Aesthetic Responses to Exact Fractals Driven by Physical Complexity

Abstract

Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a pattern. Conceptualizations such as Berlyne’s and Redies’ theories of aesthetics also provide a suitable framework for interpretation of our data with respect to the individual differences that we detect. Future studies that incorporate physiological methods to measure the human aesthetic response to exact fractal patterns would further elucidate our responses to such timeless patterns.

Theoretical Foundations of Biodiversity and Mental Well-being Relationships

“…As such, the defining feature of fractals is self-similarity; a "shape is made of smaller copies of itself…same shape but different size" (Frame et al n.d.). This self-similarity can be identified and quantified by the fractal dimension, D. The equation for fractal dimension, D, is log(N R )/log(1/S R ), where N equals the number of line segments in the pattern, S is the scale factor, and R is the number of recursions of the pattern (Bies et al 2016). For example, a fractal line will have a fractal dimension D score that is between 1.0 and 2.0, whilst a fractal surface will have a D score between 2.0 and 3.0 (Hagerhall et al 2004).…”

“…Further support for preference for mid-range D scores was found in Bies et al's (2016) study investigating preferences for statistical (fractals that do not repeat exactly but have the same statistical qualities, like those found in nature) or exact (fractal patterns that repeat precisely, created by a computer programme) fractals. For statistical fractals mid-range D scores were preferred, whilst for exact fractals a higher D score was preferred (Bies et al 2016). Interestingly, the mid-range D score of 1.3 is most prevalent in nature (Hagerhall et al 2004(Hagerhall et al , 2015, and found in species-rich habitats (Stevens 2018).…”

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

“…The crux of the problem, perhaps, is that D is a general parameter that quantifies complexity in a variety of patterns, whereas β is limited (at least in practice) in its ability to quantify some patterns' complexity. For example, Fourier analysis is poorly suited to describe the complexity of patterns including strange attractors and some line fractals (e.g., dragon fractals and Koch snowflakes), which have been used by vision researchers to study aesthetics [25,31,34,35] and perceived complexity [47]. This provides a strong impetus to convert to D when forming general conclusions.…”

“…This is especially important in aesthetics research, where there have been claims of universality in preference for patterns of moderately low complexity [23,26,[34][35][36][37]. To test this hypothesis, it is necessary to be able to translate the units of measurement of researchers who alternately use D [22,25,26,28,29,31,[33][34][35][36][37][38][39][40], β [14,24,30,32,[41][42][43][44][45][46], or, infrequently, both [23,27]. The crux of the problem, perhaps, is that D is a general parameter that quantifies complexity in a variety of patterns, whereas β is limited (at least in practice) in its ability to quantify some patterns' complexity.…”

“…Further consideration of the relationship between D and β is both important and timely because new studies are being performed using fractals to investigate a variety of behaviors including aesthetics [22][23][24][25][26][27], navigation [28], object pareidolia (perceiving coherent forms in noise) [29,30], sensitivity [24], and associated neural mechanisms [31][32][33]. This is especially important in aesthetics research, where there have been claims of universality in preference for patterns of moderately low complexity [23,26,[34][35][36][37].…”