skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Order-fractal transitions in abstract paintings

Abstract

In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-order transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks. -- Highlights: •We determined the degree of order in Jackson Pollock paintings using the Hausdorff–Besicovitch dimension. •We detected a fractal-order transition frommore » Pollock’s paintings between 1947 and 1951. •We suggest that Jackson Pollock could have painted Teri’s Find.« less

Authors:
 [1];  [2];  [3]
  1. Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970, Porto Alegre, RS (Brazil)
  2. Department of Applied Physics, CINVESTAV-IPN, Carr. Antigua a Progreso km.6, Cordemex, C.P.97310, Mérida, Yucatán (Mexico)
  3. Department of Informatics, Universidad Politécnica de Puebla, 72640 (Mexico)
Publication Date:
OSTI Identifier:
22560338
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 371; Journal Issue: Complete; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; CULTURAL OBJECTS; DATA ANALYSIS; DIMENSIONS; FRACTALS; HISTORICAL ASPECTS; METRICS; ORDER-DISORDER TRANSFORMATIONS

Citation Formats

Calleja, E.M. de la, E-mail: elsama79@gmail.com, Cervantes, F., and Calleja, J. de la. Order-fractal transitions in abstract paintings. United States: N. p., 2016. Web. doi:10.1016/J.AOP.2016.04.007.
Calleja, E.M. de la, E-mail: elsama79@gmail.com, Cervantes, F., & Calleja, J. de la. Order-fractal transitions in abstract paintings. United States. doi:10.1016/J.AOP.2016.04.007.
Calleja, E.M. de la, E-mail: elsama79@gmail.com, Cervantes, F., and Calleja, J. de la. 2016. "Order-fractal transitions in abstract paintings". United States. doi:10.1016/J.AOP.2016.04.007.
@article{osti_22560338,
title = {Order-fractal transitions in abstract paintings},
author = {Calleja, E.M. de la, E-mail: elsama79@gmail.com and Cervantes, F. and Calleja, J. de la},
abstractNote = {In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-order transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks. -- Highlights: •We determined the degree of order in Jackson Pollock paintings using the Hausdorff–Besicovitch dimension. •We detected a fractal-order transition from Pollock’s paintings between 1947 and 1951. •We suggest that Jackson Pollock could have painted Teri’s Find.},
doi = {10.1016/J.AOP.2016.04.007},
journal = {Annals of Physics},
number = Complete,
volume = 371,
place = {United States},
year = 2016,
month = 8
}
  • A fractal lattice F is defined here to comprise all points of the form a + ma' + m/sup 2/a'' + ... + m/sup q/a/sup (q)/, where q is a nonnegative integer and a,a',...,a/sup (q)/ epsilon A, where A is a finite set of points in some Euclidean space. Provided m is not too small (in particular, m must be at least 2), the dimension of F is shown to be D = log n/log m, where n is the number of points in A. It is shown further that an Ising model on F, with a ferromagnetic pair interactionmore » r/sup -..cap alpha../ between spins separated by a distance r, has a phase transition if D < ..cap alpha.. < 2D. On the other hand, for ..cap alpha.. > 2D, provided a certain condition which rules out periodic lattices is satisfied, there can be no finite-temperature transition leading to spontaneous magnetization.« less
  • It is the purpose of this paper to point out that the creation of fractal basin boundaries is a characteristic feature accompanying the intermittency transition to chaos. (Here ''intermittency'' transition is used in the sense of Pomeau and Manneville (Commun. Math. Phys. 74, 189 (1980)); viz., a chaotic attractor is created as a periodic orbit becomes unstable.) In particular, we are here concerned with type-I and type-III intermittencies. We examine the scaling of the dimension of basin boundaries near these intermittency transitions. We find, from numerical experiments, that near the transition the dimension scales with a system parameter /ital p/more » according to the power law /ital d//congruent//ital d//sub 0//minus/k/vert bar/p/minus/p/sub I//vert bar//sup /beta// with /beta/=1/2, where /ital d//sub 0/ is the dimension at the intermittency transition parameter value /ital p/=/ital p//sub I/ and /ital k/is a scaling constant. Furthermore, for type-I intermittency/ital d//sub 0//lt/D, while for type-III intermittency /ital d//sub 0/=D,where /ital D/ is the dimension of the space. Heuristic analytic argumentssupporting the above are presented.« less
  • A hierarchy of fractal geometrical exponents D(l), based upon l-rank orientational fluctuations, is proposed; D(0) = D is the usual fractal dimension. The first three D(l) are calculated via computer simulation for a growth model with a tunable fractal dimension for several values in the range 3>D>1, and for bond percolation. The new exponents are used to discuss fractal structure. The second-order light-scattering intensity is evaluated for the growing fractal clusters, and is shown to be sensitive to the higher order D(l).
  • The atomic structure of metallic glasses has been a long-standing scientific mystery. Unlike crystalline metals, where long-range ordering is established by periodic stacking of fundamental building blocks known as unit cells, a metallic glass has no long-range translational order, although some degrees of short- and medium-range order do exist.1,2,3 Previous studies1,2,3,4 have identified solute-centered clusters, characterized by short-range order (SRO) in favor of unlike bonds, as the fundamental building blocks of metallic glasses. However, how these building blocks are connected or packed to form the medium range order (MRO) remains an open question.1,2,3 Here, based on neutron and x-ray diffractionmore » experiments, we propose a new packing scheme - the self-similar packing of atomic clusters. We show that MRO has the characteristics of a fractal network with a dimension of 2.38, and is described by a power-law correlation function over the medium-range length scale. Our finding provides a new prospective of order in disordered materials and has broad implications for understanding the structure-property relationship in metallic glasses, particularly those involving change in length scales due to phase transformation and mechanical deformation.« less