The potted fractal and the history.

A very easy experiment that you may comfortably do in your kitchen. Let some water rest in a pot or other container for a sufficiently long time at a low temperature. I can't say exactly how much time is necessary, but leaving the pond for a whole day in the refrigerator would suffice. I happened to look at the results after casually leaving some water in a pot for a winter's night in the kitchen. The salt dissolved in water is going to aggregate in an irregular structure that floats on the surface, consisting of an incredible sequence of clusters, which resembles the image that I quoted in the title. That structure has the properties of a fractal, and arises from a process of gradual growth due to the fortuitous meeting of the salt particles dissolved in water and with the floating growing object. As new particles are drifting into contact with a particle already established, a chemical bond is formed and the new molecular grain of salt aggregates to the solid and increases it.

This process is described mathematically as the Diffusion Limited Aggregation (DLA in English - further details and animations are available on these pages of the Hong Kong Polytechnic University), which may well represent the physical process of crystallization far from the equilibrium point. Yes because under conditions of chemical and physical equilibrium there are no interactions between particles: all possible binds would be established, and the water molecules in Brownian motion oppose to the salt aggregation process. The process take-off consists of an accidental event: a pair of particles, which met first, begins to set other grains around them. This event accidentality makes it impossible to generate the same geometry if we were to repeat the experiment a second time: whatever the fortuitous circumstance that led to the formation of the first granular aggregate, this one will characterize the subsequent development and the form of the final structure. It is impossible, in retrospect, to understand why the solid has taken a specific form and not another equally plausible, because the structure will be fully determined by its history.

Strange attractions.

The most used way to graphically show the evolution of a dynamic system is to draw its trajectory in the phase space: a space with as many dimensions as state variables (or even degrees of freedom) in the system. Each point of this space represents a very specific state in which the system can come to find itself. The phase space of the pendulum - for instance - consists of the Cartesian coordinates system identified by two variables: the angle from the vertical and the angular speed of the pendulum.
Not all points of the infinite phase space (also known as state space) are possible points for the system: indeed, most of them are impossible points and the system will never reach them. Mathematically this means that those points (combination of status variables) are not a solution of the differential equations governing the dynamics.
If we consider all possible points of the system in a chronological order, we will find that it draws a trajectory line in the phase space, which looks like the flight of a fly. The study of these trajectories tells much about systems.
Generally speaking the development of a trajectory depends on the initial state and thus by the initial conditions.
Left free to evolve a system displays normally a short-term dynamic, temporary, known as transient stage, and a long-term dynamic conditions. The latter can be a null state (think of a spring that is discharged, or the swings of a pendulum at the end of its oscillations, when it doesn't receive additional forces) or a steady state. The current in a transformer is periodic indefinitely, for example.
The state we are interested to is the chaotic one. In this case, different initial conditions cause different trajectories, which never overlap nor cross both a given point. The chaotic trajectories are an infinite development of a curve in the phase space that do not ever repeat twice the same even for a tiny tract.
Should we draw it, we would see that this curve never intersects itself. If you think about it, this fact is closely related to the impredicibility of chaos: should an overlap ever occur periodicity would settle up, because from that moment on the curve will repeat itself, giving rise to a periodic state that allows easy predictions of the system dynamics.
The bizarre geometric figure drawn by this trajectory is called strange attractor: the name betrays the surprise of those who saw it for the very first time and did not know how else that rich dynamic could be called. The set of all states that give raise to trajectories evolving with this curious geometry is called the attractor's basin of attraction.

A chaotic system can produce more than one strange attractor, with basins of attraction hardly distinguishable from each other or even overlapped - such as riddled basins, which are fractal basins. In these cases, a dynamic system might jump from one to another attractor due to external causes and perturbations. My words fail to depict the complexity that can surface from a similar case.

But the question is not purely theoretical, rather it is very real and closely concerns the world we live in. An example might be the climate, which so often deserves the honor of the first columns in these years.

The geometry of chaos and the dynamics of fractals.

An enigmatic title, a phrase that represents a sort of two-faced palindrome, conceptually not literally, because it wants to highlight the duality between the fractals and the deterministic chaos. Once again I quote from a famous mathematician:

Following the mathematics practice and representing the dynamics in geometrical terms, one can well focus the disconcerting dual nature of chaos.
Associated with each dynamic system, there is its own geometric space, the phase space [...]. In chaotic systems the lines of flow (or state trajectories), head towards complex forms such attractors. [...] The geometry of these attractors combines the concept of chaos and the concept of fractal.

Ian Stewart - What shape is a snow flake? (translated from Italian).

The mathematics of chaos and fractal geometry are fundamental discoveries of the twentieth century, which allowed the review of basic concepts of many branches of science - not to say all the knowledge - much to be presented today as the ignition spark of a revolution epistemological still underway, whose significance is still difficult to understand.
Concepts such as chaos and fractal, abstruse to the most today, should become a basis for anyone who wants a key to our fascinating universe.

Forms and cognition: the "laws of nature."

For those who felt a certain embarrassment when hear that man has discovered the "laws of nature" and that they are nothing more than mathematical formulas, whose harsh aridity often appears intuitively inadequate to play all the symphonies of the universe. The human mind goes tirelessly searching for forms. To survive in a hostile world, we have developed a sensitivity to configurations, which we use to predict what will happen.
Sometimes [...] configurations actually exist in reality and reveal important truth about the universe. We call them laws of nature. And this is what science does, bringing to light the secret configurations that make the universe. For human beings using mathematics is the most effective way to think about configurations. We therefore believe that the laws of nature are mathematical laws.

Ian Stewart - What shape is a snowflake? (translated from Italian edition).