Fractals have immeasurably widened our ability to describe and understand nature. Fractal analysis provides us with models of reality much more realistic than Euclidean geometry with its integer exponents and smooth shapes. Fractals are characterised by a fractional scaling exponent, the fractal dimension, which describes the irregularity and intricacy of the object. Fractal scaling requires that the size-frequency distribution of objects has a power-law dependence on size over a wide range of scales.

Fractals are shapes whose roughness and fragmentation remains essentially unchanged at smaller scales. Monofractals or single fractals are homogeneous fractals, that means they are represented by an irregularity depicted by a single scaling exponent and fractal dimension. Simplified, a fractal dimension of a monofractal object can be calculated as the quotient of the log-transformed size of the object and the log-transformed measuring scale.

Mathematical or deterministic fractals

Subsets of mathematical fractals are infinitesimally subdivisional, each subset, however small, containing no less detail than the complete set. Mathematical fractals exhibit exact self-similarity across all spatial or temporal scales. An example of a set with non-integral dimension is the one-scale Cantor set (Fig. 6). This set is produced from the unit interval [0,1] by successively removing the middle one-third of each of the equal-sized line segments remaining (Fig. 6). This process is iterated ad infinitum. The result is an infinite number of clustered points in the interval between 0 and 1. The total length of the line segments in the `n`th iteration is `epsilon_n` = `(2/3)^n`. The number of line segments after the `n`th iteration is `N(epsilon) = 2^n`, each of length `epsilon` = `(1/3)^n`, for `n = 1,..., N`. Defined by the decrease of the coverage with each iteration, the Kolmogorov capacity dimension of the Cantor set is given as `D_0 = - lim_[epsilon   ->  0] [log N(epsilon)] / [log epsilon] = log 2 / log 3 = 0.631`. Thus, for Cantor sets the fractal dimension is in the range of `0 < D < 1` and, as it is only composed of points, it can be described as a 'dust'. It follows that the topological dimension of the Cantor set is `d_T = 0`.

Natural or random fractals

Unlike mathematical fractals, natural or random fractals are only self-similar (scale-invariant) in a statistical sense, because all natural structures are truncated at certain scales. By relating the size of the random variations to the scale, enlargements of small parts of an object have the same statistical distribution as the whole set. Often, random sets - such as epilithic biofilm patches, the outline of lichen, Mycelium (Fig. 7), bacterial culture growth, detritus particle, etc. - are statistically self-similar not only for a given value of the scaling ratio `epsilon`, but for all scaling ratios above some lower and below some upper cut-off value. Thus, randomness and self-similarity are not mutually exclusive concepts. Natural fractals look qualitatively the same over a range of scales, but the dimension of these fractal objects is restricted to this specific scale range. Depending on the natural pattern or process to be assessed, and if it is self-similar or self-affine, different estimators of the fractal dimension are used such as those based on e.g., variational method, curve-length method, root-mean-squares method, Poincaré sections method, rescaled-range analysis, Fourier and wavelet power-spectrum (e.g., Schmid 2000).

Particle fragmentation

In soil and aquatic systems, organic particles such as detritus (Fig. 8), serve as food and habitat for a variety of small-sized uni- and multicellluar organisms. The size-frequency distribution of those particles often closely follows a power-law function, suggesting the fractal nature of particle fragmentation-processes (Schmid & Schmid-Araya 2007). The size-frequency distribution of objects or fragments in a statistically self-similar system is defined as `N_P(E >= epsilon) = c epsilon^-delta` (Mandelbrot 1983; Schmid 2000), where `N_P` is defined as the number of organic particles/particle aggregates larger than or equal to a specific size `epsilon`, `c` is a constant of proportionality and `delta` is the fragmentation fractal dimensionality of the particle-size distribution. This fractal dimensionality implies that similar, often biological, mechanisms govern the particle breakdown across a wide range of scales (Schmid & Schmid-Araya 2007).

References

Mandelbrot, B.B. 1983.  The fractal geometry of nature. W.H. Freeman & CO, New York.
Schmid, P.E. 2000.  Adv. Ecol. Res. 30, 339.
Schmid, P.E. & Schmid-Araya, J.M. 2007.  Body size and scale invariance: multifractals in invertebrate communities. In Hildrew, A.G., D.G. Raffaelli & R. Edmonds-Brown (eds) Body Size: The Structure and Function of Aquatic Ecosystems. Cambridge University Press, Cambridge. 140.

Fractality in nature often manifests itself not just in terms of a single fractal dimension (monofractal) but as many interwoven fractal sets, all of them associated with their own scaling exponent, resulting in an entire spectrum of exponents (multifractal). While the applicability of a monofractal implies scale-invariance, multifractal applicability does not. The pertinence of multifractal statistics to natural patterns and processes provide important clues to the underlying physical processes.

