At the recent Decoding Formal Club meeting, organized by Oski and sponsored by Cadence, there was a presentation on computational origami. That sounds like a great marketing name for formal verification, but in fact it was indeed what it sounds like. I freely admit that this is a bit off topic—but it was at a Formal Verification event, that's close enough for me.

There is a surprising amount of mathematics associated with origami, paper folding. Origami has a long history going back a few hundred years to Japan. I'm sure you did it as a child. Dr Robert Lang has found ways to get people to pay him to do it as an adult. His presentation was called *From Flapping Birds to Space Telescopes.*

There is obviously an art aspect to origami, but there is also a serious engineering aspect, too. Anything that needs to be "small for the journey, expand at the destination" is perfect for the origami approach. This ranges from unfolding solar panels on satellites, to folding mirrors on space telescopes, to bullletproof kevlar shields that can be unfolded by one person, to medical stents that can be inserted small and expand to open up an artery.

One of the things that you would like to know is whether you can actually fold a design or not. You know where the creases need to go, but is there a sequence of folds that will get you there. It turns out that origami is mathematically pretty simple. There are, in fact, just four rules:

- The pattern you get with all the folds must be two-colorable like the four-color theorem for maps, but basically for origami you have to be able to color everything in black and white, with no two polygons of the same color touching along an edge (corners are okay).
- At every vertex where folds come together, the number of mountain (up) folds - the number of valley (down) folds must be 2 or -2. Or to be even more mathematical M-V = ±2.
- The alternate angles at any vertex must add up to a straight line. Since the pattern is two colorable, take all the black polygons and all the white. The angles of the white at the vertex must add up to 180° and the same for the black (well, they have to add up correctly if the white do). Or putting it another way, every other angle around any vertex must add up to 180°.
- No self intersection at overlaps. If there are two folded sheets at some edge, either they wrap around each other, or one is stacked on top of the other, but the two cannot alternate since that requires one piece of paper to go through another. Same for a single sheet meeting a folded sheet, either it goes over the top of the folded sheet, or underneath, but not straight through the middle.

In fact, it turns out that the first rule is redundant. If M-V=±2 then the number of creases meeting at a vertex must be even, and thus the area is locally two colorable, and this extends to the entire pattern. So if rule 2 is met, then rule 1 is automatically met.

## Examples

The talk contains a fair bit of information on how to design the folding patterns for insects, and a program that he created to automate some of the process. Here are a few examples. The crease pattern on the left gives rise to the object on the right.

Robert said that the most complex thing he ever created was this cactus. He said it took him seven years. It is a single piece of paper, red on one side and green on the other. Yes, the pot and the cactus are the same piece of paper.

## Space Telescopes

We were promised space telescopes and here they are. This is the design for the Eyeglass Telescope, which has a 100m (football-field-sized) lens (it is a refracting, not a reflecting telescope). Since it has to go into a rocket to get it up there, it is the perfect example of something that has to be "small for the journey, expand at the destination. The diagram on the left shows the basic design that was developed, on the right is the 5m prototype.

## Watch for Yourself

You can watch Dr Lang's talk on this video. This wasn't taken at the Decoding Formal Club meeting, but it is essentially the same talk (about an hour):

Or, if you want something shorter, you can join over 2M people who have already seen it, and watch his TED talk (16 minutes).

## More Information

Usually, at this point in a post, I point you to a page on our website with information about some Cadence product. Well, we don't actually produce origami design tools! But since this was presented at a a formal verification event, here's the JasperGold product page. The Jasper User Group is coming up too, on November 7 and 8. Here's the information page for that, including a link for registration (free).

But the best places for more information are Robert Lang's book, Origami Design Secrets, Mathematical Methods for an Ancient Art, and his own website.

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