THE LAW OF PERFORATION by Alan Lipfriend                                   May 1985

I was very interested to read Professor Kington's paper in The Times (17.04.85) on perforated holes. This is a subject I have been working on for many years but, since Prof. Kington's paper is the first publication on this topic, I am quite happy to accept his title  Kington's Law of Perforation. However, I do not accept that the principle  "If you puncture a piece of paper with a line of small holes, that line of holes becomes the strongest part of the paper"  is of general application.

Prof. Kington is, of course, well known as an extremely accurate observer of human experience but, after years of research, I think I am fairly close to being able to prove the mathematical basis for the principle of Kington's Law.

Fig. 1 below shows a blank cheque with the stub to the left of the Line of Perforations (LP)

P = force applied to the right hand end of cheque

Θ = angle of force to the midline of the cheque

AA, BB and CC are sections across the cheque

a = distance of a particular section from the end of the cheque at which the force P is applied

w = width of the cheque

It can easily be proved that if θ is equal to or less than 37° the stress across any particular section of the cheque is uniform when the distance of the section from the end at which the force is applied is equal to the width of the cheque i.e. if a = w the stress is uniform across section AA. But when the stress, which travels from section AA to BB to CC, reaches the LP, in certain cases, there is an astonishing result.

I make it clear that my work on this aspect has not yet been completed but I am confident that the following conclusion will be reached in Fig. 2 below

d = diameter of the Perforation

s = space between two adjacent Perforations

There is a critical ratio for d/s at which it becomes impossible to tear the paper. It is fairly easy to establish in the laboratory that if you put a line of perforations under a microscope in good light and study any particular perforation, you see nothing. My theory is that when d/s equals the critical value, the perforations act as black holes. The effect of this is that the stress, which became uniform at AA and worked its way across to BB and CC, when it reaches LP is absorbed by the black holes and then there is nothing left to tear the paper between the perforations. I have also established that this critical ratio depends on a number of parameters, including the thickness of the paper, the number of perforations per unit width of paper and the co-efficient of elasticity of the paper.

Of course, if the ratio d/s increases beyond the critical value, the strength of the LP becomes less than the rest of the paper. Some years ago at an international conference I met a very wealthy Professor from the Cayman Islands who had persuaded his bank to provide him with a cheque book where the perforations were of such a size that d = w.

Apparently he had told his bank that he acted as a financial consultant to many frail old ladies and the cheque book was printed as he had requested and the bank computer now continues churning out the same instructions for the manufacture of his cheque books and every time I meet this man he seems to be wealthier and wealthier.

Prof. Kington's final example  that of the boxes of paper tissues is somewhat more complicated than a simple application of the Law of Perforation. I agree that it is almost impossible to tear the top of the box along the LP but this is substantially because of two separate and additional principles. One is due to the fact that since the top layer of the box of tissues is always peach coloured, this acts to reinforce the LP. I firmly believe that the cause of this strengthening of the LP has a basis similar to that behind the new Post Office colour sorting machines. But the main point arises from the shape of the LP.

The general principle that if the ratio of d/s is critical, the perforations act as black holes remains. But if the shape of the LP is elliptical then the ratio of d/s becomes critical over a range of values.

For example, if the critical value for d/s for a straight LP is X then, if the LP is elliptical, the critical value lies within plus or minus 13% of X.This is a particular application of a principle I have called the Law of Oblate Spheroids.

There is a further example that I would like to draw to the attention of Prof. Kington. That is the perforated tax disc. As most of us know the tax disc is really a circle of information printed on a square piece of paper with an LP forming the circumference of the circle. The ordinary citizen buys one of these at the Post Office and is then required to attach the disc to the inside of his windscreen. Many motorists use a thin plastic container for the paper tax disc. But how does one remove the tax disc from the square piece of paper? This I concede is not quite as difficult as removing the top of the tissue box, but that is because of the absence of any peach coloured material. The Department of the Environment is to be congratulated on the fact that, come what may, they have never used peach coloured tax discs.

I have often tried to conform to the requirement of windscreen exhibition without attempting the tear along the LP by bending the surrounding paper along the LP. This is rather difficult and, even when I have managed to do this I have never been able to get the bent over portions into the plastic container. But what I can't understand is why the tax disc is always circular. There is nothing in the relevant acts which requires this and if the tax disc was rectangular or square, is there any need for the surrounding piece of paper, which has to be torn off  or bent over? We would be left with the relatively simple situation of a straight LP and our hope that the ratio d/s was not critical. I am, at the moment, also engaged in studying the feasibility of a square or rectangular tax disc  with no surrounding paper! And I hope then to go on to the possibility of having an adhesive face to the tax disc. I say adhesive face because it cannot be at the back, since we all put the tax disc on the inside of the windscreen. But if we have a square or rectangular tax disc with adhesive on one side, we come back to the difficulties Prof. Kington has referred to with the stamp and I doubt whether the Post Office union would allow its members to be subjected to the overwhelming frustrations of having to separate tax discs attached to each other by a series of black holes on a wet Thursday afternoon at the end of each month

My research continues.

(c) Alan Lipfriend 1985                                                                                            {Ref 1007}


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