GEOMETRICAL CURIOSITY.
(Vol. viii., p. 468.)
Mr. Ingleby's question might easily be the foundation of a geometrical paper; but as this would not be a desirable contribution, I will endeavour to keep clear of technicalities, in pointing out how the process described may give something near to a circle, or may not.
When a paper figure, bent over a straight line in it, has the two parts perfectly fitting on each other, the figure is symmetrical about that straight line, which may be called an axis of symmetry. Thus every diameter of a circle is an axis of symmetry: every regular oval has two axes of symmetry at right angles to each other: every regular polygon of an odd number of sides has an axis joining each corner to the middle of the opposite sides: every regular polygon of an even number of sides has axes joining opposite corners, and axes joining the middles of opposite sides.
When a piece of paper, of any form whatsoever, rectilinear or curvilinear, is doubled over any line in it, and when all the parts of either side which are not covered by the other are cut away, the unfolded figure will of course have the creased line for an axis of symmetry. If another line be now creased, and a fold made over it, and the process repeated, the second line becomes an axis of symmetry, and the first perhaps ceases to be one. If the process be then repeated on the first line, this last becomes an axis, and the other (probably) ceases to be an axis. If this process can be indefinitely continued, the cuttings must become smaller and smaller, for the following reason. Suppose, at the outset, the boundary point nearest to the intersection of the axes is distant from that intersection by, say four inches; it is clear that we cannot, after any number of cuttings, have a part of the boundary at less than four inches from the intersection. For there never is, after any cutting, any approach to the intersection except what there already was on the other side of the axis employed, before that cutting was made. If then the cuttings should go on for ever, or practically until the pieces to be cut off are too small, and if this take place all round, the figure last obtained will be a good representation of a circle of four inches radius. On the suppositions, we must be always cutting down, at all parts of the boundary; but it has been shown that we can never come nearer than by four inches to the intersection of the axes.
But it does not follow that the process will go on for ever. We may come at last to a state in which both the creases are axes of symmetry at once; and then the process stops. If the paper had at first a curvilinear boundary, properly chosen, and if the axes were placed at the proper angle, it would happen that we should arrive at a
regular curved polygon, having the two axes for axes of symmetry. The process would then stop.
I will, however, suppose that the original boundary is everywhere rectilinear. It is clear then that, after every cutting, the boundary is still rectilinear. If the creases be at right angles to one another, the ultimate figure may be an irregular polygon, having its four quarters alike, such as may be inscribed in an oval; or it may have its sides so many and so small, that the ultimate appearance shall be that of an oval. But if the creases be not at right angles, the ultimate figure is a perfectly regular polygon, such as can be inscribed in a circle; or its sides may be so many and so small that the ultimate appearance shall be that of a circle.
Suppose, as in Mr. Ingleby's question, that the creases are not at right angles to each other; supposing the eye and the scissors perfect, the results will be as follows:
First, suppose the angle made by the creases to be what the mathematicians call incommensurable with the whole revolution; that is, suppose that no repetition of the angle will produce an exact number of revolutions. Then the cutting will go on for ever, and the result will perpetually approach a circle. It is easily shown that no figure whatsoever, except a circle, has two axes of symmetry which make an angle incommensurable with the whole revolution.
Secondly, suppose the angle of the creases commensurable with the revolution. Find out the smallest number of times which the angle must be repeated to give an exact number of revolutions. If that number be even, it is the number of sides of the ultimate polygon: if that number be odd, it is the half of the number of sides of the ultimate polygon.
Thus, the paper on which I write, the whole sheet being taken, and the creases made by joining opposite corners, happens to give the angle of the creases very close to three-fourteenths of a revolution; so that fourteen repetitions of the angle is the lowest number which give an exact number of revolutions; and a very few cuttings lead to a regular polygon of fourteen sides. But if four-seventeenths of a revolution had been taken for the angle of the creases, the ultimate polygon would have had thirty-four sides. In an angle taken at hazard the chances are that the number of ultimate sides will be large enough to present a circular appearance.
Any reader who chooses may amuse himself by trying results from three or more axes, whether all passing through one point or not.
A. De Morgan.