General fractal topography: an open mathematical framework to characterize and model mono-scale-invariances

  • Yi JinEmail author
  • Xianhe Liu
  • Huibo Song
  • Junling Zheng
  • Jienan Pan
Original Paper


In this work, we reported there are two kinds of independent complexities in mono-scale-invariance, namely to be behavioral complexity determined by fractal behavior and original one wrapped in scaling object. Quantitative characterization of the complexities is fundamentally important for mechanism exploration and essential understanding of nonlinear systems. Recently, a new concept of fractal topography (\(\mathrm {\Omega }\)) was emerged to define the fractal behavior in self-similarities by scaling lacunarity (P) and scaling coverage (F) as \(\mathrm {\Omega }(P, F)\). It is, however, a “special fractal topography” because fractals are rarely deterministic and always appear stochastic, heterogeneous, and even anisotropic. For that, we reviewed a novel concept of “general fractal topography”, \({\varvec{\Omega }}(\mathrm {{\mathbf {P}}}, \mathrm {{\mathbf {F}}})\). In \({\varvec{\Omega }}\), \(\mathrm {{\mathbf {P}}}\) is generalized to a set accounting for direction-dependent scaling behaviors of the scaling object G, while \(\mathrm {{\mathbf {F}}}\) is extended to consider stochastic and heterogeneous effects. And then, we decomposed G into \(\mathrm {{\mathbf {G}}}\left( G_+, G_-\right) \) to ease type controlling and measurement quantification, with \(G_+\) wrapping the original complexity while \(G_-\) enclosing behavioral complexity. Together with \({\varvec{\Omega }}\) and \(\mathrm {{\mathbf {G}}}\), a mathematical model \(F_\mathrm {3S}\left( {\varvec{\Omega }}, \mathrm {{\mathbf {G}}}\right) \) was then established to unify the definition of deterministic or statistical, self-similar or self-affine, single- or multi-phase properties. For demonstration, algorithms are developed to model natural scale-invariances. Our investigations indicated that the general fractal topography is an open mathematic framework which can admit most complex scaling objects and fractal behaviors.


Scale-invariance Fractal dimension Hurst exponent Porous media Fractal networks Fractal topography Self-sameness self-similarity self-affinity (3S) 

List of symbols


Space fractal dimension


Space fractal dimension in x-direction

\({\mathcal {D}}\)

Property fractal dimension


General Space fractal dimension


Mass fractal dimension


Hurst exponent


General Hurst exponent


Scaling lacunarity


Scaling lacunarity in x-direction

\(\mathrm {{\mathbf {P}}}\)

Scaling lacunarity set


Effective scaling lacunarity, scalar quantity of \(\mathrm {{\mathbf {P}}}\)


Scaling coverage

\(\mathrm {{\mathbf {F}}}\)

Scaling coverage set

\(\langle F\rangle \)

Expectation of scaling coverage


Scaling object


Regime wrapping original complexity in G


Regime enclosing behavioral complexity in G


Characteristic linear size of G

\(\mathrm {{\mathbf {l}}}\)

Vector of l

\(\mathrm {{\mathbf {L}}}\)

Range of \(\mathrm {{\mathbf {l}}}\)


Characteristic linear size of G in x-direction


Effective characteristic size of G

\({\mathcal {P}}\)

A desired property

Greek symbols

\(\mathrm {\Omega }\)

Special fractal topography

\({\varvec{\Omega }}\)

General fractal topography

\(\varphi \)


\(\rho \)

Density of desired property \({\mathcal {P}}\)

\(\epsilon \)

A dimensionless constant



Maximum values


Minimum values


Compliance with ethical standards

Conflict of interest

This manuscript has not been published in part or in entirety and is not under consideration by another journal. All authors have been informed and consent to this submission. We declare no competing interests (financial or non-financial) or other interests that might be perceived to influence the results and/or discussion reported in this paper. In this research, all funding sources were correctly included and there are no animals involved. We have read and understood your journal’s policies, and we believe that neither the manuscript nor the study violates any of these.


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Authors and Affiliations

  1. 1.School of Resources and EnvironmentHenan Polytechnic UniversityJiaozuoChina
  2. 2.Collaborative Innovation Center of Coalbed Methane and Shale Gas for Central Plains Economic RegionJiaozuoChina

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