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### The Dimensionality Of Origami

English
##### by Peter Budai

 Below I will try to define origami generally in its technical sense, regarding its relationship with dimensionality (of space). I have no special goal with this but since humankind likes to define things, why not take a closer look on origami as well? First of all, Let me explain some important expressions used later on in the definitions. Symbols "n" and "k" represent non-negative integers. "Material" will mean generally the thing we work with. Most of the time this is paper, but it can be for example metal as well or potentially anything that is sensible to make origami with. It is clear that water may be out of question as well as a thin layer of glass, but who would try to fold water or such? Out of these reasons I will not go into trying to define which material is foldable and which is not. Suppose I mean "foldable material" when saying "material". The word "folds" will stand for the phenomena used to manipulate the material. The prefix "quasi-" is used because in reality we do not have, for example, "really" 2-dimensional material in its strongest sense. Instead, we have a quasi-2-dimensional material, something that is majorly characterized by its area and whose height (3rd extension) and higher-dimensional extensions are negligable in relation with the foldability of the given material (1). More generally, when saying "quasi-n-dimensional material", I mean something whose (n+1)th, (n+2)th, ... , (n+k)th extensions are negligable (this includes that they might be equal to zero as well). As a consequence, "quasi-" has to be used for both the folds and spaces as well. (2) Now how may we define origami in its form we all practice it? Things in our most seemingly 3-D world we like to call "3-D", so let us first call normal origami "3-D origami" (later I will explain my problem with this kind of naming). Having dealt with this issue, here is the first go at the definition: 3-D origami is the manipulation of quasi-2-D material using quasi-1-D folds to move around in quasi-3-D (or higher) space. Why "or higher" is mentioned? We could now go into the issue of playing around with 4-D but there is another way as well. For this, let us see how 2-D origami looks like using the same definition: 2-D origami is the manipulation of quasi-1-D material using quasi-0-D folds to move around in quasi-2-D (or higher) space. To understand the problem, take a wire and break it at a point. The wire is the "quasi-1-D material " and the break is the "quasi-0-D fold". What you did so far could have been made in a "quasi-2-D space" as well. Now break the wire again at another point so that the plane of movement is not the same as the one of the previous break. From the condition given it is obvious that the two breaks could not be done in "quasi-2-D space". Instead, they can be done in a "quasi-3-D space". Thus the "or higher" expression in the definition. (3) The question comes that if the definition could be applied to define 3-D and 2-D origami, why could not it define origami more generally? I see no reason why we should not construct a general definition: (n)-D origami is the manipulation of quasi-(n-1)-D material using quasi-(n-2)-D folds to move around in quasi-(k)-D space (where k>=n). Can 1-D origami be defined then? The answer is no, simply because (-1)-D folds do not exist. This means that according to our definition, the first possible origami is 2-D origami. But isn't that awkward? We define n=0,1,2.... dimensions so it would be appropriate to define n=0,1,2...-dimensional origami as well, instead of n=2,3,...-dimemensional origami. The root of the problem is that we have chosen a wrong naming convention. We named after the dimension of quasi-lowest-dimensional space. Rather the quasi-dimension of the applied folds should be chosen as the base for naming. The general definition according to this is: (n+2)-D origami is the manipulation of quasi-(n+1)-D material using quasi-(n)-D folds to move around in quasi-(k)-D space (where k>=n+2). Therefore we can start with 0-D origami, which is what we called "2-D-origami" so far, then comes 1-D origami, which is what used to be "3-D origami" until now, and so on. I admit that calling "normal" origami "1-D origami" sounds very strange for the first time, still I think it is not that hard to get used to the idea that we name after the dimensions of the fold and not the dimensions of space. Of course it's realtive, and we could even name after the dimensions of the material as well. However, I would suggest the first option (folds) because it creates a sensible system of "(n)-dimensional origami". One could start nitpicking that the meaning of "origami" is " paperfolding" and we defined origami as folding of any material. (4) That is true. To prevent this, the name of the definition could be changed to "(n)-dimensional folding". (5) (1) One could fold a 5 cm high, 4 sqare meter sheet of polyfoam in half but could not do this with a sheet of wood with the same size parameters because it will break. For this, a lot thinner piece of wood; paper is needed. (2) If the ideal situation is presumed, that is, we assume that there are true 1-D, true 2-D materials, all the "quasi-" prefixes can be left out. It is a shame that life is not ideal... (3) Note that just breaking the wire will not be equivalent to "flat folding". For that the wire needs to be aligned back to itself after each break. (4) Even if "origami" is used generally for example for "foil folding" as well, not only and exclusively for "paper folding". (5) Another advantage of this: now nobody can complain about why to name (n)-dimensional folding after the dimensions of folds.

Author: Peter Budai

Last updated: 20. January 2002.