Art, Origami & Education
About me  
Akira's Children  
Angle trisection  
Contact John  
Division 3rd 5th 7th  
Fold Difficulty  
Forgiving folds  
History Notes  
In the Beginning  
Minimal Folding  
OIL : Origami instruction language  
Records & Curiosities  
Santa Claus  
Scanner Pictures  
Suggesting movement  
Therapy & Education  
UK Sites  
site design : nick robinson  

Measuring fold difficulty and area ratios

(1) ThIs article is based on papers I published in the BOS magazine in 1976, under the title " It's all relative"

(2) There are two themes running through this paper both of which have interested me (and at times annoyed me!) over a number of years.

A. Is there a simple way to indicate how difficult a model is going to be to fold?

I first tackled this when preparing the early BOS Library catalogues - it is not really enough to give the title of a model and the author - some idea of the likely difficulty of the folding would be useful as well. Accordingly I introduced an index of fold areas' which was simple to calculate and on the face of it did reasonably order the models according to accepted difficulty.

However the research was not really adequate and I remained doubtful that the proposed index was as good as it could be.

B. What is the relationship of the size of the finished model to the area of the paper from which it was folded?

The usual diagrams are almost hopeless on this score - not surprisingly authors tend to vary the size of the model in their drawings - particularly in the final moves and thus the scale changes.

It is often impossible, therefore, to see what the size of the final model will be for a given sheet of paper. This is a darn nuisance at anytime but when one is trying to prepare exhibition models or pre-folded examples for teaching it is downright annoying.

Having two such themes as this it is natural to wonder if they are related,' i.e. is it true, or not, that the difficulty of folding is usually related to the relative size of the model? In the end it became clear that "it's all relative!"

(3) To bring the two themes into sharp focus I set myself five objectives:-

(i) Find the 'best' measure of the relative size of the model to the original paper area. 

(ii) Find the 'best' measure of the difficulty of the model. 

(iii) Find the 'best' relationship between the folding and the relative model size. 

(iv) Examine different ways of showing the ratio of the model to the paper size and fold difficulty. 

(v) Note any interesting points along the way.


I am using 'paper' here merely for convenience .. I refer, of course, to the regular shaped material from which one has started in its flat fully opened pristine state.

It will also become clear that I have limited myself to 'flat' models or at least models that can be adequately displayed as flat. This is not because I do not consider 3D should be studied, merely that I had to limit the work somewhere.

Naturally the idea of finding a 'best' of anything means looking at a number of possibilities and deciding in some objective fashion (if possible) which, if any of those possibilities is superior. In any work of this kind the problem is to try to ensure that one is considering a reasonable spectrum of possibilities and is using an adequate sample on which to test them.

I would not regard what I have to report here as any thing more than tentative since the number of models I have examined is small and the measurements I have made likely to be of limited application, nevertheless I shall be content if I have encouraged other folders to ponder the problems posed here and consider the solutions. If they can carry further research I shall be delighted.

(4) Let me now consider some of the ways I could measure the relative size of the model to that of the paper.

(i) Use the ratio of the model length to that of the paper. Diagram 1

I have preferred to use the maximum length of the 'posed' or display position of the model when placed in the 'frame' of the paper to that of the side of the paper.

Ratio (L) = paper length/ model length

(ii) Use the ratio of the 'enclosing' or framing area to that of the paper. Diagram 2

By 'enclosing' I mean the rectangle which will exactly frame or enclose the 'posed' or displayed model (considered as flat).

By now dividing the area of this rectangle by the paper area I have found the relative area

(iii) Use the ratio of the 'model area' to that of the paper. Diagram 3 

By 'model area' I mean the area of the model itself when in a flat 'posed' or displayed position. This is shown in Diagram 3 notice how both ears are displayed since they have been formed separately and are therefore capable of yielding model area I have therefore considered it necessary to include them. It is only in this way that it is possible to be consistent in both the problem of relative size and fold difficulty. In any case presumably the extra folding and reduction in size necessary to secure separable points or projections for legs, feet, ears, etc., must have been intentional and should therefore be included in the measurements. 

Diagram 3 Model Area 

It is a tedious business to estimate the model area and I have done so by using a grid of 1/10" squares and counting those squares fully inside the model or having a half or more of the model inside (see Diagram 3). This technique was suggested to me by Mick Guy and certainly seems the simplest. Where a line runs approximately parallel to the grid but does not lie exactly on a line it is necessary to count every other square or some such fraction.

