# Relationship between Fractal Dimension and Spectral Scaling Decay Rate in Computer-Generated Fractals

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−D}

_{(Surface)}, lies in the range between these two Euclidean dimensions: 2 < D

_{(Surface)}< 3. Taking a vertical slice through this terrain (i.e., taking its intersection with the xz or yz plane) creates a fractal “mountain” profile (see Figure 1c) quantified by D

_{(Mountain Edge)}= D

_{(Surface)}− 1. Similarly, taking a horizontal slice creates a fractal “coastline” (see Figure 1d) with D

_{(Coastal Edge)}= D

_{(Mountain Edge)}. To measure D

_{(Surface)}, mathematicians and natural scientists typically determine D

_{(Mountain Edge)}or D

_{(Coastal Edge)}(and then add 1) because the measurements involved are easier and faster to implement than for measurements of D

_{(Surface)}

_{.}

_{V}(f) = 1/(cf

^{β})

_{V}(f) is the spectral density (power), f is the spatial frequency, and β and c are constants.

_{H}(x), with x in an E-dimensional Euclidian space, which has a fractal dimension D and spectral density S

_{V}(f) ∝ 1/f

^{β}, for the fluctuations along a straight line path in any direction in E-space” is provided by the equation

_{(Mountain Edge)}.

_{(Mountain Edge)}has not been observed empirically for either 1-dimensional (E = 1, D ≤ 2) or 2-dimensional (E = 2, D ≤ 3) fractals. To experimentally discern this relationship, we use two methods for generating fractals—namely, midpoint displacement fractals (in which D serves as the input parameter) and fractal Fourier noise (in which β is the input). We find that, when measured along a straight-line path, the relationship described by Voss [5] holds for fractals with both E = 1 and 2. However, the relationship described by Voss [5] does not extend to the observed relationship between measurements of D and β when β is measured by the standard method of vision researchers (in a 2-dimensional Fourier space, which we will hereafter call β

_{(Surface)}). A new equation that extends the relationship between D and β to multi-dimensional Fourier spaces has the potential to enhance discourse among mathematicians, who are experts in the geometry of fractals, physicists, who are experts in surfaces and textures, and vision scientists, who are experts in animals’ sensation and perception of geometric shapes, surfaces, and textures.

## 2. Materials and Methods

#### 2.1. Midpoint Displacement Fractals

#### 2.1.1. One-Dimensional Midpoint Displacement Fractals

^{−2(3 − D)(R + 1)}, where D is the fractal dimension and R is the current level of recursion. This process is shown schematically for an exact midpoint displacement fractal that is not affected by random perturbations in Figure 2. As in the schematic, the scaling factor of the fractals generated for this study was held constant for each vertex at a given level of recursion, and changed with each level of recursion. The vertices at each recursion served as endpoints in the next level of recursion for R recursions in order to generate time-series data.

#### 2.1.2. Two-Dimensional Midpoint Displacement Fractals

#### 2.2. One- and Two-Dimensional Fractal Fourier Noise

_{(input)}(see Figure 6e–g). Values of β

_{(input)}were paired with phase maps to create 2-dimensional fractal images as well (see Figure 7e–g).

#### 2.3. Measurement of the Box Counting Dimension

#### 2.3.1. Box Counting Analysis of 1-Dimensional Fractals: D_{(Mountain Edge)}

^{3}) to L/(2

^{n}

^{− 3}) were averaged to compute D, while the counts from the grid scales outside this range (the larger and smaller boxes, where n = {0, 1, 2, n − 2, n − 1, and n}), were not used. The embedding dimension of the series of points is 1, but the embedding dimension of the fractal mountain edge is 2, so we averaged these values of D, computed for pairs of grid sizes, and added 1 to report the fractal dimension of the 1-dimensional fractals, D

_{(Mountain Edge)}, which span the range 1 < D

_{(Mountain Edge)}< 2.

#### 2.3.2. Box Counting Analysis of xy Slices of 2-Dimensional Fractal Coastlines: D_{(Coastal Edge)}

_{(Coastal Edge)}, which span the range 1 < D

_{(Coastal Edge)}< 2.

#### 2.4. Fourier Decomposition and Measurement of β

#### 2.4.1. Spectral Scaling Analysis of 1-Dimensional Fractals: β_{(Mountain Edge)}

_{(Mountain Edge)}, of the time series.

