Let us see how the language works for a preliminary fold. Here is the O.I.L. for the fold.
Let me now translate:-
This is the definition matrix. The boundary points are angles on the boundary of the flat model. We number them from top to bottom and left to right as we go. The number by itself as in 18.104.22.168. indicates a right angle. (see (2)). Thus our square is arranged with a point uppermost. if points are in the same row then they 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 < before 1 and 2 indicates an angle less than 90 deg. (if it was larger it would be shown as >1 or >2 (for example). These signs 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 and 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 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
T = ½ (1-- 2 )
Now consider a fold
This is a valley fold - all layers together i.e. 2 is put over P (See above diagram).
which is a mountain fold with all layers together (see above
diagram) - If we wish to 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 in 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.
Now let us carry on from the preliminary fold and form a Bird base.
Now for the explanations. Notice (P) appears in the definition matrix.
Where a letter is enclosed In brackets Indicates a location point either on or inside the boundaries of the model, (or even outside of the boundaries).
This point must be defined further If necessary, and so we had P = 4+ (4--3).
This means locate P as the distance from 4 given by the length of the line from point 4 to point 3 or add to point 4 the length of line (4--3) i.e. 4+ (4--3). Two points enclosed in brackets with a dash Indicate a line. We know that (P) lies on the line from point 1 to 4 and Is above points 2 and 3. In fact one can locate it by folding in the edge (4--3) along the line (4--1).
Now we need to make four reverse folds in all and thus we have four fold instruction lines.
Here is one of the fold instructions which tells us to make a fold between layers 3 and 4 so as to put point 2 on to point P. Thus 4 will be a valley fold and 3 a mountain as a reverse fold is made.
The four fold commands are a demonstration incidentally that a the two petal folds in the bird base are really four reverse folds.