Abstract
Fractals are patterns that show self-similarity at different levels of scale. Typically they appear in nature and this degree of similarity is approximate or statistical. However, artificial or exact fractals have also been studied and the advantage of these stimuli is the ability to more carefully control the relationships that occur across various hierarchies. In two experiments we studied the perceived beauty of a novel class of exact visual fractal in which we introduced reflection, rotation, translation, and random symmetries that repeated at a local and global levels. Rotation and reflection were consistently preferred to translation and randomness. Only reflected patterns were preferred at a vertical orientation. For all other symmetries there was no difference in preference between vertical and horizontal. In a second experiment we progressively eliminated the salience of local symmetry through opaque shading . Perceived beauty decreased with an increase in shading . For these patterns greater discriminability of their fractal quality makes them more aesthetically appealing.
A visual fractal is a pattern that repeats at different levels of spatial scale. The details of a fractal look similar at different magnifications. These patterns are pervasive throughout the natural world and can be found in the branching structures of rivers, in the contours of landscapes, in the flow patterns of clouds, and in the shorelines of continents. All fractals have several features in common, including scale invariance (Williams, 1997). This can be exact, as we see in some geometric fractals whose elements appear identical at each scale, or approximate, as we see in statistical fractals in which the elements are similar but not identical. Their complexity can be quantified with a measure known as the fractal dimension (D). The similarity dimension of a fractal refers to how much space it occupies. Simple fractals that deviate only slightly from a line occupy values just greater than one. Those that come close to filling up a two-dimensional space with more meandering contour will have a value closer to two. In addition, fractals are generated by the repetitions of an operation, which can be natural, such as erosion, or deterministic, such as the iteration of a mathematical equation.
A number of studies show that human observers find images with fractal statistics visually pleasing. When given a choice between fractal and non-fractal patterns, there is an overwhelming majority preference for fractals (Taylor, 1998, 2003). Sprott (1993) found that appealing evaluations were provided to patterns with a low to moderate fractal dimension. Aks and Sprott (1996) found aesthetic preferences correlated with a fractal dimension of 1.26. Spehar et al. (2003) found that observers displayed a consistent preference for fractal images regardless of whether they were generated by a natural process, by mathematics, or by human hand. Peak preferences in their study were found for a fractal dimension of about 1.3 for natural images and simulated coastlines and about 1.5 for paintings by the artist Jackson Pollock.
A question that arises is whether the preference for fractals in this range is universal or not. Street et al. (2016) found greater consistency in preference for mid-range D fractals but also found individual differences explained the variance in this responding. They additionally found that females had a higher preference (D = 1.6) than did men and attribute this to an evolutionary division of labor model in which foraging requires a greater understanding of complex visual scenes than hunting (Silverman et al., 2007). A universal preference for mid range fractals and the presence of individual differences based on ratings of pleasantness, interest, and complexity has also been found (Viengkham & Spehar, 2018).
One explanation for these preferences stems from the similarity between such images and natural scenes. Paintings and natural scenes in particular have been found to share similar image statistics, including power spectra distributed as 1/f2, sparse spatial structure, and similar edge co-occurrence statistics (Graham et al., 2006). Redies (2007) suggests that an artist creates a work of art so that it induces a specific resonant state in the visual system. This resonant state is based on the adaptation of the visual system to natural scenes. In this view, looking at a painting activates our brain the way it does when we are looking at natural scenes. The sense of naturalness induces a positive aesthetic quality.
Symmetry is a property of some fractals. In exact fractals, like Moore curves, Sierpinski triangles and H-fractals, this symmetry can be very similar or identical. In natural fractals it is usually statistically similar. The relation between different types of symmetry and perceived beauty in non-fractal patterns has been the subject of longstanding research attention. Some studies show that symmetric faces are judged as attractive (Rhodes, 2006; Rhodes et al., 1998), ostensibly because they signal immunocompetence, although this account has been disputed (Zaidel & Cohen, 2005). Bertamini et al. (2019) found a preference for bilateral symmetry for shapes and faces but not landscapes. Symmetric abstract graphic patterns are liked more than their asymmetric counterparts (Bauerly & Liu, 2008; Jacobsen & Hofel, 2003). Reflected shapes are preferred to rotated or translated ones and rotated shapes are preferred to random ones based on the Implicit Association Test (Makin et al., 2012).
The relationship between symmetry and fractal properties has not been as thoroughly investigated. Rainville and Kingdom (1999) looked at the ability to detect mirror symmetry embedded in fractal noise. Performance was facilitated in images with power spectra that characterize natural scenes. Modeling of their data show that humans are not ideal observers when detecting symmetry and do so only over a few cycles of spatial scale rather than the entire range. They conclude that this is probably an efficient strategy, as it reduces the associated neural costs. The fractal qualities of Rorschach inkblots with vertical mirror symmetry have been studied (Taylor et al., 2017). The number of induced images associated with these inkblots decreased with a fractal dimension ranging from 1.10 to 1.25. They found the same decreasing trend for computer generated fractals over a larger dimensional range of 1.00–2.00. The number of visual associations was thus greater to less complex visual patterns.
