Symmetropy of Fault Patterns: Quantitative Measurement of Anisotropy and Entropic Heterogeneity

Nanjo, Kazuyoshi; Nagahama, Hiroyuki; Yodogawa, Eiji

doi:10.1007/s11004-005-1559-z

Cited by 6 publications

(7 citation statements)

References 20 publications

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“…We regard the AE events as circles with finite diameter. Following previous studies (e.g., Nanjo et al, 2000;2005;2006), the spatial patterns of AE were covered with 2 5 ×2 6 cells. If we find a part of or whole of one or more of circles in a cell of (i, j), then x ij = 1 , otherwise x ij = 0.…”

Section: Methods I: Fractal Analysis and Walsh Analysismentioning

confidence: 99%

“…Nanjo et al, (2000) have examined the fracturing process based on entropy-like measure of symmetry, i.e., symmetropy (Yodogawa, 1982). This symmetropy approach is a useful tool for the quantification of anisotropy, asymmetry and entropic heterogeneity of fracturing patterns (Nanjo et. al., 2005;2006).…”

Section: Introductionmentioning

confidence: 99%

“…Such a critical comparison of a large scale (a whole) versus a small scale (parts) will better enable us to find out an essential feature of a geometrical pattern. In the previous study (e.g., Nanjo et al, 2000;2005;2006), however, only a whole symmetropy of fracturing patterns has been considered. That is, the relationship between a whole symmetropy and a partial symmetropy (PS) has not been discussed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

Yamasaki

Nanjo

2009

Physics of the Earth and Planetary Interiors

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We have shown that the symmetry of fracturing process in a macro scale can be quantified by using the concept of symmetropy (an entropylike measure of symmetry). Here we extend this approach to examining the symmetries in a range from small (partial) scales to larger (whole) scales: This approach is called partial symmetropy (PS). To check its applicability, we consider one illustrative example, the temporal change of the spatial patterns of acoustic-emission events in a well-documented rock fracture experiment. Our results are summarized as follows: (i) The PS enables us to distinguish the nucleation phases from the other phases such as the pre-nucleation and the propagation phases; (ii) The variation of the PS shows that the fracturing process is a associated with a type of phase transitions from the subcritical state to the critical state; (iii) The scale dependence of the PS reveals the presence of the sandwich structure that consists of order and non-order in the evolution of the fracturing pattern; (iv) Within the framework of nonextensive Tsallis entropy we develop the PS concept, and show that the degree of nonextensivity on a large scale increases immediately after the nucleation. The results shown

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Section: Methods I: Fractal Analysis and Walsh Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

Yamasaki

Nanjo

2009

Physics of the Earth and Planetary Interiors

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“…In physical process, the Curie symmetry principle has been known as an aspect of the causality relationship between the symmetry of the cause and that of the resultant effect, and allows us to predict possible properties and to forbid impossible ones (e.g., Jaeger, 1920). The Curie symmetry principle is expressed in terms of the symmetropy as follows (Nanjo et al, 2005): symmetropy evaluating the cause S cause is equal to or smaller than symmetropy evaluating the resultant effect S effect :…”

Section: Discussionmentioning

confidence: 99%

Symmetry and entropy of biological patterns: Discrete Walsh functions for 2D image analysis

2011

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To quantify symmetry and entropy inherent in the discrete patterns such as spatial self-organization in cell sorting and mussel bed ecosystems, we introduce the discrete Walsh analysis. This analysis enables us to estimate the degree of the complicated symmetry, and to extract the symmetry from the pattern that seems to be asymmetric. The results obtained in this paper are summarized as follows. (I) The geometrical patterns of the cell sorting become systematic with the predominance of the particular symmetry. This implies that not only the entropy but also the particular symmetry can decrease in the biological process. (II) The magnitude of the symmetry is related to the absolute value of the adhesion, and the type of the symmetry is related to the sign of the adhesion. That is, centro-symmetry dominates in the cell sorting pattern caused by large negative adhesion, and double symmetry dominates in the pattern caused by large positive adhesion. (III) Spatial self-organization in mussel bed is accompanied by the decreasing of the centro-symmetry. This implies that the positive "adhesion" between mussel individuals increases with time. (IV) In the biological process, the Curie symmetry breaking occurs at intervals.

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“…This paper concentrates on the geometric symmetry in the discrete pattern caused by the CA's rule, and suggests a holistic approach to the four classes of the CA from the viewpoint of symmetropy (an object related to symmetry and entropy). For this, we use the discrete Walsh analysis (Figure 1(a); see also p. 573 in [1]), which has been used to express two-dimensional discrete patterns as the superposition of the four types of basic symmetry [9,10] (Figure 1(b); the details are given in Section 3). The Walsh approach is useful for analyzing the phase transition of discrete patterns, such as microfracturing and cell-cell adhesion [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Symmetry and Entropy of One-Dimensional Legal Cellular Automata

Yamasaki¹,

Nanjo²,

Chiba³

2011

ComplexSystems

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The one-dimensional legal cellular automata (CAs) used in Wolfram's original classification from a viewpoint of symmetropy (an object related to symmetry and entropy) are quantified. For this quantification, the discrete Walsh analysis that expresses the two-dimensional discrete pattern in terms of the four types of symmetry is used. The following was found. (1) The relationship between symmetry and entropy of the CA patterns corresponds to the three qualitative classes of CAs: II, III, and IV. (2) The change in symmetropy shows that class IV (complex) exists between class II (periodic) and class III (chaotic). As an application of these findings, the scale dependence of the symmetropy of the CA patterns is considered, and it is shown that class IV is useful for drawing complicated patterns when the system must keep the number of cells low.

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Symmetropy of Fault Patterns: Quantitative Measurement of Anisotropy and Entropic Heterogeneity

Cited by 6 publications

References 20 publications

A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

Symmetry and entropy of biological patterns: Discrete Walsh functions for 2D image analysis

Symmetry and Entropy of One-Dimensional Legal Cellular Automata

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