Ratio = Model Squares divided by No. of squares in the original paper.

(5) Of the three methods described here each has its strengths and weaknesses.

Ratio of model length to paper length

This is useful when considering display space, particularly if one is clear about the appearance of the final model: it is also very simple to calculate. Problems arise when one considers shapes such as a triangle pentagon or various rectangles - it is not obvious how or where to measure the papers length.

In addition how should one measure the 'length' of a model - is it always the maximum when displayed as it were in a frame? What does one do about geometric or abstract shapes? Should one take the maximum length one could draw on the paper (say the hypotenuse of a rectangle) and express as a ratio the maximum length which would connect two points to be found on the model itself?

On the whole I felt these problems were too complex to solve and rejected this ratio or its possible variants.

(ii) Ratio of the Enclosing Area to the paper area.

This is very attractive in that it gives the relative 'frame' which fits the model. It is visually easier to see than to realise numerically. As a simple way of showing in diagrammatic form the relative size this is, I think, much superior to (i)

From the point of view of display space etc., it is a very useful way of describing the relative size. In effect It incorporates the various 'line lengths' discussed in (i) in one simple shape.

There are, however, one or two snags In searching for the relationships of folds to relative size it seems more sensible to consider the actual model area than some regular shape which will 'frame' it

The size of the frame will also be changed by simply extending a leg or wing and hence the ratio will vary according to the display posture. This is not altogether a satisfactory state of affairs for a measure to be used in studying relationships.

(iii) Ratio of Model Area to that of paper.

This would seem much the best way of expressing the actual 'paper' areas. It does not depend on the choice of a frame shape or on the chance display arrangement. It is, perhaps, the most attractive measure from the point of view of establishing other relationships. The objection to it is that it is not immediately clear what the model size in space will be from a particular piece of paper. A model with long thin projections a la Sekoda will have a small area but may well need a relatively large 'frame'.

On balance I favour the use of the model area ratio for analysis and the 'enclosing' area (or frame area) ratio for indicating in a visual way the size relationship.  

(6) How I can turn attention to the second theme - of the indicating the difficulty of folding and consider some of the ways I might do this.

(i) Counting the 'fold' areas.

This is the method I used in early editions of the Library catalogue. The procedure is illustrated in Diagram 4. Total 6 note:- as second ear is 'free' it is counted as displayed.

Notice the second ear is arranged so that it is 'displayed' a point that has already been discussed. A fold area is one that is bounded by the edges of the model and an edge (or edges) of an overlaying fold. This will be clear from the diagram or, if not, by folding the bunny (see Randlett's 'Art of Origami').

Diagram 4........... Fold areas ..........................Diagram 5 .......Outline changes

(ii) Counting the Outline Changes

In this method one simply counts the number of times that the outline of the model changes direction ,that is, an angle occurs. It is made clear in Diagram 5. For convenience I have numbered the lines connecting the overline changes rather than the change points themselves but this gives exactly the same results. Total 10. 

(iii) Fold Analysis

In this approach I go directly to the heart of the matter and count the various folds used to make the model , I have preferred to follow exactly the sequence given in the publications I have consulted, although there are clearly many different sequences which are possible for any given model and this may affect the fold analysis.

In Diagram 6 a simple model is analysed and then presented in a summary table giving the count of each type of fold and the total.

Diagram 6 Fold analysis

 = 2 Mountain..................= 2 Valleys.......... = 2 Squashes........... Total 6

I must recognise that folds differ considerably in difficulty. A book fold is much easier than sinking the 'cone' of a frog base. In the one case I are only manipulating one fold in the other about 24 (depending on your definition of manipulating). Can I therefore convert the Fold Analysis to some form of index which will give in one number the fold difficulty?

In order to tackle this problem I look at two ideas. One was to count the folds which needed manipulation in a given move and the other was to time the fold itself. 

In order to obtain the timings I decided to attempt a large number of folds in succession using different positions for the folds and obtain my average time. I did not hurry the folding but attempted to maintain the same tempo throughout and maintain the same accuracy of location and creasing. In this way I hoped to reduce the effect, if not eradicate it, of using myself as a sample. I could then divide through by a basic time and reasonably hope that the resulting numbers would give a standard time - difficulty index which might have some more general validity. 