Bies et al. (2016) obtained linear preference trends that increased with dimension for exact midpoint displacement fractals with high degrees of symmetry. In a second study they used artificially generated fractals with asymmetry, reflective, and radial symmetry. These included dragon fractals, Sierpinski carpets and Koch snowflakes. At low levels of recursion (number of times the pattern repeats at different levels of spatial scale) and moderate to high dimensionality, symmetry drove high preference ratings. In contrast, the majority of responders needed much more recursion to like patterns that lacked mirror or radial symmetry.
In the current study we examine further this class of exact symmetric fractals (see Figure 1). They contain little noise or variability compared to natural fractals and are limited to two levels of recursion. Each pattern is made up of a kernel or seed pattern that can be subject to a symmetric transformation. The resulting symmetry then serves as a block or construction element that is used to build the same general pattern at a secondary more global level. The advantage of using such patterns is their simplicity. We can control for dimensionality and the exact type of symmetry, either random (asymmetric), translated, reflected, or rotated, with the same symmetry present at both scales. We can also vary the orientation of such patterns to examine whether vertical or horizontal placement of the symmetry axis affects aesthetic judgment. A number of studies show that detection of vertical bilateral symmetry is sometimes easier than for other orientations which may equate to a preference difference (Friedenberg & Bertamini, 2000; Palmer & Hemenway, 1978; Wenderoth, 1994).
Examples of Fractal Symmetry Patterns Used in Experiment 1 Shown for Each Type of Symmetry. The left column is vertically oriented, the right column shows the same example at a horizontal orientation.
Research on the perceived beauty of symmetric patterns tends to show a general trend in which reflection is preferred to rotation, which in turn is preferred to translated or random patterns (Friedenberg, 2018; Makin et al., 2012). However this work has typically used dot or contour stimuli that appear at one level of spatial scale only. Less studied is the case of symmetric fractals, where these symmetries repeat at several spatial scales. In a second experiment we reduced the salience of local symmetry by adding in opaque shading to our patterns. There are two alternate predictions for the second experiment. If simpler, global forms are preferred where the fractal or gestalt properties are less visible, and local symmetry is diminished, then patterns with increased shading will be preferred. If instead, more complex forms containing local and global symmetric structure are present and the fractal and gestalt nature of the stimuli are more salient, then patterns with decreased shading will be preferred.
Experiment 1
Method
Participants
Thirty-six undergraduates participated in order to fulfill a class requirement. There were 16 males and 20 females. Vision was normal or corrected to normal. Average age of the students was 19.2 years. All participants voluntarily agreed to participate and signed a consent form. American Psychological Association ethical standards and data confidentiality were adhered to. All materials used to conduct the research are available to other investigators for purposes of retesting.
Stimuli
Each global square pattern was created by a local square pattern. The local pattern in turn was made up of a 4 × 4 array of small squares. Each symmetry condition was created by taking half of this local array (2 × 4) and randomly filling it with black elements, resulting in four black and four white squares. This “seed” pattern formed the basis of each version. In the random condition the patterns in the two halves were generated independently of one another. In the translated condition, one half was the exact repeat of the other. For reflection it was a mirror image reflection and for rotation it was a 180° rotation (Figure 1).
The size of the global square pattern was 15 cm. The size of the local square pattern was 3.75 cm and the size of the smallest squares was 0.93 cm. A black border of two point line thickness surrounded the global square in order to make it more visible. each pattern was presented on a white background in the center of the computer screen and viewed at an average distance of 60 cm.
Each of the four symmetry conditions was presented at two orientations, vertical and horizontal. Vertical in this context was relative to the axis separating the seed pattern from its corresponding half. Horizontal was a 90° rotation of the vertical. There were eight different versions for each symmetry condition, each made from a unique starting seed. Thus in total there were four symmetries, two orientations, and eight versions crossed to yield sixty-four total stimuli (4 × 2 × 8 = 64). These constituted a single block of trials. Each participant viewed three blocks of trials, so there were 192 trials in an experiment session.
The dimensionality of a fractal can be calculated as: D = log LE/log (1/E), where D is the fractal dimension, L is estimated length and E is the length of the ruler or measuring unit. A log-log plot of these two produces a straight line. The slope of this line is equivalent to D. Shallow slopes have smaller D values and smoother, less meandering contours. Steeper slopes have higher D values and more convoluted complex contours.
We calculated the fractal dimension for each of the 32 patterns excluding orientation, since orientation does not change this value. This was done using the Fraclac plugin for the ImageJ analysis software. The bin counting method was used. In this technique a set of successively smaller square grids are overlaid on top of the image. The amount of contour in each sized grid at the different scales is counted. We set the initial size of the largest grid to be 45% of the total image size or region of interest, and a minimum size of two pixels. Image type was set to binary. Because the images were all generated in the same manner, their D values were similar with a small variance (M = 1.36, SD = 0.04). We are thus able to control for dimensionality by keeping it constant across conditions in this study.