#### 2.4.2. Spectral Scaling Analysis of 2-Dimensional Fractal Intensity Images: β_{(Surface)}

_{(Surface)}, of the image (see Figure 9a–c).

## 3. Results

#### 3.1. Relationship between D_{(Mountain Edge)} and β_{(Mountain Edge)} for 1-Dimensional Fractals

^{20}. The measurement technique described in Section 2.3.1 over-estimates D by a progressively smaller amount as D approaches 2, as shown in Figure 10a. This is a minor measurement error. Accordingly, the best linear fit, D

_{(Input)}= 0.91 + 0.16 × D

_{(Mountain Edge)}, for which R

^{2}= 0.97 (Figure 10a, black line), deviates from the unity line (Figure 10a, blue line) by only a small amount.

^{20}. The measurement technique described in Section 2.4.1 does well at approximating β (as shown in Figure 10b), with the best linear fit, β

_{(Input)}= 0.0003 + 0.9999 × β

_{(Mountain Edge)}, for which R

^{2}= 1.00 (Figure 10b, black line), overlapped by the unity line (Figure 10b, blue line).

_{(Mountain Edge)}into the experimentally determined regression equation stated above in this section and computing the expected D

_{(Input)}, which we call D

_{(Adjusted)}(as shown in Figure 10c). The best linear fit—D

_{(Adjusted)}= 2.48 − 0.50 × β

_{(Mountain Edge)}—for which R

^{2}= 0.97 (Figure 10c, black line), is close to Voss’s approximation (Equation (5)), given E = 1 (Figure 10c, red line). We conclude that Voss’s approximation for fractals with E < D < E + 1 and 1 < β < 3 is accurate when E = 1. We also note that the difference between Equation (5) and our regression equation is inconsequential, and both overlap our measurements across the range of D and β.

#### 3.2. Relation of D_{(Mountain Edge)} and D_{(Coastal Edge)} for 2-Dimensional Fractals

^{11}. For each fractal, the dimension of the mountain profile was measured according to the technique described in Section 2.3.1. These measures of D

_{(Mountain Edge)}were averaged together for each image. Again, this under- and over-estimates D by a progressively larger amount as D approaches 2 and 1, respectively, as shown in Figure 11a. The best linear fit—D

_{(Input)}= 0.38 + 0.77 × D

_{(Mountain Edge)}, for which R

^{2}= 0.87 (Figure 11a, black line)—deviates from the unity line (Figure 11a, blue line) in a manner similar to that observed for 1-dimensional fractals. When the coastal edge of the image was measured according to the technique described in Section 2.3.2, we observe a similar trend, with the best linear fit—D

_{(Input)}= 0.28 + 0.81 × D

_{(Coastal Edge)}, for which R

^{2}= 0.97 (Figure 11b, black line)—deviating from the unity line (Figure 11b, blue line) in a manner similar to that observed for the dust measurement technique. These measures of D, averaged mountain edges and coastal edge values, are reasonable approximations of each other, with the best linear fit D

_{(Mountain Edge)}= 0.07 + 0.93 × D

_{(Coastal Edge)}, for which R

^{2}= 0.86 (Figure 11c). Both of these measures provide an accurate means by which to compute the fractal dimension of an image.

#### 3.3. Relation of β_{(Mountain Edge)} and β_{(Surface)} for 2-Dimensional Fractals

_{(Input)}ranging from 1 to 3, in steps of 0.1, with side length 2

^{11}pixels. Measuring the spectral decay of a 2-dimensional Fourier analysis as described in Section 2.4.2 provides measured β

_{(Surface)}values that are consistent with the specified input β

_{(Input)}values, with the best linear fit β

_{(Surface)}= 0.12 + 0.95 × β

_{(Input)}, for which R

^{2}= 0.9999 (see Figure 12a). Having verified the generation process with an analysis in native space, we measured the β of these 2-dimensional fractals along a straight line path, β

_{(Mountain Edge)}. We averaged the β

_{(Mountain Edge)}measurements for each row of each image, as described in Section 2.4.1, to allow us to follow the definition put forth by Voss [5]. We found that these values of β

_{(Mountain Edge)}differ from the specified input β values (see Figure 12b), with an offset as evidenced by the best linear fit β

_{(Mountain Edge)}= −0.39 + 0.82 × β

_{(Input)}, for which R

^{2}= 0.998 (Figure 12b, black line).

