[5] In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory.

[10] He first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations, doing so in 1905.

[12][13] Poincaré also laid the seeds of the discovery of radioactivity through his interest and study of X-rays, which influenced physicist Henri Becquerel, who then discovered the phenomena.

There he studied mathematics as a student of Charles Hermite, continuing to excel and publishing his first paper (Démonstration nouvelle des propriétés de l'indicatrice d'une surface) in 1874.

In 1887, he won Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies.

Among the specific topics he contributed to are the following: The problem of finding the general solution to the motion of more than two orbiting bodies in the Solar System had eluded mathematicians since Newton's time.

Indeed, in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem.

One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."

[34] Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "luminiferous aether"), could be synchronised.

[40] In 1881 Poincaré described hyperbolic geometry in terms of the hyperboloid model, formulating transformations leaving invariant the Lorentz interval

[43] It was shown that the Poincaré sphere possesses an underlying Lorentzian symmetry, by which it can be used as a geometrical representation of Lorentz transformations and velocity additions.

While studying the conflict between the action/reaction principle and Lorentz ether theory, he tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included.

However, Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid.

If the energy during its propagation remained always attached to some material substratum, this matter would carry the light along with it and Fizeau has shown, at least for the air, that there is nothing of the kind.

Finally in 1908[54] he revisits the problem and ends with abandoning the principle of reaction altogether in favor of supporting a solution based in the inertia of aether itself.

Poincaré proved a formula relating the number of edges, vertices and faces of n-dimensional polyhedron (the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.

In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on).

In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies.

He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works.

Poincaré was interested in the way his mind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris.

His method of thinking is well summarised as: Habitué à négliger les détails et à ne regarder que les cimes, il passait de l'une à l'autre avec une promptitude surprenante et les faits qu'il découvrait se groupant d'eux-mêmes autour de leur centre étaient instantanément et automatiquement classés dans sa mémoire (accustomed to neglecting details and to looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he discovered, clustering around their center, were instantly and automatically pigeonholed in his memory).Poincaré is credited with laying the foundations of special relativity,[10][9] with some arguing that he should be credited with its creation.

[79] He is said to have "dominated the mathematics and the theoretical physics of his time", and that "he was without a doubt the most admired mathematician while he was alive, and he remains today one of the world's most emblematic scientific figures.

"[80] Poincaré is regarded as a "universal specialist", as he refined celestial mechanics, he progressed nearly all parts of mathematics of his time, including creating new subjects, is a father of special relativity, participated in all the great debates of his time in physics, was a major actor in the great epistemological debates of his day in relation to philosophy of science, and Poincaré was the one who investigated the 1879 Magny shaft firedamp explosion as an engineer.

[80] Due to the breadth of his research, Poincaré was the only member to be elected to every section of the French Academy of Sciences of the time, those being geometry, mechanics, physics, astronomy and navigation.

[84] Bell noted that if Poincaré had been as strong in practical science as he was in theoretical, he might have "made a fourth with the incomparable three, Archimedes, Newton, and Gauss.

"[85] Bell further noted his powerful memory, one that was even superior to Leonhard Euler's, stating that:[85] His principal diversion was reading, where his unusual talents first showed up.

In temporal memory - the ability to recall with uncanny precision a sequence of events long passed — he was also unusually strong.Bell notes the terrible eyesight of Poincaré, he almost completely remembered formulas and theorems by ear, and "unable to see the board distinctly when he became a student of advanced mathematics, he sat back and listened, following and remembering perfectly without taking notes - an easy feat for him, but one incomprehensible to most mathematicians.

[91] Nominators included Nobel laureates Hendrik Lorentz and Pieter Zeeman (both of 1902), Marie Curie (of 1903), Albert Michelson (of 1907), Gabriel Lippmann (of 1908) and Guglielmo Marconi (of 1909).

He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic and not analytic.

In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.