Introduction to multifractals

The multifractal approach is an extension of the monofractal concept to include intricate, often convoluted, structures such as heterogeneous fractals. One of the properties of multifractals is a spectrum of dimensions of which the capacity dimension `D_0` is only one. Multifractal statistics offers a framework to analyse and quantify intricate systems such as those in nature. Multifractals are found in all fields of the sciences as diverse as, for instance, quantum mechanics (e.g., Richardella et al. 2010), biochemistry (e.g., Dewey 1997), aquatic sciences (e.g., Schmid & Schmid-Araya 2007; Seuront 2010) and are widely applied in economy, technology and engineering (e.g., Li & Mao 2012; Ţălu & Stach 2014).

To understand multifractals in the context of aquatic systems, let us consider a simple example. The pore-space distribution in streambed sediments serves as habitat for many different species and varies considerably among different locations. These variations have important implications, for instance, for environmental predictions, functional relations in food-webs, water quality assessment, and natural resource management. The image of a streambed cross-section is composed of pixels of pore and particle size-classes (Fig. 9). In the monofractal case, a square of a superimposed lattice (Fig. 9) is counted as occupied by pores, regardless of whether it is entirely covered by pore-class pixels or just by a single pore-class pixel. In contrast, the multifractal approach accounts for the pore area or volume contained in each square or circle (Schmid & Schmid-Araya 2019). Thus, we divide the image of the sediment cross-section into `n` squares of size `epsilon`. For each square `i` of size `epsilon` the fraction of pore space in that square is calculated as `p_i = m_i` / `M`, where `m_i` is the number of pore-class pixels, that is the pore area or 'mass', contained in a square and `M` is the total number of pore-class pixels in `n` squares of size `epsilon`.

Multifractal analysis

The starting point in the analysis of multifractals is to define a weighted measure as `z_i(q,epsilon) = p_i^q`, where `z_i` represents the proportion of pore-space in squares of size `epsilon`. The `q` values are the statistical moments of the measure. Based on the weighted measure, we can define the generating function as `z(q,epsilon) = sum_[i=1]^[n(epsilon)] z_i(q,epsilon)`. Multifractality may be intrinsic to these gravelly sediments if the log-log transformed relation of the power-law function `z(q,epsilon) prop epsilon^[-tau(q)]` is highly significant for each of various `q`- moments. `tau(q)` is the Rényi exponent that can then be expressed in the following equation as `tau(q)= - lim_[epsilon -> 0] [log z(q,epsilon)]/log epsilon`.
Various formalisms have been developed to depict the multifractal properties in terms of the singularity spectrum, `f(alpha)` that describes the fractal dimension `f` of a subset with a given Lipschitz-Hölder exponent `alpha` or Rényi's generalised dimensions `D_q` (e.g., Schroeder 1996; Schmid & Schmid-Araya 2007). Using a Legendre transformation from the variables `q` and `tau` to the variables `alpha` and `f` we can define the relation between `tau(q)`, `alpha` and `f(alpha)` as `alpha(q) = - d/[dq] tau(q)` and `f(alpha(q)) = q alpha(q) + tau(q)`, respectively (e.g., Halsey et al. 1986), where `f(alpha(1)) = alpha(1) = D_1` (see below). A singularity spectrum with a wide parabolic shape, as shown in Fig. 10 for the pore-space distribution, is typical for habitats exhibiting multifractal properties.
The generalised dimensions, which are based on the concept of generalised entropies (Rényi 1955), can be expressed as `D_q = [tau(q)]/(q-1) = lim_[epsilon -> 0] [log z(q,epsilon)]/[(q-1) log epsilon]` for all `q` - moments `!= 1`. For `q` = 1, the dimension is given as `D_1 = [d tau(q)]/[d q] = lim_[epsilon -> 0]sum_[i=1]^[n(epsilon)] [z_i(1,epsilon) log z_i(1,epsilon)]/log epsilon`. Thus, the ecologically most relevant dimensions are `D_0`, the Kolmogorov capacity, `D_1`, the information and `D_2` the correlation dimension (Hentschel & Procaccia 1983). `D_1` provides information about the degree of heterogeneity in the distribution of the measure and `D_2` is associated with the correlation function and quantifies the average distribution density of the measure. In the monofractal case, `D_0` is similar or equal to `D_1` and `D_2`, while multifractals exhibit `D_0 > D_1 > D_2`. The pore-space distribution of the sediment cross-section displays a multifractal spectrum with `D_0` = 1.68, `D_1` = 1.34 and `D_2` = 1.29 (Fig. 10). We can define the spectrum width as the difference between `D_2` and `D_0` (Schmid & Schmid-Araya 2019). A wide spectrum width implies a heterogeneous distribution in pore space in this streambed section.

Mathematical multifractals

While a Cantor set, generated from segments of equal length, is defined by a single scaling factor, multifractals are described by two scaling factors, one for the supporting fractal and one for the probability weights. To understand the logic of mathematical multifractals, let us modify the Cantor set (Fig. 11). The set has a length `l_0 = 1` with a probability weight `p_0 = 1`. The basic difference between the construction of a mono- and multifractal set is that the initial set is divided into segments of unequal lengths. Applying the generator of a two-scale Cantor set consists of splitting the bar into two rescaled lengths, for instance, one of length `l_1 = 0.50` and the other of `l_2 = 0.25` with the middle piece cut out. Each of the segments has a probability, the larger segment has a probability of `p_1 = 0.67` and the smaller one of `p_2 = 0.33`.