_{(Input)}. We found that the mountain profile from a fractal terrain with an arbitrary value of β

_{(Surface)}, β

_{i}, is rougher (i.e., has a larger contribution of fine structure) than a mountain profile from a 1-dimensional fractal with β

_{(Mountain Edge)}= β

_{i}(see Figure 13). We then took measurements for an ensemble of 100 random phase maps around β = 0, which show that our measures converge when there is equal power across frequencies (see Figure 12c). An exploratory analysis on a new set of images with 0 ≤ β

_{(Input)}≤ 4.5 allowed us to empirically determine that fractal Fourier noise terrains with β

_{(Input)}values in the range 1.8 < β

_{(Input)}< 3.8 consistently give β

_{(Mountain Edge)}values in the range 1 < β

_{(Mountain Edge)}< 3 (see Figure 12d). We found that β

_{(Input)}and β

_{(Mountain Edge)}are relatable by the regression equation β

_{(Mountain Edge)}= -0.64 + 0.93 × β

_{(Input)}, for which R

^{2}= 0.997 (Figure 12d, black line), across the range 1 < β

_{(Mountain)}< 3 and 1.8 < β

_{(Input)}< 3.8.

_{(Mountain Edge)}and β

_{(Surface)}approximately converge at β = 0 (as expected), because for white noise, there is equal power across frequencies. This would be trivial if the two measures followed the unity line (Figure 12d, blue line), but certifies that our otherwise non-equivalent measures accurately describe white noise. We note that our β values exhibit slight measurement errors, such that classical Brownian traces (β

_{(Mountain Edge)}= 2, β

_{(Surface)}= 3) have empirically determined means of (2.14, 2.95). In the absence of measurement error, the empirically determined range 1.8 < β

_{(Input)}< 3.8 would be 2 < β

_{(Surface)}< 4.

#### 3.4. Relation of β to D for 2-Dimensional Fractals

_{(Input)}ranging from 0 to 5 in steps of 0.1 with side length 2

^{11}pixels.

#### 3.4.1. Relation of β_{(Mountain Edge)} to D_{(Coastal Edge)} of 2-Dimensional Fractals

_{(Coastal Edge)}and β

_{(Mountain Edge)}. We performed the Fourier analysis described in Section 2.4.1 on each row of each image, and measured the rows’ fractal dimension with the technique described in Section 2.3.2. The relationship between these measures is described by a best linear fit—β

_{(Mountain Edge)}= 5.18 – 2.00 × D

_{(Coastal Edge)}, for which R

^{2}= 0.99 (Figure 14a, black line)—which approximates Voss’s [5] equation (Figure 14a, red line, which is Equation (5)). This confirms Voss’s [5] assertion that measuring along a straight-line path will provide measures of D and β that are related by Equation (5).

#### 3.4.2. Relation of β_{(Surface)} to D_{(Coastal Edge)} for 2-Dimensional Fractals

_{(Surface)}, which captures the radial scaling properties of the images. When plotted against D

_{(Coastal Edge)}, we observe that the relationship between these measures is described by a best linear fit, β

_{(Surface)}= 6.24 − 2.14 × D

_{(Coastal Edge)}, for which R

^{2}= 0.99 (Figure 14b, black line). The observed relationship agrees with the data from a smaller set of images previously reported by Spehar and Taylor [23] (Figure 14b, green line). Significantly, this observed relationship between β

_{(Surface)}and D

_{(Coastal Edge)}agrees with Equation (11) which we present below, and will allow conversion across measures of D and β in multidimensional spaces.

## 4. Discussion

#### 4.1. Mathematical Relationships between Ds and βs

_{(Surface)}, is not that which Voss [5] described. Voss’s [5] equation (Equation (5)) applies for the measure we call β

_{(Mountain Edge)}. However, vision researchers typically use β

_{(Surface)}. Whereas the difference between these two spectral decay rates is nonexistent for white noise, where β = 0, these measures are substantially different in the range over which these noises are fractal. Before commenting further on the different measures of β, we will first summarize the relationships for the fractal images discussed in this paper.

