It is several days since Saintee (President of the Raccoon Republic - where is that?) asked what were the Properties of Origami. In response to request for clarification he wrote: "The way the paper folds - or - getting the same model with two or more ways to do it".
There hasn't been much of a response and perhaps the theme has gone cold by now. Even so, it has been churning round in my mind and I should like to raise it again. I should like to help if I can.
I suppose that the difficulty turns in part on Sainatee's use of the word "properties". What does Sainatee mean by the word "properties" in this context? I haven't looked it up in the dictionary, but it is one of those words which has a multitude of meanings. Before Sainatee's question can be answered, we really need to know the definition of the word "properties" in this context.
Does it mean the way the paper physically folds and creases - how the crease forms and how it is "remembered" by the paper or other material? Perhaps related to this is the question, often asked: why does paper always crease in a straight line?
I suspect, however that Sainatee is asking about the mathematical patterns which the folding produces. This fascinates many folders and it is surprising the way that the same patterns reproduce in very widely differing styles of folding. If this is what is being asked, there are several books and articles that discuss the matter. As a start, I would suggest Peter Engle's "Folding the Universe, Origami from Angelfish to Zen", where among many other aspects of paperfolding he discusses folding patterns. In fact the whole of the introduction to the book may be considered to be a discussion of many of the properties of origami. There are, however, several other writers who have discussed crease patterns, including Robert Lang (particularly in articles he has written apart from his books. Kunihiko Kasahara also discusses them, as for example in "Origami Omnibus". Kunihiko Kasahara features the work of Jun Maekawa in several of his books and Jun Maekawa has also written articles about folding patterns.
Perhaps I may mention my own consideration of basic crease patterns, in which I divide them into radial patterns, grid patterns and (parallel) pleated patterns. They are found in folding from simple to complex. All through, radial patters are overwhelmingly used for living plants and animals whereas grid patterns are used for inanimate objects. These ideas are not my own: they were put forward by Murray and Rigney in "Fun with Paper Folding" in 1928, but Murray and Rigney used the terms "diagonal folding" and "square folding", which I believe to be very misleading; hence my substitution of the terms "radial" and "grid". Koshi Uchiyama also discusses basic crease patterns in his book "Origami Zukan" of 1958. The Mathematics of crease patterns is, however, a vast subject and the books I have suggested are only a few among many. I have said nothing at all about the crease patterns of origami tessellations or the patterns of Shuzo Fujimoto. Nevertheless, a study of the subject even at an elementary level can give a much clearer understanding of this aspect of the properties of paperfolding and of paperfolding generally.
Perhaps, as well as trying to define what is meant by the word "properties" we should ask about the definition of origami. Joseph Wu has come up with an excellent definition of Origami. Unfortunately, I have to confess that I cannot remember its precise form and have been unable to trace it among the thousands of postings to Origami-L recently - perhaps someone can remind me. My own suggestion for a definition is simply: "The Art and Science of Folding", which is comprehensive enough include all styles of folding and all kinds of materials. But perhaps some may find it a little too imprecise.
To conclude this cursory glance at what may be meant by "the Properties of Origami", I can think of nothing that throws better light on what origami is and what it is not than John Smith's article: "Origami Profiles". This can be found at John's own web site at: http://www.users.waitrose.com/~pureland/
Alternatively it may be found under "Theory" at the British origami Society Website at: http://www.britishorigami.info
John's article deserves the closest study, because it show how the nature of origami varies with the constraints that each individual folder seeks to impose on his or her folding.
While you are looking at the British Origami Society site, you may also like to look at Lister's List (also under the heeding "Theory"). In it you will find a number of my impromptu postings to Origami-L in which I consider a number of matters which may throw light on the properties of paperfolding.
On the other hand....you may find that my musings only serve to obscure the subject!
Anyway, Saintee's questions deserve careful thought. The very process of thinking about them extends our understanding of paperfolding and remote though all this theorising my seem at first, it will be of great help to help us when we come to the actual process of folding.
David Lister Grimsby, England.