Usual requirements for bulk and fissure energies are considered in obtaining the interdependence among external stress, thickness and area of crack polygons in desiccated films. The average area of crack polygons increases with thickness as a power-law of 4/3. The sequential fragmentation process is characterized by a topological factor related to a scaling finite procedure. Non-sequential overly tensioned (prompt) fragmentation is briefly discussed. Vermeer's painting, Girl with a Pearl Earring, is considered explicitly by using computational image tools and simple experiments and applying the proposed theoretical analysis. In particular, concerning the source of lightened effects on the girl's face, the left/right thickness layer ratio (≈1.34) and the stress ratio (≈1.102) are evaluated. Other master paintings are briefly considered.

This study explores the thermodynamic properties of doped graphene using an adapted electronic spectrum. We employed the one-electron tight-binding model to describe the hexagonal lattice structure. The dispersion relation for graphene is expressed in terms of the hopping energies using a compositional parameter that characterizes the different dopant atoms in the lattice. The focus of the investigation is on the impact of the compositions, specifically the presence of dopant atoms, on the energy spectrum, entropy, temperature, and specific heat of graphene. The numerical and analytical results reveal distinct thermodynamic behaviors influenced by the dopant composition, including topological transitions, inflection points in entropy, and specific heat divergences. In addition, the use of Boltzmann entropy and the revision of Nernst’s theorem for doped graphene are introduced as novel aspects.

In desiccated films, particularly with old paintings, molecular bonds may break to create intricate patterns of macroscopic cracks. The resulting directions of the cracks quantifiably enable an evaluation of the entropy and degree of disorder in the network. Experimental tests on prepared samples and a two-interacting-variables model allow the evolution of entropy to be tracked. Calculations were performed, primarily using data from the painting Girl with a Pearl Earring by Vermeer, revealing that the left side of the girl’s face features a crack structure with higher entropy (or less order) than the right side. Other old paintings were considered. The extrapolation of experiments to these old paintings confirms that saturation still is not reached.