_{(Mountain Edge)}= 1 + (3 − β

_{(Mountain Edge)})/2

_{(Mountain Edge)}= β

_{(Surface)}− 1

_{(Mountain Edge)}< 3 and 2 < β

_{(Surface)}< 4. Combining Equations (6) and (7) gives:

_{(Mountain Edge)}= 1 + (4 − β

_{(Surface)})/2

_{(Surface)}can then be obtained using:

_{(Surface)}= D

_{(}_{Mountain Edge)}+ 1 = D

_{(}_{Coastal Edge)}+ 1

_{(Coastal Edge)}in our investigations. However, we expect that, if a coastal edge was unraveled by the process described by Zahn & Roskies [48], its β value will equal β

_{(Mountain Edge)}because D

_{(Coastal Edge)}is equivalent to D

_{(Mountain Edge)}and E = 1 applies to both the mountain and coastal edges.

#### 4.2. Distinguishing βs

_{(Mountain Edge)}< 2), we have 1 < β

_{(Mountain Edge)}< 3. However, we have shown that β measured in a single variable-space (i.e., along a straight line path as β

_{(Mountain}

_{Edge)}) diverges from β measured in a two-variable space (i.e., across a plane as β

_{(Surface)}) to an extent that is characterized by Equation (7) for the range over which 2-dimensional noise is fractal. For D surface (2 < D

_{(Surface)}< 3), we have 2 < β

_{(Surface)}< 4 and 1 < β

_{(Mountain Edge)}< 3. The fact that the β values measured by 1- and 2-dimensional Fourier transforms differ for fractal noises holds crucial consequences. The fractal structure of a terrain is quantified by β

_{(Mountain Edge)}. Visual inspection of Figure 13 makes it immediately apparent that its value is significantly smaller than β

_{(Surface)}for fractals of topological dimension E = 2. Given that β

_{(Input)}matches β

_{(Surface)}rather than β

_{(Mountain Edge)}, it is likely that many vision researchers have been misjudging the fractal content of their fractal terrains, or adapting them by an intuitive sense of the image’s roughness. Equation (7) provides a formal justification for adjusting the β of 2-dimensional fractals.

_{i}s) shown in Figure 6a–c and phase matrix shown in Figure 6d). In contrast, a 2-dimensional fractal pattern is generated from a matrix of amplitudes and a matrix of phases (for illustration, see the visualization of the amplitude matrices corresponding to three different input Betas (β

_{i}s) shown in Figure 7a–c and phase matrix shown in Figure 7d).

_{V}(f) by f to increase the embedding dimension by 1 (from 1 to 2) requires a subtraction of 1 from β, as denoted by the following equation:

_{V}(f) × f = (f

^{−β}

_{i}) × f = 1/f

^{(β}

_{i}

^{− 1)}

#### 4.3. A Generalized Equation to Relate Ds and βs

_{(Mountain Edge)}and F = 2 for β

_{(Surface)}). This new equation (Equation (11)) allows for conversion from D to both of the Fourier measurement techniques that can describe static images, and provides an extension of Mandelbrot’s [1,4], Voss’s [5], and Knill et al.’s [14] relationships that can describe spectral decay in dynamic fractal Brownian stimuli generated with Fourier, midpoint displacement, and other equivalent methods. Equation (11) extends Voss’s [5] equation (Equation (5)) by generalizing the term for β from β

_{(Mountain Edge)}.

#### 4.4. Importance of the Relationship between D and β for Current and Future Research

_{(Volume)}= 2.2. This is because Equation (11) implies the optimal range of values of β

_{(Volume)}for such an application would be 4 < β

_{(Volume)}< 4.4, because E = 3 and F = 3, where fractal noises are in the range 3 < β

_{(Volume)}< 5.

_{(Mountain Edge)}, they cite Knill et al. [14], who provided the relationship between D and β for surfaces. This highlights the difficulty associated with discerning the optimal range of β in aesthetics and vision research. More problematic is that because of the relative convenience of the respective algorithms’ implementation, others report a combination of D

_{(Mountain Edge)}and β

_{(Surface)}values [23], for which there is no clear conversion provided in the published literature.

_{(Coastal Edge)}< 2, whereas the navigated environment has complexity in the range 2 < D

_{(Surface)}< 3, and the visually perceived scene has a spectral decay that likely falls off at a rate in the range 2 < β

_{(Surface)}< 4. While it is mathematically no less appropriate to describe all of these in terms of β

_{(Mountain}

_{Edge)}, it is easier to interpret results described in units that reflect the experienced dimensional space precisely and explicitly. There is convenience to be gained by using this more general equation (Equation (11)) and its variables’ boundary conditions rather than more specialized equations, such as those which have been put forth previously [5,14,21]. Equation (11) stems from a recognition that β varies with the number of variables with which the Fourier transform is applied. As such, it is important that we define which β is being used (β

