Using Nodes as a Method Of Classification and Measurement of Difficulty
I published this research in1990 in "Patterns in Paper" (BOS 32)
The identification of nodes and the simple process of counting the crease lines which meet at a node suggests that nodal classification could be used as a means of identifying folds and of expressing their level of difficulty.
Measuring the difficulty of a fold
The simplest approach we can adopt is to count the number of creases that connect to each nodal point on the basis that the more complex a model the more nodal creases it will have. This ignores the problem of locating a fold and of the increasing difficulty which may arise as the number of folds at a node or the total number of creases involved in a model increases. (In other words the relationship assumed in the simplest model is linear whereas, in fact, it may be non linear).
To take the actual example for a Bird Base
The crease pattern of a Bird Base has
Since the fold lines all reach the boundary directly then the boundary nodes are necessarily created when the folds are made for the internal nodes. It appears adequate, therefore, in this case to only count the internal nodes ( ie inside the paper boundary), thus simplifying the task.
In our simplest approach, therefore we have 6 * 1 + 4 * 4 = 22 folds. (* means multiply)
Compare this with other bases put in order of nodal count.
As a check on the idea that the number of internal nodal folds will indicate the level of difficulty I have used some of my previous work in this area. (B. 0. S. No. 61). A Bird Base consists in it's simplest form of one valley fold followed by 6 reverse folds. This gives a time to fold estimate of 6 * 12 secs plus 5 seconds = 77 seconds. A fish base is formed by 1 valley + 2 reverse folds giving a time of 29 seconds. Thus in terms of timing the Bird Base takes 77/29 = 2.66 times as long as the Fish Base.. The method of nodal counting gives a ratio of 22/8 =2.75 which is reasonably close. In addition the rank order of difficulty given by the nodal count does seem to accord with experience. In the above nodal calculations I have only counted the nodal creases in the final flat form of the base.
USING NODES AS A METHOD OF CLASSIFICATION,
In my original work on this idea I thought in terms of a vector which gave the frequencies for each nodal count Thus one could assume that the order of node counts was 2,4,6,8,10 (assuming internal nodes that could be flattened) and then all that would be needed was to have a vector which gave the frequency for each nodal position. Thus the Bird Base would be 04100000 etc This vector could very easily be sorted on a computer and thus serve as a method of recording classifying and identifying folds. I am assuming that the nodal patterns would not have many cases where the frequency was exactly the same for different models. Further work convinced me that the method would become very clumsy if one had to include boundary node counts and if the number of nodes exceeded 9. In the end I came to the view that the best way to proceed was to set up a data base which would include only the actually occurring number of creases at a node and the frequency Thus such a set for the Bird Base including the boundary nodes would be 1 4 2 2 ; 3 2 ; 4 4 ; 6 1. with the number of creases at the node appearing first then the frequency. Multiplying out the pairs and adding them up gives us 36. The data base would also include other relevant matters such as the author, first date of publication, what the models title is, and what it represents etc.
The use of the nodal count has two advantages ;..
1 It gives an indication of the difficulty of the
Thus by doing the nodal analysis on the model we arrive at the nodal count. By referring to the data base we can find those models which have the same count and then compare the actual distribution and what is represented by the model etc.
This classification can cope with 3D models although the folding difficulty may not be sufficiently indicated by the nodal counts. A problem that needs to be solved is how to easily identify the nodes when the paper is opened up, should one include the folds which are made to assist in the folding but are not required in the final model? Obviously a great deal of further work is needed to resolve these and other problems.