Sadi Carnot stands as a symbol of the “Ingénieur-Savant” who have profoundly enriched the fertile ground of French innovation. The discipline of thermodynamics, which he founded, today underpins information theory, the development of future quantum computers, and artificial intelligence, as well as climate science, pillars upon which rest both our industrial policies and the future of our societies. Sadi Carnot formulated the second law of thermodynamics in his treatise, “Reflections on the Motive Power of Fire “(1824), that initially received little attention, but emerged from obscurity years later thanks to the works of Émile Clapeyron, William Thomson (later Lord Kelvin), and Rudolf Clausius. Sadi Carnot drew upon a remarkably wide range of knowledge, nourished by an eclectic and inquisitive mind. Taken by illness only a few years after publishing his Reflections, he remains forever preserved in the brilliance of his genius. The explanation of thermodynamics through geometric models was initiated by seminal figures such as Gibbs, Reeb, Carathéodory and Souriau. We shall trace the narrative of the geometric models, from Carathéodory (1909) to Souriau (1969), that have sought to re-establish Sadi Carnot’s thermodynamics upon new foundational variational principles. As observed by Vladimir Arnold, “Every mathematician knows it is impossible to understand an elementary course in thermodynamics. The reason is that thermodynamics is based, as Gibbs has explicitly proclaimed, on a rather complicated mathematical theory, on the contact geometry« . A seminal contribution of Carathéodory lies in the introduction of a differential equation that governs the infinitesimal changes in state functions as the system undergoes infinitesimal transitions between states. This equation is of paramount importance as it embodies the first and second laws of thermodynamics in a single, unified mathematical expression. Carathéodory’s axiomatization is intricately linked with the language of differential geometry, later interpretated as contact geometry and has greatly influenced Misha Gromov geometer who introduced the concept of Carnot-Carathéodory spaces. One of the pivotal results of Carathéodory’s axiomatization is his theorem, which asserts that, given certain conditions the second law of thermodynamics, as formulated by Clausius, can be derived directly from the fundamental thermodynamic relation. While Carnot thermodynamics was primarily concerned with systems in equilibrium, Carathéodory’s axiomatization extends far beyond this, offering a more general framework that accommodates systems in non-equilibrium states. Only recently, however, has the Souriau’s Symplectic Foliation Model, introduced within the domain of geometric statistical mechanics, provided a geometric definition of entropy as an invariant Casimir function on symplectic leaves, specifically, the coadjoint orbits of the Lie group acting on the system, where these orbits are interpreted as level sets of entropy. We present a symplectic foliation interpretation of thermodynamics, based on Jean-Marie Souriau’s Lie Groups Thermodynamics. This model offers a Lie algebra cohomological characterisation of entropy, viewed as an invariant Casimir function in the coadjoint representation.