Key points are not available for this paper at this time.
Fractal analysis has become an essential tool in multiple scientific disciplines, including physics, biology, neuroscience, medicine, biomedical engineering, materials science, economics, and environmental sciences, to name a few. The concept was first proposed by Mandelbrot (1983), based on the self-similarity of structures across different scales that are not definable by Euclidean geometry (1)(2)(3).However, despite its widespread application, the precise definition and interpretation of fractalrelated concepts remain inconsistent across disciplines (2,(4)(5)(6)(7). Terms such as "fractal dimension", "self-similarity", "self-affinity", "roughness", "scaling laws", the role of embedding dimension and topological dimension in spatial contexts are frequently applied in contexts where their mathematical rigor is uncertain (8)(9)(10). This has led to a semantic drift in scientific discourse, creating barriers to effective interdisciplinary communication (11,12). Addressing this, a differentiation between fractal analysis and fractal synthesis has been suggested, where fractal analysis focuses on single-dimension metrics to measure the complexity of an image or time series, whereas fractal synthesis combines local and global dimensions, entropy, and spatialtemporal dynamics to establish a more comprehensive understanding of complex systems. To take this further, this manuscript addresses the technical foundations and methodologies required to reliably interpret fractal descriptions by addressing assumptions of stationarity, selection of scale range, and use of surrogate data testing.Before the advent of fractal analysis, Richardson (1961) explored the relationship between scale and length in coastlines, demonstrating that coastline length increases as measurement scale decreases, following a power-law relationship (13). The exponent in this relationship quantifies the rate at which length changes with scale, forming the foundation of fractal dimension analysis (3,14). This early work laid the foundation for modern fractal analysis, which employs techniques such as box-counting, dilation, mass-radius, and the caliper method (5,(15)(16)(17). These methods all serve as analytical tools that estimate object size or mass with the measurement scale and collectively form the basis of fractal analysis. The key exponent in these relationships, termed the fractal dimension (DE), characterizes the complexity of an object's scaling properties. However, there is ongoing debate regarding what constitutes a "dimension" in mathematical and physical sciences. While the Hausdorff dimension is mathematically rigorous, other fractal dimensions, such as Minkowski and Kolmogorov dimensions, are widely used in applied fields despite not meeting strict mathematical definitions (18,19).Many traditional fractal analysis methods, such as box-counting, assume a well-defined structure within an image or dataset, which does not always translate well to nonlinear and self-affine timeseries data. Multifractal detrended fluctuation analysis (MF-DFA) has been introduced to account for these variations but remains sensitive to preprocessing techniques and data resolution, leading to the proposal of diverse multiscaling or multifractal applications (20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30).A fundamental aspect of fractal analysis when applying DFA or box-counting is the selection of an appropriate scaling range. A shortcoming of DFA is that it assumes stationarity within each detrending window (see below for mathematical treatment). However, this is not always found in physiological time series, which leads to incorrect inferences of fractality. Testing goodness-of-fit and ensuring the correct polynomial detrending order to avoid under-or overfitting provides robust results (31).Fractal analysis has been employed across a wide spectrum of scientific and technological endeavors as well as in the arts. One of the possibly best-known applications of fractal analysis in the arts was the identification of a fractal-like pattern in the artistic work of Jackson Pollock and the Mandelbrot set (32)(33)(34)(35). Differences in different schools of Orthodox iconography and fractal analysis of scribal and ink identification also have a connection to fractal analysis (36,37). Connected to the arts is the built-up environment (38,39). From the very large scales to the molecular scale, solar and space physics use fractal approaches to model solar flare distributions, solar wind turbulence, and magnetospheric dynamics. In thermodynamics, fractal analysis deals with the molecular or atomic level (40)(41)(42). From thermodynamics, combustion theory can be understood, where fractals describe flame front irregularities and turbulent eddies (43)(44)(45)(46). In materials science, fractal models describe grain boundary growth and porous structure distributions (47). Liquid crystal textures exhibit fractal patterns during phase transitions (48). Statistical physics incorporates fractals to describe anomalous diffusion and scaling in critical phenomena (49). In geophysics and environmental sciences, fractals appear in models of porous media, glaciology (crevasse networks and ice-sheet roughness), and sedimentary layering (50,51). Earthquake prediction research uses fractal statistics to characterize fault systems and seismicity patterns (52). Ocean dynamics, river basin distribution patterns, and tsunami wave modeling leverage fractal structures to capture nonlinear wave propagation and coastline complexity (53,54). Climate science is another related area of research utilizing fractal analysis principles (55)(56)(57). In finance, fractal analysis is applied to market time series to understand volatility clustering and multifractal structures in asset returns. This rather limited overview of the wide-ranging applications highlights the need for a standardized methodological framework across disciplines. From the previous paragraphs and the citations, it becomes apparent that, for instance, the term "roughness" is quite common to describe surface complexity, yet roughness as discussed later is not a fractal property. The current paper concentrates on physiological processes and morphology related to fractal analysis and argues for a consistent use of definitions.Beyond spatial applications, fractal analysis has been increasingly applied to physiological timeseries data that can be analyzed by several different methods, briefly highlighted in this section (36,58). Similar to geometric fractals, time-series fractals exhibit scale invariance in their temporal fluctuations, which can be analyzed using a variety of techniques such as detrended fluctuation analysis (DFA), Hurst exponent estimation, wavelet transforms, Higuchi algorithm, diverse entropy-based methods, and symbolic dynamics (30,(59)(60)(61)(62)(63)(64)(65)(66)(67)(68)(69)(70)(71)(72)(73)(74)(75). 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Jelinek et al. (Wed,) studied this question.