_{(Line)}, β

_{(Surface)}, β

_{(Volume)}, etc.) for easier interpretation of results and to facilitate the communication of future endeavors in interdisciplinary fractals research.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

β: | Spectral slope |

D: | Fractal dimension |

E: | Euclidian dimension |

F: | dimensional space of the Fourier transform |

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**Figure 1.**Plots of a fractal terrain with D = 2.5 and its intersection with axial planes; (

**a**) “Surface” plot of a fractal terrain; (

**b**) intensity “image” of the terrain; (

**c**) “mountain edge” profile of an “xz-slice” or “yz-slice” of the terrain; (

**d**) “coastal edge” of an “xy-slice” of the terrain.

**Figure 2.**Illustration of the generation of 1-dimensional midpoint displacement fractals. (

**a**) Cartoon graph of a scaling plot in log–log coordinates that determines the rate of scaling of midpoint displacements across recursions for high (solid line) and low (dashed line) D fractals; (

**b**) Schematics of recursions 0–2 are shown for low (dashed line) and high (solid line) D exact midpoint displacement fractals. Gray arrows indicate displacements that occur with each recursion in (b).

**Figure 3.**Plots of 1-dimensional statistical midpoint displacement fractals. (

**a**–

**c**) Fractal traces that vary in D, such that D = 1.2, 1.5, and 1.8, are generated from a single set of random numbers that contribute to the variable length and direction of displacement of the midpoints at each recursion.

**Figure 4.**Illustration of the generation of a 2-dimensional midpoint displacement fractal as the heights (indicated by grayscale intensity) of particular points are specified over eight recursions. (

**a**–

**d**) The second, fourth, sixth, and final recursions are shown. In (

**a**–

**c**), white space indicates points for which height has not yet been specified.

**Figure 5.**Plots of 2-dimensional midpoint displacement fractals. (

**a**–

**c**) Surface plots of fractals generated from a single set of random numbers that vary in D, such that D = 1.2, 1.5, and 1.8; (

**d**–

**f**) Grayscale intensity map images of the surface plots in (

**a**–

**c**).

**Figure 6.**Generation of 1-dimensional Fourier noise fractals. (

**a**–

**c**) Power (y-axis) as a function of frequency (x-axis) for β

_{(Input)}= 2.6, 2, and 1.4; (

**d**) Set of random phases (y-axis) as a function of frequency (x-axis); (

**e**–

**g**) fractal traces resulting from the pairing of the phases in panel (

**d**) with power spectra from (

**a–c**) respectively.

**Figure 7.**Generation of 2-dimensional Fourier noise fractals. (

**a**–

**c**) Power (z-axis) as a function of frequency in xy coordinates for β

_{(Input)}= 2.6, 2, and 1.4; (

**d**) Set of random phases (z-axis) as a function of frequency in xy coordinates; (

**e**–

**g**) fractal terrains resulting from the pairing of the phases in panel (

**d**) with power spectra from (

**a–c**) respectively.

**Figure 8.**Edge extraction procedure for 2-dimensional midpoint displacement fractals. (

**a**–

**c**) Grayscale intensity map images of fractals generated from a single set of random numbers that vary in D, such that D = 1.2, 1.5, and 1.8; (

**d**–

**f**) Binary images resulting from the threshold procedure applied to the terrains shown in (

**a**–

**c**); (

**g**–

**i**) Coastal edges extracted from the binary images shown in (

**d**–

**f**).

**Figure 9.**Fourier decomposition of 2-dimensional fractals. (

**a**) Fractal surface generated with the inverse Fourier method; (

**b**) Power spectrum of the Fourier decomposition of the terrain shown in (

**a**); (

**c**) Power spectrum shown in (

**b**) with low spatial frequencies centered.

**Figure 10.**1-dimensional fractal measurements. (

**a**) Midpoint displacement fractals’ D

_{(Mountain Edge)}measurements plotted against their D

_{(Input)}values; (

**b**) Fourier noise fractals’ β

_{(Mountain Edge)}measurements plotted against their β

_{(Input)}values; (

**c**) Fourier noise fractals’ D

_{(Mountain Edge)}measurements, adjusted by the linear fit from panel (

**a**), plotted against their β

_{(Mountain Edge)}measurements. In each panel, the best linear fit for the data is shown with a black line. In panels (a,b), unity is represented by the blue line. In panel (