To analyse this fractal object we can define the generating function as `Z(tau,q) = sum_[i=1]^[N(l)] p_i^q / l_i^tau`, where `N` is the total number of segments with size `l_i` and probability `p_i` of the `i`th segment. At the first level of the modified Cantor set the sum of the generator is given as  `Z_1(tau,q) = n_1 p_1^q / l_1^tau + n_2 p_2^q / l_2^tau`, where `n_i` is the number of segments at level `i`. To assess the extremes of the singularity spectrum, the minimum and maximum Lipschitz-Hölder exponent is given as `alpha_min` = `log p_1 / log l_1` and `alpha_max` = `log p_2 / log l_2`, respectively. Thus, the Lipschitz-Hölder exponent of this two-scale Cantor set ranges from  `alpha_min = 0.585` to `alpha_max = 0.792`, with `f_max = f(alpha(0)) = 0.694` (Fig. 12). The segment distribution of this modified Cantor set displays a spectrum of generalised dimensions with `D_0` = 0.694, `D_1` = 0.689 and `D_2` = 0.683 (Fig. 12).

Lacunarity

Lacunarity is a concept introduced by Mandelbrot (1983) to describe the distribution of gap sizes in a fractal sequence. The word stems from lacuna which is the Latin word for ditch or gap. Lacunarity analysis derives from fractal mathematics and allows to determine the texture associated with patterns of spatial dispersion across multiple scales in one- to three dimensions. Lacunarity is a counterpart to fractal dimension as it depicts and quantifies aspects of patterns that exhibit scale-dependent changes in structure. In addition to the multifractal spectrum, the pore-space may be fully characterised by its lacunarity (Fig. 13). Here, we estimate lacunarity from the variations in pixel density given as `Lambda(epsilon,gamma)=(sigma/mu)^2` in each image for different scales `eta`and orientations `gamma` of a square lattice. Although there are several approaches to assess lacunarity, we define here the grand mean lacunarity, `Lambda(lambda)`, as the deviation from translational and rotational invariance by depicting the size distribution of voids in a sediment structure across a range of scales and orientations as `Lambda_epsilon(lambda)=sum(Lambda(epsilon,gamma))/N_epsilon` (Fig.13)and `Lambda(lambda)=sum(Lambda_epsilon(lambda))/N_gamma` where `Lambda_epsilon(lambda)` is the mean lacunarity across scales and `N_epsilon` and `N_gamma` are the total number of different scales and lattice orientations, respectively. Thus, lacunarity can be estimated as the average across the spatial scale range covered by the size and shape composition of a 'habitat structure'. Habitat structures which may display identical monofractal dimensions are distinguishable by the differences in their lacunarity values. Higher lacunarity values (Fig. 13) characterise irregular and patchy arrangements of voids (variant fractals), while habitats with lower lacunarity are indicative for more regularly spaced voids, often combined with translationally and rotationally invariant fractal features. Lacunarity displays transitions from statistically self-similar to self-affine patterns, and together with multifractality, may best define all elements of habitat complexity (Schmid & Schmid-Araya, 2019; Schmid-Araya & Schmid, 2019). The mean lacunarity of the pore-space distribution of the streambed sediments (shown in Fig.9) is `0.30+-0.01` SE, which pertains to a patchy arrangement of different pore sizes.

References

Dewey, T.G. 1997 Fractals in molecular biophysics. Oxford University Press, Oxford.
Halsey, T. C. et al.1986.  Phys. Rev. A, 33, 1141.
Hentschel, H. & Procaccia, I. 1983.  Physica D, 8, 435.
Li, D. & Mao, J.- F. 2012. PIER, 126, 399.
Mandelbrot, B.B. 1983.  The fractal geometry of nature. W.H. Freeman & CO, New York.
Rényi, A. 1955. Acta Math. Hung. 6, 285.
Richardella, A. et al. 2010. Science 327, 665.
Schmid, P.E. & Schmid-Araya, J.M. 2007.  Body size and scale invariance: multifractals in invertebrate communities. In Hildrew, A.G., D.G. Raffaelli & R. Edmonds-Brown (eds) Body Size: The Structure and Function of Aquatic Ecosystems. Cambridge University Press, Cambridge. 140.
Schmid, P.E. & Schmid-Araya, J.M. 2019. Application of multifractal analyses to habitat complexity research in stream systems: a cross-scale approach. In: Tokeshi, M. (eds) Habitat complexity in aquatic systems: ecological perspectives.
Schmid-Araya, J. M. & Schmid, P. E. 2019. Habitat complexity and meiofauna: a review and new insights on food-search dynamics of organisms across habitat patches of varying complexity. In: Tokeshi, M. (eds) Habitat complexity in aquatic systems: ecological perspectives.
Schroeder, M. 1996.  Fractals, chaos and power laws. Minutes from an infinite paradise.W.H. Freeman and Company, New York.
Seuront, L. 2009. Fractals and multifractals in ecology and aquatic science. CRC Press, Taylor & Francis Group, NY.
Ţălu, Ş. & Stach, S. 2014. Polym. Eng. Sci. 54, 1066.