## Thursday, January 17, 2008

### Difficult Divisions in Origami

Easy origami models generally confine themselves to equal divisions which are easy to obtain by simple folding methods, that is, binary divisions in the 2-4-8-16 series. The origami difficulty index is raised when it is necessary to divide the paper into equal non-binary sections.

There are a number of ways of doing this, most of them highly mathematical. In fact, origami may be described, perhaps a little loosely, as a branch of applied mathematics. According to someone named Julie, quoted on the Origami USA O-list lately, you know you are addicted to origami if "you confuse your geometry teacher with origami proofs". To which Andrew Hudson replied: "I did that once. It was awkward. They just don't understand..." Andrew is doubtless referring to those unusual mathematicians who have no background in origami. Of course I realize that there may be some dispute in some quarters over which of the two groups is better described by the adjective.

Other methods of dividing into equal non-binary sections involve trial and error and approximations.

The end result, in these cases, is almost invariably a piece of starter paper with lines which are unrelated to the creases required for the integrity of the target model which may mar the visual impact of that model. Pinch creases and "soft" folding can help reduce this problem but it does not eliminate it entirely.

My favorite method of dividing into non-binary divisions is to use a template of parallel and equally spaced lines which are a little smaller than the divisions I want to make on my model paper. I line one corner of my model paper up against the bottom of the "zero" line and move the horizontally opposite corner of my model paper up until it hits the line corresponding to the number of divisions I wish to make. The paper is held down firmly with one hand while the other brings the corner over to the line marked "one". Line up the edges at the top and the bottom and crease carefully. The resulting crease will provide a relevant division in the middle section of the paper which can be used to obtain the other creases in the series by simple binary folding.

The reason for lining the corner up with the first, instead of the penultimate, line in the required series is that this gives the greatest degree of edge matching area and thus improves the accuracy of the fold.

I like to turn the paper upside down and repeat this process on the opposite end. This gives me two middle section creases to work with.

Here is a sample of the paper division helper template I constructed for my own use.