**c**), Voss’s approximation (Equation (5)) is represented by the red line. Data are colored to distinguish adjacent input values such that each datum's color is determined by D

_{(Input)}in panel (

**a**) and β

_{(Input)}in panels (

**b**,

**c**).

**Figure 11.**2-dimensional fractal D measurements. (

**a**) Midpoint displacement fractals’ D

_{(Mountain Edge)}measurements plotted against their D

_{(Input)}values; (

**b**) Midpoint displacement fractals’ D

_{(Coastal Edge)}measurements plotted against their D

_{(Input)}values; (

**c**) Midpoint displacement fractals’ D

_{(Coastal Edge)}measurements plotted against their D

_{(Mountain Edge)}measurements. In each panel, unity is represented by the blue line, while the best linear fit for the data is represented by the black line. Data are colored to distinguish adjacent input values such that each datum's color is determined by D

_{(Input)}in panels (a–c).

**Figure 12.**2-dimensional fractal β measurements. (

**a**) Fourier noise fractals’ β

_{(Surface)}measurements plotted against their β input values; (

**b**) Fourier noise fractals’ β

_{(Mountain Edge)}measurements plotted against their β

_{(Input)}values; (

**c**) Fourier noise fractals’ β

_{(Mountain Edge)}measurements plotted against their β

_{(Surface)}measurements, showing that the measures converge at β = 0; (

**d**) Fourier noise fractals’ β

_{(Mountain Edge)}measurements plotted against their β

_{(Input)}values. In each panel, unity is represented by the blue line, while the best linear fit for the data is represented by the black line. Data are colored to distinguish adjacent input values such that each datum's color is determined by β

_{(Input)}in panels (a–d).

**Figure 13.**Mountain profiles from 1 and 2-dimensional fractal Fourier noise. (

**a**) 1-dimensional fractal with β

_{(Input)}= 1.5; (

**b**) 1-dimensional fractal with β

_{Input}= 2.5; (

**c**) 2-dimensional fractal with β

_{(Input)}= 2.5; (

**d**) 1-dimensional fractal mountain edges from the terrain in (c).

**Figure 14.**2-dimensional fractal measurements of Fourier noise (

**a**,

**b**). (

**a**) Fourier noise fractals’ β

_{(Mountain Edge)}measurements plotted against their D

_{(Coastal Edge)}measurements; (

**b**) Fourier noise fractals’ β

_{(Surface)}measurements plotted against their D

_{(Coastal Edge}) measurements. In each panel, the best linear fit for the data within the region that was shown to exhibit fractal scaling (identified with gray lines) is shown with a black line. In panel (a), Voss’s [5] equation (Equation (5)) is shown with a red line. In panel (b), Spehar & Taylor’s [23] data is shown with a green line and our extension of Voss’s [5] equation (Equation (11)) is shown with a red dashed line. Data are colored to distinguish adjacent input values such that each datum's color is determined by β

_{(Input)}in panels (a,b).

**Figure 15.**Real and imaginary frequency components of 1- and 2-dimensional fractal Fourier noise plotted in 2- and 3-dimensional spaces with color changing with frequency, such that higher frequency components are shown in cooler colors. (

**a**,

**b**) Amplitude-frequency plots of a 1-dimensional noise with low frequency components centered such that the amplitude of the higher frequency components fall at the edges of the plot; (

**c**,

**d**) Amplitude-frequency plots of 2-dimensional noise with low frequency components centered such that larger concentric circles indicate higher frequency components.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bies, A.J.; Boydston, C.R.; Taylor, R.P.; Sereno, M.E.
Relationship between Fractal Dimension and Spectral Scaling Decay Rate in Computer-Generated Fractals. *Symmetry* **2016**, *8*, 66.
https://doi.org/10.3390/sym8070066

**AMA Style**

Bies AJ, Boydston CR, Taylor RP, Sereno ME.
Relationship between Fractal Dimension and Spectral Scaling Decay Rate in Computer-Generated Fractals. *Symmetry*. 2016; 8(7):66.
https://doi.org/10.3390/sym8070066

**Chicago/Turabian Style**

Bies, Alexander J., Cooper R. Boydston, Richard P. Taylor, and Margaret E. Sereno.
2016. "Relationship between Fractal Dimension and Spectral Scaling Decay Rate in Computer-Generated Fractals" *Symmetry* 8, no. 7: 66.
https://doi.org/10.3390/sym8070066