Flat folding question

[HopefulFolds]

When you see a box pleated CP, how do you decide which creases to eliminate in order for the model to be flat folded? I know that some aren't meant to be flat folded, but when I draw them I am trying to make it flat foldable. I am talking about which creases that radiate from a point in the center of the paper

[FlareglooM]

Maybe this helps:

There are four mathematical rules for producing flat-foldable origami crease patterns:

1. crease patterns are two colorable
2. Maekawa's Theorem: at any vertex the number of valley and mountain folds always differ by two in either direction
3. Kawasaki's theorem: at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even. Explaining Picture
4. a sheet can never penetrate a fold.

Source: http://en.wikipedia.org/wiki/Flat-folda ... at_folding

I think the third one can help you out.

[anonymous person]

Also, the number of creases meeting at a vertex must always be even.

[origamimasterjared]

The most useful one I find is a corollary to the M = V±2 theorem.

If M = V+2 then M+V = V+V+2 = 2(V+1) => M+V is even
If M = V-2 then M+V = V+V-2 = 2(V-1) => M+V is even

So the total number of creases intersecting at any interior point must be even.

This is the easiest to check, as all you have to do is check to see if you have any points in the interior where an odd number of creases intersect. An odd number of creases means you have either too many or too few at that point. In most of the box-pleat CPs I've seen, the problem is too few. A lot of people leave out the hinge creases.

The Kawasaki 180˚ one is the second thing to check. If you are using simple box-pleating, where the creases are all at 45˚ or 90˚ i.e. no pythagorean stretches, this shouldn't be a big issue.