Iván Sánchez

Fractals in dentistry.

OBJECTIVES To systematically review applications of fractal geometry in different aspects of dental practice. In this review, we present a short introduction to fractals and specifically address the following topics: treatment and healing monitoring, dental materials, dental tissue, caries, osteoporosis, periodontitis, cancer, Sjögrens syndrome, diagnosis of several other conditions and a discussion on the reliability of FD determinations from dental radiographs. SOURCES Google Scholar, Ovid MEDLINE, ScienceDirect, etc. (up to August 2010). STUDY SELECTION The review considered original studies, reviews and conference proceedings, published in English or Spanish. Abstracts and posters were not taken into account. CONCLUSIONS Fractal geometry has found plenty of applications in several branches of dental practice. It provides a way to quantify the complexity of structures. Whereas one desires to study a radiograph, an histological section or the signal from a transducer, there are several methods available to determine the degree of complexity using fractal analysis. Several pathological conditions can alter the complexity of anatomical structures, and this change can be detectable with the help of fractal parameters. Although during the last two decades there have been plenty of works on the field, reported cases having enough reproducibility, with different groups showing similar results are not very common. Further replications are needed before we can establish statistically significant correlations amongst fractal parameters and pathological conditions.

Towards a relevant set of state variables to describe static granular packings

We analyze, experimentally and numerically, the steady states, obtained by tapping, of a two-dimensional granular layer. Contrary to the usual assumption, we show that the reversible (steady state branch) of the density-acceleration curve is nonmonotonous. Accordingly, steady states with the same mean volume can be reached by tapping the system with very different intensities. Simulations of dissipative frictional disks show that equal volume steady states have different values of the force moment tensor. Additionally, we find that steady states of equal stress can be obtained by changing the duration of the taps; however, these states present distinct mean volumes. These results confirm previous speculations that the volume and the force moment tensor are both needed to describe univocally equilibrium states in static granular assemblies.

Contact angle entropy and macroscopic friction in noncohesive two-dimensional granular packings

We study the relationship between the granular contact angle distribution and local particle friction on the macroscopic friction and bulk modulus in noncohesive disk packings. Molecular dynamics in two dimensions are used to simulate uniaxial loading-unloading cycles imposed on the granular packings. While macroscopic Mohr friction depends on the granular pack geometric details, it reaches a stationary limit after a finite number of loading-unloading cycles that render well-defined values for bulk modulus, grain coordination, porosity, and friction. For random packings and for all polydispersities analyzed, we found that as interparticle friction increases, the bulk modulus for the limit cycle decreases linearly, while the mean coordination number is reduced and the porosity increased, also as approximately linear functions. On the other hand, the macroscopic Mohr friction increases in a monotonous trend with interparticle friction. The latter result is compared to a theoretical model that assumes the existence of sliding planes corresponding to definite Mohr-friction values. The simulation results for macroscopic friction are well described by the theoretical model that incorporates the local neighbor angle distribution that can be quantified through the contact angle entropy. As local friction is increased, the limit entropy of the neighbor angle distribution is reduced, thus introducing the geometric component to granular friction. Surprisingly, once the limit cycle is reached, the Mohr friction seems to be insensitive to polydispersity as has been recently reported.

Characterization of invariant patterns in a slowly rotated granular tumbler

We report experimental results of the pattern developed by a mixture of two types of grains in a triangular rotating tumbler operating in the avalanche regime. At the centroid of the triangular tumbler an invariant zone appears where the grains do not move relative to the tumbler. We characterize this invariant zone by its normalized area, Ai, and its circularity index as a function of the normalized filling area A. We find a critical filling area so that only for A>Ac invariant zones are obtained. These zones scale as Ai∼(A−Ac)2 near Ac. We have obtained a maximum in the circularity index for A≈0.8, for which the shape of the invariant zone is closer to a circular one. The experimental results are reproduced by a simple model which, based on the surface position, accounts for all the possible straight lines within the triangle that satisfy the condition of constant A. We have obtained an analytic expression for the contour of the invariant zone. Experimentally, we obtained a displacement in Ac that we explain in terms of a finite width of the avalanche region. This displacement is needed only to correct the size of the invariant zone, not its shape.

Faraday Wave Patterns on a Triangular Cell Network

We present experimental observations of the Faraday instability when an air/water interface is split over a network of small triangular cells for exciting frequencies in the range \(10\le f\le 30\) Hz. Just above the threshold for instability, waves appear on the water surfaces within all individual cells. After a transient state, adjacent cells progressively synchronize and self-organize to produce a regular pattern covering the whole grid. Collective cell behaviour is seen to lead to four different patterns depending on the forcing frequency range. Beyond \({\approx }{28}\) Hz, adjacent cells no longer interact as the vibration wavelength becomes smaller than half the altitudes of the triangular cells and so the waves remain constrained within individual cells in the form of localized harmonic oscillons.