It was obvious that many people can fold faster and more accurately than I can but I was hopeful that the relative index of time would be similar,

In Diagram 7 I give the results of my work. In all I made 155 folds of 7 different types (e.g. petal, squash reverse etc.). In addition I included tucking in and opening out which were needed in my model studies. These 10 or fold types were sufficient to account for all the sequences I found in my fold analysis of various models. This is quite a tribute to our President Robert Harbin who first made clear, named and identified these basic 'bricks' of Origami. 

In order to 'standardise' the Index I have divided each Average time in secs. by the time required for a mountain or valley fold. Thus, as discussed above, I hope to preserve the relative difficulty while removing my own particular speed of folding.

Diagram 7. Fold-Time Index of Difficulty


Av. Time in secs



















Rabbits Ear




Sink (4 layers)




Sink (8 layers)




Open Out




Tuck in with fold




Tuck in no fold




Since the index was based on time it is reasonable to assume that the index will give us a scale which is additive (i.e. we can add the indices) and in which the relative difficulties are expressed (a ratio scale). 

As a check on this scale I examined the relationship with the number of folds requiring manipulation in each fold type. The relationship was reasonable but it was evident that the index reflected the time saved in the location of a fold - thus a squash fold which involves 3 fold manipulations took less time than 3 mountain or valley folds. 

(7) It is evident that the methods of counting fold areas or outline changes is of limited application. There are many geometrical folds which have only 4 outline changes but are of considerable complexity. We may reduce fold areas by turning a model inside out and thus hiding fold complexity. The simple fold summary and count does not reflect the different difficulty of the folding and I am satisfied that the Fold Index is likely to be the best single measure of difficulty with a reasonably clear meaning.

(8) In order to throw light on the relative merits of fold difficulty measurement and on the relationships, if any, between the model size ratio and difficulty I decided to study a number of models in detail. In the end I analysed twenty models ranging from the simple House to Rhoads Elephant & Nakano's Pointer.

In Diagram 8 I give an example of the Information collected for one such model, Rhoad's Elephant. 

Most of this is self explanatory but the following notes may be helpful: - 

(a) Ratio A (area) = 2500 divided by 139 = 17.99. The same procedure is followed for the other ratios. 

(b) The fold index is simply the result of multiplying the index in Diagram 7 by the corresponding number of such folds e.g. 11 squashes x 2.58 = 28.38 These were then added together to give the total index of 141.44.

By multiplying by the Valley/Mountain time of 4.84 secs I obtained my 'standard time' for the model of 684.57 secs or about 11 ½ mins. I would expect to take this time If I knew each move and each location and the required sequence. Obviously this time would need adjustment for different paper sizes and materials but I have done no work on this point. (I used 6" paper in my studies).

Diagram 8

An example of Model Analysis 

Rhoads Elephant 

Original Area


Outline Changes


Model Area


Fold Area


Ratio (area)


No. of Folds


Paper Length


Fold Index


Model Length


Fold Secs.


Ratio (length)


Enclosing Area


Ratio (enclosing area)










Rabbit's Ear




Open up


Tuck in


 (9). With the large set of measurements it is now possible to enquire into the 'best' relationships.

(a) The time-index appears in my studies to be the best single measure of fold difficulty but it does require a lot of analysis and calculation. I examined how well it was predicted by Outline changes and Fold Areas. The best was Outline Changes which was significantly better than Fold Areas for the sample considered. 

The method I used was a statistical technique called linear regression. To judge how good is a prediction one considers the total variation in the measurement one is predicting and calculates the proportion explained by the other measure. Over 80% of the variation in the time-index was explained by the regression equation Indicating a very strong relationship. (For those who wish to pursue this I will give the results each time. In this case: -

r = .9468 n = 20

y = 17.416 + 3.2525x

where y = Time Index and x = Outline Changes).

This result is probably valid for the kind of models I included in my studies which were nearly all animals. It is unlikely to hold for geometric models in general.

(b) I can now consider the problem of predicting the relative model size (ratioA). It is necessary to introduce at this point two conjectures:- 

(i) Model size is determined by the reduction due to overlap.

(ii) A reverse fold and the equivalent mountain valley fold give the same overlap.

(The same thing appears to be true for other folds).

The first conjecture is demonstrated by simply taking a square of paper and folding it in half. The area is now ½ of what it was and this is due to the overlap.

The second is clear, I think, when we consider the folding of the leg of a bird.I may obtain the same display size by either mountain/valley folding or by reverse folding.

