Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1408.2448

Dynamical Characterization of Fractal Objects: Determination of the Fine Fractal Topology Using the Energy Cover

Luiz Bevilacqua·Marcelo Miranda Barros·Gil Márcio A. Silva
View original source ↗Read source document ↗Export metadata ↓

Abstract

The foundation of the theory presented here has already been proved to be effective for the case of curves belonging to the Koch family. The present paper extends the investigation to more complex curves, namely randomly generated curves and the Weierstrass-Mandelbrot curve. The analysis is focused on numerical experiments. The results obtained with the numerical analysis allow advancing some interesting proposition concerning the fine fractal structure of plane curves. The method uses the dynamical response of appropriate harmonic oscillators built up according to the geometry of the fractal set. For the plane motion three fundamental periods are determined. Each period plotted against the characteristic length of the corresponding curve in a log-log scale estimates the fractal dimension. It is shown that each degree of freedom corresponds to a distinct energy covering associate to a certain topological property. Using this technique it is possible to distinguish the fine fractal structure of plane curves. The dynamic technique is proved to work also for the identification problem, that is, to determine the fractal characteristics embedded in some sample of a given fractal set. It is proposed the classification of fractal structures into at least two categories, perfectly fractal and partially fractal. The affine similarity can be precisely detected using this method.

Explore related subjects

math.MG

Keep this discovery

BibTeXRIS

Luiz Bevilacqua, Marcelo Miranda Barros, Gil Márcio A. Silva. 2014-08-05. Dynamical Characterization of Fractal Objects: Determination of the Fine Fractal Topology Using the Energy Cover. https://arxiv.org/abs/1408.2448

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles

Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings $\mathscr{T}_h$ and $\mathscr{T}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\textit{ASPmr}\{\text{A-tile}, \text{B-tile}\}$.

math.MG↗
arXiv2026

The mean distance to a simple closed curve on the sphere

Kimberling's Problem 10 asks for a simple closed curve of prescribed length $L$ (in particular, $L=4\pi$) on the unit sphere minimizing the mean geodesic distance $\mathcal{J}$ from a point of the sphere to the curve. For a positive integer $n$, put $\vartheta_{n}=\pi/(2n)$ and $L_{n}=2\pi/\sin\vartheta_{n}$. We show that the minimum of $\mathcal{J}$ over rectifiable simple closed curves of length at most $L_{n}$ equals $\vartheta_{n}-\tan(\vartheta_{n}/2)$, that it is attained only by curves of length exactly $L_{n}$, and that the sphere-filling ropes $\beta^{n,k}$ of Gerlach and von der Mosel attain it. Kimberling's case is $n=3$: at $L=4\pi$ the minimum is $\pi/6+\sqrt{3}-2=0.255649\ldots$, attained by an explicit six-arc curve and by its mirror image. For $L\le2\pi$ we determine $J(L)$, the infimum of $\mathcal{J}$ over curves of length $L$, exactly: it equals $\pi/2-L/(2\pi)$, attained precisely by the circles of length $L$. At the lengths $L_{n}$ we do not classify all minimizers, but show that every one of them bisects the sphere into two disks of area $2\pi$ and inradius $\vartheta_{n}$ whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for $n=1$, and the $\beta^{n,k}$ are, up to congruence, the only ones of thickness at least $\sin\vartheta_{n}$. For arbitrary $L$ the function $J$ is nonincreasing, and together with the above this brackets it between two explicit values.

math.MG↗
arXiv2026

The topology of Gromov--Hausdorff space

We prove that the Gromov--Hausdorff space is homeomorphic to the Hilbert space. This paper is divided into four parts. In Part I, we construct an assignment of a full-support probability measure to every nonempty compact metric space that respects isometries and is continuous for simultaneous Hausdorff convergence of the spaces and weak convergence of the measures. In Part II, we use these measures to construct finite-dimensional local models whose induced pseudometrics approximate the original distances uniformly and whose norms and point maps vary continuously up to orthogonal changes of coordinates. In Part III, we use the local models to prove that the Gromov--Hausdorff space is an absolute retract for all metrizable spaces. In Part IV, we establish a discrete approximation property and conclude that the space of isometry classes of nonempty compact metric spaces is homeomorphic to the real separable infinite-dimensional Hilbert space.

math.MG↗