J. L. Aragón

Size-dependent symmetry breaking in models for morphogenesis

A general property of dynamical systems is the appearance of spatial and temporal patterns due to a change of stability of a homogeneous steady state. Such spontaneous symmetry breaking is observed very frequently in all kinds of real systems, including the development of shape in living organisms. Many nonlinear dynamical systems present a wide variety of patterns with different shapes and symmetries. This fact restricts the applicability of these models to morphogenesis, since one often finds a surprisingly small variation in the shapes of living organisms. For instance, all individuals in the Phylum Echinodermata share a persistent radial fivefold symmetry. In this paper, we investigate in detail the symmetry-breaking properties of a Turing reaction–diffusion system confined in a small disk in two dimensions. It is shown that the symmetry of the resulting pattern depends only on the size of the disk, regardless of the boundary conditions and of the differences in the parameters that differentiate the interior of the domain from the outer space. This study suggests that additional regulatory mechanisms to control the size of the system are of crucial importance in morphogenesis.

Turbulent Luminance in Impassioned van Gogh Paintings

We show that the patterns of luminance in some impassioned van Gogh paintings display the mathematical structure of fluid turbulence. Specifically, we show that the probability distribution function (PDF) of luminance fluctuations of points (pixels) separated by a distance R compares notably well with the PDF of the velocity differences in a turbulent flow, as predicted by the statistical theory of A.N. Kolmogorov. We observe that turbulent paintings of van Gogh belong to his last period, during which episodes of prolonged psychotic agitation of this artist were frequent. Our approach suggests new tools that open the possibility of quantitative objective research for art representation.

Analytic expressions for the vertex coordinates of quasiperiodic lattices

Abstract Using the generalized dual method, closed analytical expressions for the coordinates of quasiperiodic lattices, derived from periodic or quasiperiodic grids, are given. The obtained formulae constitute a useful and practical tool to generate and perform calculations in quasi periodic structures.

The decomposition of an orthogonal transformation as a product of reflections

In this work, an algorithm to decompose a given orthogonal transformation as a product of reflections through hyperplanes is presented. This in fact constitutes a constructive proof of a Cartan theorem, valid over any field K=Q, R or C. Clifford algebras are used to explicitly calculate the reflections that decompose a given orthogonal transformation. Our algorithm may have application in fields such as computer graphics or crystallography, and can also play an important role in orthogonal elimination and the solution of equations systems. An explicit example is provided and we apply our results to the crystallographic problem of coincidence lattices.

Coincidence lattices in the hyperbolic plane

The problem of coincidences of lattices in the space R(p,q), with p + q = 2, is analyzed using Clifford algebra. We show that, as in R(n), any coincidence isometry can be decomposed as a product of at most two reflections by vectors of the lattice. Bases and coincidence indices are constructed explicitly for several interesting lattices. Our procedure is metric-independent and, in particular, the hyperbolic plane is obtained when p = q = 1. Additionally, we provide a proof of the Cartan-Dieudonné theorem for R(p,q), with p + q = 2, that includes an algorithm to decompose an orthogonal transformation into a product of reflections.

Quasicrystalline and rational approximant wave patterns in hydrodynamic and quantum nested wells

The eigenfunctions of nested wells with an incommensurate boundary geometry, in both the hydrodynamic shallow water regime and quantum cases, are systematically and exhaustively studied in this Letter. The boundary arrangement of the nested wells consists of polygonal ones, square or hexagonal, with a concentric immersed, similar but rotated, well or plateau. A rich taxonomy of wave patterns, such as quasicrystalline states, their crystalline rational approximants, and some other exotic but well known tilings, is found in these mimicked experiments. To the best of our knowledge, these hydrodynamic rational approximants are presented here for the first time in a hydrodynamic-quantum framework. The corresponding statistical nature of the energy level spacing distribution reflects this taxonomy by changing the spectral types.

Eutactic star closest to a given star

A eutactic star is a set of M vectors in Rn (M>n) that are projections of M orthogonal vectors in RM. Eutactic stars have remarkable properties that have been exploited in several fields such as crystallography, graph theory, wavelets, and quantum measurement theory. In this work we show that given an arbitrary star of vectors, there exists a closest eutactic star in the Frobenius norm. An algorithm for calculating this star is presented. Additionally, the distance between both stars provides a new measure of eutacticity.

Self-reciprocal redundant vector stars

Abstract In this work it is shown that a star of M vectors in Rn (M > n) is self-reciprocal, in the sense that it coincides with its Mackay generalized reciprocal star, if and only if it forms an eutactic star. We also show that eutactic stars behave in ways that resemble the behavior of orthonormal sets. A characterization of eutactic stars based on the Moore-Penrose pseudoinverse is presented.

Polyhedral truncations as eutactic transformations.

An eutactic star is a set of N vectors in Rn (N > n) that are projections of N orthogonal vectors in RN. First introduced in the context of regular polytopes, eutactic stars are particularly useful in the field of quasicrystals where a method to generate quasiperiodic tilings is by projecting higher-dimensional lattices. Here are defined the concepts of eutactic transformations (as mappings that preserve eutacticity) and of vector radiations (vectors that stem from the vectors of an eutactic star), which are used to describe and parameterize polyhedral truncations. The polyhedral truncations preserve eutacticity, a result of relevance to the faceting and habit-forming characteristics of quasicrystals.

A measure of regularity for polygonal mosaics in biological systems

BackgroundThe quantification of the spatial order of biological patterns or mosaics provides useful information as many properties are determined by the spatial distribution of their constituent elements. These are usually characterised by methods based on nearest neighbours distances, by the number of sides of cells, or by angles defined by the adjacent cells.MethodsA measure of regularity in polygonal mosaics of different kinds in biological systems is proposed. It is based on the condition of eutacticity, expressed in terms of eutactic stars, which is closely related to regularity of polytopes. Thus it constitutes a natural measure of regularity. The proposed measure is tested with numerical and real data. Numerically is tested with a hexagonal lattice that is distorted progressively and with a non-periodic regular tiling. With real data, the distribution of oak trees in forests from three locations in the State of Querétaro, Mexico, and the spiral pattern of florets in a flowering plant are characterised.ResultsThe proposed measure performs well and as expected while tested with a numerical experiment, as well as when applied to a known non-periodic tiling of the plane. Concerning real data, the measure is sensitive to the degree of perturbation observed in the distribution of oak trees and detects high regularity in a phyllotactic pattern studied.ConclusionsThe measure here proposed has a clear geometrical meaning, establishing what regularity means, and constitute an advantageous general purposes alternative to analyse spatial distributions, capable to indicate the degree of regularity of a mosaic or an array of points.