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An Origami Instruction Language

by John Smith

Note: The following is first section of John Smith's booklet, An Origami Instruction Language, published by the British Origami Society and is included here by permission from the author. Any reproduction of this material by any means is prohibited without the express consent of the author. For further information, please consult the booklet and/or contact the author. I have made every attempt to preserve the original flavour of the text, making only such changes as were necessary to display this in HTML form (including re-drawing the diagrams). --Joseph Wu.

Reference: Smith, John. An Origami Instruction Language. B.O.S. Booklet No. 4, first published December 1975. © 1975 by John S. Smith.

Our existing methods for writing down instructions for origami models have both advantages and disadvantages. Amongst the advantages is that one can quickly check one's progress against a picture of each stage and identify a surface fold easily. The disadvantages are that it requires considerable skill to prepare good drawings and they are expensive to reproduce. Further, the present instructional system is not really precise enough about the layers to be manipulated in a fold or the exact location required for a fold. There is, therefore, a case for considering whether a written language can be developed which does not depend upon the expertise of a draughtsman and which is precise. As a by-product such a language will of necessity enable us to study more closely the structure of folds and aid in classifying and identifying fold sequences and complexities.

I presented an origami draft of such a language in my paper "The Nature of Paper Folding" (Library reference J509, 1971). Since then I have been developing the ideas and they are now at the point that I believe it useful to publish the present state of the language and to see how members feel about it. I must thank Simon Williams for his most valuable proposal which enabled me to simplify some of the typical problems of representing reverse folds.

The language has two parts which occur in each step.

  1. Definition matrix - a formal statement in rows and columns uniquely identifying the points on the boundary or elsewhere of the model so far and defining any new locations needed for the next fold/s. To a certain extent the matrix can be used to check the accuracy of what has been folded so far.
  2. Fold Instructions - Usually these consist of identifying the two points which are to be brought into coincidence by a fold (or folds). On occasions two lines may be specified or a mixture of points and lines. It is always assumed that the smallest number of creases will be made to flatten the model with the coincidence as specified (e.g. one fold only if all layers are involved). If necessary the line along which a fold is to be made can be specified.

A Demonstration

Let us see how the language works to fold a preliminary fold. Here is the O.I.L. for the fold.

Let us now translate:-

Fold Instruction Commentary
( (1) is the step number )

This is the definition matrix. The boundary points are angles on the boundary of flat model. We number them from top to bottom and left to right as we go. The number by itself as in indicates a right angle. (see (2) ). Thus our square is arranged with a point uppermost. If points are in the same row then the are exactly in the same line horizontally on the paper. If in a column then they are exactly in line vertically.
Here is a fold instruction. It means put point 1 on to point 4 and make a crease. The arrow shows a fold instruction. The number 1 over the arrow means that the top layer only is involved (we count layers involved in a fold from the top). As it is over the arrow it means put 1 on top or over 4 i.e. a valley fold.
This line shows the end of a fold. Notice how each step consists of a definition matrix followed by a fold (or folds).
Here is our second step and we start again with a definition matrix and each time we start our numbering afresh.

The line above 1 and 2 indicates an angle less than 90 degrees (if it was larger it would be shown as 1 or 2 for example). These lines are called qualifiers.

Now our fold instructions:- notice that as our numbering uniquely identifies a boundary point we do not need to show the qualifiers again. Our first fold says put point 1 to point 3. But we are required to fold between the two layers of paper as shown by the arrow passing between layer numbers. Therefore our top layer (=1) will be a mountain fold and our second (lower layer) will be a valley fold. Therefore they appear as; 2 above the arrow thus fold 1 over 3 (valley) for second layer; 1 is under the arrow thus requiring us to put 1 under 3 for the first layer (mountain). Clearly this is a reverse fold. Now repeat for the other side i.e. 2--->3. Simon Williams introduced the idea of folding into or between layers which is typical of reverse or sink folds.
It may be worthwhile to spend a little more time on this particular problem of folds made between layers - compared with folds made with layer or layers treated as one as it is an important concept in the language. Consider a square folded in half thus:-
the line from (1-2) is bounded (it has no open edges at all - or if you prefer only one edge). The rest of the boundaries have two edges.

If we write this down in O.I.L. we have

Now consider a fold
This is a valley fold - all layers together i.e. 2 is put over P. (See aboove diagram). Then try which is a mountain fold with all layers together (see above diagram). If we wish to fold but between layers 1 and 2 we have a reverse fold.

Such a move involves a valley and mountain fold and the reversal of the bounded line.

This might be appreciated more easily by cutting the line (1-T). Now we can make two separate folds.

If, however, we have the bounded line (1-T) then this must be reversed.

Therefore the instruction

necessarily means in this case three folds to be simultaneously manipulated as follows:-


& the reversing of (1-T)

In order to keep the language as compact and simple as possible the instruction only is used because the mountain and valley folds required by the coincidence of 1 to P can only be secured by the reversal of the bounded layer. Remember the minimum number of folds to achieve point coincidence is always assumed.

Shows end of fold.

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