold the most complicated bug. But are there any applications in real world?
Miura-Ori origami folding pattern is used in space exploration. This folding pattern makes tessellation which allows for example to open a map just with one move. Here is how it works.
Origami is used in Solar sail - NASA uses it to observe far away planets. Of course, there is another agency which would love to turn it to the other direction...;-)
Heart implants, origami stents, protein folding, folding membranes... Paul Rothemund used DNA folding to develop cancer drugs which would be folded and unfold only when they have reached a cancer cell they should kill.
Origami is not an invented past time thing - nature knows how to fold - proteins, DNA, earwigs fold their wings.
To learn more and access software for mathematical origami design explore Robert Lang's official website.