Does reality really look like our visual impression? A little maths/physics to bend your mind

Clouds are not spheres, mountains are not cones and lightening does not travel in a straight line. The complexity of nature's shapes differs from the shapes of ordinary geometry. To describe the shapes that populate the real world, famed mathematician Bonoit Mandlebrot conceived and developed a new geometry - the geometry of fractal shapes, which has been described as the attempt to understand quantitatively the notion of roughness.

Below is a little extract from Mandlebrot's "The Fractal Geometry of Nature".

Although closer observation of an object generally leads to the discovery of of a highly irregular structure, we often can approximate its properties by continuous functions. Although wood may be infinitely 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 phenomena may be represented by regular continuous 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.

However, if we were to go further and 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 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 including 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 different for spheres containing 1,000 cubic meters and 1 cubic centimeter respectively, but but it is expected to vary only by 1 in 1,000,000 when comparing 1 cubic centimeter to one-thousandth of a cubic centimeter.

Now suppose the volume of our sphere becomes continually smaller. Instead of becoming less and less important these fluctuations in fact begin to increase. For scales at which 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 hypothetical spherule becomes of the order of a hundredth of a micron.

One step further and our spherule becomes 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 mean 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 henceforth owing to the emptiness of of intra-atomic space; the true density vanishes almost every where, except at an infinite number of isolated points, where it reaches an infinite value.