(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 {15} _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.