From these conjectures I may expect the Fold count (i.e. No. of folds) to be relative to the size ratio. In fact it is by far the best of the measures examined.

r = .89 n = 19

y = 2.4117 + .29235x

where y = model size ratio Ra; x = No. of folds.

(I have excluded tuck ins and open-ups from this analysis).

10. Some Interesting points

(i) One model stood out as a 'sore-thumb'. It had a ratioA of 5.75 compared with 6.94 for the Bunny and 3.27 for the House, yet its Fold Index was 112.98 which made it of the same order as difficulty as Neal Elias' s Lion. Rechecking it showed that the reason was simple - the model in question (Halloween Cat by Rohm) contained several cuts. Thus it revealed the consequence of cutting for a given complexity - model area will be larger than for a non-cut model of equal complexity. This consequence is obviously dependent on the cutting being exploited - cutting to no purpose would not effect the balance.

(ii) A very interesting proposal made to me was that by folding and timing say a bird base one could then calibrate the time index - that is work out the multiplier to convert the index to seconds for ones own speed of working. 

(iii) At a convention it was suggested that one could analyse the fold plan for a model and obtain an index from this. This is a very interesting idea and I refer to this in another paper in my research

(iv) In my studies of models it became evident that there is a difference in the choice of folds between authors. In one case reverse folds were preferred where another author used mountain/valley folds.

It is possible that over a reasonable sample of models the ratio of fold types will be peculiar to a particular folder and it may be possible to distinguish between them. This is almost the equivalent of statistical methods of determining authorship. I wonder if it may throw light on the true origins of origami?

(v) I have avoided 3D models in these studies. Clearly the fold-index will work but what of the size? It has been suggested that I use the interior cubic capacity ! It does seem to me that 3D models are really something special. Consider a water bomb - in Its flat form the overlap is considerable giving a large ratio. But this overlap is exploited when the model is blown up. It seems to me to be a very exciting idea that of using overlap to generate 3D. The simplest way of expressing the ratio to the original paper is probably to show the 3 views (one for each dimension) with the enclosing frames.

(vi) In my studies I also came across the need for an understanding of bases and the relative size they impose on the model resulting from their use. I do not have space to develop this idea here but given certain principles one can show the maximum relative size of a model derived from each base and this solution is unique. (These are not 3D of course !)

(vii) Using the fold-index one can see whether a different fold procedure is easier in the sense of a small index. It is possible that fold location may be the vital factor ! I took this idea much further in my presentation to COET 95

(11). Showing relative size and Indicating difficulty. 

What would I prefer to see from all of the foregoing discussion in indexing or referring to models?. 

(I) Without the diagrams I would like to have:- 

(a) Fold Index.

(b) No. of folds.

(c) length and breadth of 'enclosing' rectangle as a fraction of the length and breadth of the paper - assuming the paper is rectangular. If the paper is not rectangular I would like to have the enclosing ratio.

(d) Model area ratio

(ii) With diagrams I would like the Fold Index and the No. of folds but would settle for the actual model placed on the original paper as shown in

diagrams 9 to 12. Mick Guy has pointed out that they will take a lot of space If the model is shown clearly (or it will appear as a little blob If the paper is shown in a reasonable size). 

I accept the difficulty but so far I have not found models smaller relatively than the Grasshopper which is 1/24 of the paper area (is there a limit?). If it is a problem perhaps one could put the enclosing rectangle around the model and show the fraction of the papers length and breadth.


  12 Model Efficiency Index

Since the outline changes are a measure of the complexity of the model and the ratio of the model area to the original paper area tells us about the amount of paper 'consumed' in the model we can use these two measures to indicate the efficiency with which a model achieves its effect. In general we prefer a model to use paper as effectively as possible.

A simple index of efficiency is to divide the number of outline changes by the ratio of the model to the original area.

Rhoad's Elephant has a area ratio for the model of 17.99 and the outline change count is 45. This gives us an index of efficiency of 45 divided by 17.99, that is 2.5. If the model had been more complex but still used the same amount of paper the index would be greater. If for the same complexity less paper had been used, again the index would be greater.

As a comparison the traditional house (diagram 10), has a model ratio of 3.2 and 8 out,ine changes. This gives an index of efficiency of 2.5. the same as the elephant.

This does make the point that the index is not a measure of how successful a model is ( whatever that may mean) but how little paper is used relative to it's complexity.

golden crane award