Pythagorean Stretch

[quesoonfire]

I Know that Robert Lang has used Pythagorean stretches in his models to up the efficiency of a boxpleated cp. I was wondering how this works. Any help is greatly appreciated.

[Jonnycakes]

I have been looking into Pythagorean stretches and I am not entirely sure how to implement them. At the '07 New York convention Lang gave a talk on them, but I lost my notes from that class-does anybody know how they work?

[BluePaper]

I know how to use them , however its a rather difficult thing to explain...

It is as the title suggests, it has to do with the pythagorean thereom, triangles with sides that have ratios of 3-4-5, 5-12-13, and so on (I don't really know many others) the concept isn't all that difficult to grasp as long as you know the pythagorean thereom. Using these stretches on the other hand is a lot more hit and miss with me, I'm sure there are poeple out there who are very good at using them but I'm not one of them, for me it is mainly trying to make a connection between 2 points when space is limited. I don't use them all too often, but then again I don't do a lot of box pleating.

I hope this can clear things up a little...

[Brimstone]

I don't know it very well even though I was at the Lang conference in OUSA07, but what I can tell you is:

From lang page: Pythagorean stretch: a technique in box pleating in which two diagonally-oriented leaf vertices are allowed to move closer to each other than the square packing would normally permit. This results in the creation of axial creases and ridgeline creases at angles other than 0°, 45°, and 90° in the vicinity of the stretch but permits greater efficiency in the crease packing.

You know how in box pleating the angles are usually 45 degrees? Well the Pythagorean Stretch uses different angles, basically angles that comply with the pythagoras theorem (the one about the square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides). This method uses less area of paper and hence is more efficient than regular (45 degree) box pleating.

OIne clear example of it can be found on Kamiya's Wasp CP

[Jonnycakes]

Thanks for the info, but that is basically what I have gathered already. I know that it can be used with the triangles with integer-length sides, but I am not so sure about the rest. I experimented a little bit, and here is what I was able to do:

-I used a 3-4-5 (6-8-10 actually) triangle Pythagorean Stretch to connect two flaps of lengths 4 and 6 that overlapped. The flap lengths work out perfectly with the hypotenuse of the triangle, so it was not too hard.

-I used a 45-45-90 (degrees, that is) stretch to connect two flaps of length 6 that overlapped forming a 2x2 square (that is, if you draw the squares for both of the flaps, the overlap is a 2x2 square). The edges of the triangle that was used for the stretch did not line up with the grid, but the other creases did. It worked, and now I know how to use Pythagorean Stretches on flaps whose nodes can be connected with a 45 degree line.

In short, I know how to execute a select few situational Pythagorean Stretches, but I don't know how to universally apply them. I know there is a way to do so (Lang explained it in front of me ), but I do not remember how.

[ahudson]

I ran across a cp with both pythagorean and regular molecules in them:

http://origamipro.ucoz.ru/forum/11-92-15

It's all in Russian, so I have no idea what model it is, but that's the first time that I've seen the two and recognized what they were.

The two different molecules haven't been pleat-sunk, so you can see the structure easily.

[origami_8]

It's the metallic insect by Sebastian Arellano.
http://origami.artists.free.fr/sebastianarellano/