Chaos theory is a relatively recent and often misunderstood field of study because it is wrongly considered by laymen as tied to random events. Chaotic systems have properties similar in many ways to those of stochastic processes, for example punctual unpredictability. What is interesting is the fact that potentially controllable situations such as deterministic ones are instead difficult to interpret. It is quite natural to consider some of these situations in a text that deals with probabilities. Even though we continue to stress, these are different fields. To make their way through the many possible examples, fairly simple objects have been preferred.
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I say unto you: a man must have chaos yet within him to be able to give birth to a dancing star. ~ Friedrich Nietzsche
One of the fundamental questions that man asks himself concerns the origin of the universe. The answers can be sought in science, religions, myths. The starting point for all these reflections is always chaos. Chaos is presented in Genesis (the first book of the Bible) as a state of imperfection from which, by divine work, the Universe is shaped. For the Greeks it was Chaos who ruled a world and only his descendants gave shape to reality. Man, however, asked himself other profound questions to which he answered by building new myths. As the writer William Somerset Maugham (1944) said: “The faculty for myth is innate in the human race. .... It is the protest of romance against the commonplace of life”.
We open a small parenthesis that lends itself well to compare chaos, randomness, and order. The ancient populations sought answers by observing the movement of the stars and the shapes that could be recognized. In fact, people have always grouped the stars that appear close together and have therefore recognized (or, better, believed they could identify) the constellations: random clusters of stars that could be considered as formations with a structure. Each culture has built its own figures by giving them different names and stories according to its own traditions. We can say that different cultures have created different constellations. It is curious, however, that some constellations are common in civilizations far apart in a spatial and temporal sense. The 88 constellations are divided into three groups: the 12 constellations of the Zodiac, which are located along the ecliptic and are therefore traversed by the Sun in its apparent motion on the celestial vault during the year; 36 constellations listed by Claudius Ptolemy (about 100-175) in his Almagest, now rising to 38; the remaining constellations have been identified since 1600 in the empty spaces. Some symbols of animals such as Taurus, Scorpio, Leo indicated the quadrants of the sky corresponding to the equinoxes and solstices, as they were located in antiquity. With the precession of the equinoxes the function passed to the following constellations, Aries, Libra, Cancer (Figure 1). In many iconographies the signs of the zodiac are associated with the months of the year, often linked to representations of the agricultural work characteristic of each month.
One question arises spontaneously, does the presence of agglomerations justify the “divine” theory of constellations or do the stars have a random distribution?
If we try to “generate random stars” on a plane (Figure 2) we are amazed by the apparent presence of regularity and we can understand how the ancients built a cosmogony. Observing the starry sky the human being often tries to identify rules and patterns and in any randomly generated figure you can try to identify relationships.
In three-dimensional space the stars that form the same constellation can be separated even by enormous distances; vice versa, two or more stars that on the celestial sphere appear perhaps extremely far apart, in three-dimensional space others that belong to the same constellation can be relatively close. The example given here is not only a curiosity, but it allows us to underline the fact that random points seem to have an internal rule and present themselves with quite counterintuitive groupings and empty spaces.