The Irregular and Fragmented in Nature

Jean Perrin

familiar motions this view appears true enough, do not see that it involves considerable diffi- culties.

Mathematicians know that it is childish to try to show by drawing curves that every continuous function has a derivative.

Differentiable functions are the simplest and the easiest to deal with. But they are exceptional.

Using geometrical language, curves that have no tangents are the rule.

Regular curves, such as the circle, are interesting but quite special.

At first sight, the consideration of the general case seems merely an intellectual exercise, ingenious but artificial, the desire for absolute accuracy carried to a ridiculous length.

Those who hear of curves without tangents, or of functions without derivatives, often think at first that Nature presents no such complications, nor even suggests them.

The contrary, however, is true.

The logic of the mathematicians has kept them nearer to reality than the practical representations employed by physicists.

This assertion may be illustrated by considering certain ex- perimental data without preconception.

Consider, for instance, one of the white flakes that are obtained by salting a solution of soap.

At a distance, its contour may appear sharply defined. But as we draw nearer its sharpness disappears. The eye can no longer draw a tangent at any point.

A line that at first sight would seem to be satisfactory ap pears on close scrutiny to be perpendicular or oblique.

The use of a magnifying glass or mi- croscope leaves us just as uncertain, for fresh irregularities appear every time we increase the magnification, and we never succeed in getting a sharp, smooth impression, as given, for example, by a steel ball.

So, if we accept the latter as illustrating the classical form of continuity, our flake could just as logically suggest the more general notion of a continu- ous function without a derivative."

An interruption is necessary to draw atten- tion to Plates 10 and 11.

The black-and-white plates first mentioned in a given chapter are collected on pages that follow immediately, and are numbered as the pages on which they occur. The color plates form a special signature, whose captions are written to be fairly independent of the rest of the book.

The quote resumes.

The uncertainty as to the position of the tangent at a point on the contour is by no means the same as the uncertainty observed on a map of Brittany.

Although it would differ according to the map’s scale, a tangent can always be found, for a map is a conventional diagram. On the contrary, an essential characteristic of our flake and of the coast is that we suspect, without seeing them clearly, that any scale involves details that absolutely prohibit the fixing of a tangent.

We are still in the realm of experimental reality when we observe under the microscope the Brownian motion agitating a small parti- cle suspended in a fluid [this Essay’s Plate 13].

The direction of the straight line joining the positions occupied at two instants very close in time is found to vary absolutely irregularly as the time between the two instants is decreased. An unprejudiced observer would therefore conclude that he is dealing with a function without derivative, instead of a curve to which a tangent could be drawn.

Closer observation of any object generally leads to the discovery of a highly irregular structure.

We often can with advantage approximate its properties by continuous func- tions. Although wood may be indefinitely porous, it is useful to speak of a beam that has been sawed and planed as having a finite area.

In other words, at certain scales and for certain methods of investigation, many phe- nomena may be represented by regular contin- uous functions, somewhat in the same way that a sheet of tinfoil may be wrapped round a sponge without following accurately the latter’s complicated contour.

If, to go further, we… attribute to matter the infinitely granular structure that is in the spirit of atomic theory, our power to apply to reality the rigorous mathematical concept of continuity will greatly decrease.

Consider, for instance, the way in which we define the density of air at a given point and at a given moment. We picture a sphere of volume v centered at that point and includ- ing the mass m.

The quotient m/v is the mean density within the sphere, and by true density we denote some limiting value of this quotient. This notion, however, implies that at the given moment the mean density is practically constant for spheres below a certain volume.

This mean density may be notably dif- ferent for spheres containing 1,000 cubic meters and 1 cubic centimeter respectively, but it is expected to vary only by 1 in 1,000,000 when comparing 1 cubic centimeter to one-thousandth of a cubic millimeter.

“Suppose the volume becomes continually smaller. Instead of becoming less and less im- portant, these fluctuations come to increase. For scales at which the Brownian motion shows great activity, fluctuations may attain 1 part in 1,000, and they become of the order of 1 part in 5 when the radius of the hypo- thetical spherule becomes of the order of a hundredth of a micron.

“One step further and our spherule be- comes of the order of a molecule radius. In a gas, it will generally lie in intermolecular space, where its mean density will henceforth vanish. At our point the true density will also vanish. But about once in a thousand times that point will lie within a molecule, and the mean density will be a thousand times higher than the value we usually take to be the true density of the gas.

“Let our spherule grow steadily smaller. Soon, except under exceptional circumstances, it will become empty and remain so hence- forth owing to the emptiness of intra-atomic space; the true density vanishes almost every- where, except at an infinite number of isolat- ed points, where it reaches an infinite value.

“Analogous considerations are applicable to properties such as velocity, pressure, or