Procedure
Participants were given as much time as they needed to respond. Rating scale judgments were obtained for each trial. Beauty judgments were made using the number scale that ran across the top of the computer keyboard. This scale ran from 1 to 7, with “1” labeled as a Little Beautiful, “4” labeled as Average Beauty, and “7” labeled as Most Beautiful. Participants were instructed to use any number from the entire scale. They were additionally told there was no right or wrong answer and that they could rate the patterns in any way they wanted. This was done to reduce any experiment-induced judgment criteria or demand characteristics. If any number other than 1–7 was entered the participant would not be able to advance to the next trial. If this occurred they were instructed to re-enter an appropriate value. On average, the experiment took about 15 minutes to complete.
Results and Discussion
The data were first screened for outliers. Any responses that took longer than 7 seconds were considered to be moments of inattention and removed prior to analysis. These constituted three percent of the tail end of the distribution. Beauty ratings were transformed using the formula (Score−Minimum)/(Max−Min) × 100, where Max was seven and Min was one. This transformation was done purely for ease of interpretation and does not affect the shape of the distribution or any of the assumptions of the statistical tests.
A two-way analysis of variance (ANOVA) was performed with effect sizes calculated as partial eta squared. There was a significant main effect of Symmetry F(3, 140) = 61.65, p < 0.01, ηp2 = 0.49. The main effect for Orientation was not significant F(1, 70) = 1.66, p = 0.19, ηp2 = 0.009, but the Symmetry by Orientation interaction was, F(3, 280) = 7.78, p < 0.01, ηp2 = 0.11). Figure 2 shows the bar plot of this interaction. As can be seen, there was no difference in orientation for random, translation, or rotation, but there was a statistically significant difference for reflection with the vertical orientation rated higher.
Mean Normalized Beauty Ratings for Each Symmetry Type and Orientation in Experiment 1. Error bars show ±1 standard error of the mean.
Looking at Symmetry, we see that rotation was preferred the most of all types, followed by reflection, translation, and random. The means and standard errors for these four conditions are shown in Table 1, along with a Tukey Least Square Means Test (Q = 2.59, alpha = 0.05). Rotation differed significantly from reflection. Reflection also differed statistically from translation, but there was no difference between translation and random.
Tukey Least Squared Means Differences HSD Test With Means and Standard Errors for Symmetry in Experiment 1.
Note. Levels not connected by the same letter are significantly different.
Based on the literature one might have expected reflection to have been preferred most. It is unclear why rotation was instead liked most of all. Our reflected patterns seem more unified or integrated between the local and global levels which may make them simpler in overall appearance. They are the only pattern we used that has complete global symmetry across the entire pattern. This global symmetry may make the local symmetry less noticeable, whereas in the rotated patterns the organization at each level of spatial scale maintains more of its distinctiveness. This interpretation suggests that patterns in which the fractal characteristics are salient are preferred. Those that are more of a unified gestalt and the fractal nature is less obvious may be liked less.
Another interesting result is that translation was not preferred more than random. If order or regularity is liked more than randomness, then translation should be liked more than random. Friedenberg (2018) found that reflected contours were judged more beautiful than 180° rotated contours, which were judged more favorably than translated or random contours, with no difference between these last two types, so this latter effect is replicated here. Also, it should be noted that the random patterns we use here contain order and are not truly random in two senses. First, the local square repeats itself in the two-dimensional plane and so has inherent translation. Second, there is scale symmetry, so the random pattern repeats at different levels of spatial scale. For these reasons our randomly generated patterns do have some degree of regularity. This probably explains why there is no difference in ratings between them and the translated condition.
For the interaction effect, the only difference was a preference for vertical over horizontal symmetry for reflection, a finding that has been found in previous research and which may be rooted in the primacy of vertical reflection in biology (Grammer & Thornhill, 1994). There was no difference between the 0° and 90° orientation conditions for any of the other three cases, explaining the absence of a main effect for Orientation. One might have predicted a vertical preference for translation because many friezes and decorative patterns repeat motifs horizontally, making this a more familiar type of repetition, but this was not the case. For rotation, there is no a priori reason to expect an orientation difference. Rotations are about a point, so shouldn't vary in terms of salience when the local pattern is spun about either orientation.
Experiment 2
Fractals are an example of hierarchical stimuli since they contain information at local and global levels (Robertson, 1996). Much work has been done on the attention to and identification of items embedded in such displays. A central tenet of such work has been the notion of global precedence, whereby global, low spatial frequency forms are accessed and processed before local higher spatial frequency items (Blanca et al., 2002). In a classic study by Navon (1977) identification of local letters is facilitated if they make up a larger letter that is identical (an H made up of Hs) and interfered with if it is different (an H made up of Ss). For our fractal patterns we can ask the question of whether there is a preference for the local or global level. If global precedence holds it means that the overall pattern is processed first and the local patterns secondarily. This could have implications for the judged aesthetics of such patterns.
In this second experiment we progressively shaded the patterns from the first study by filling in the empty spaces of the local square (see Figure 3). As the opacity of these empty squares increases, the salience of the local pattern diminishes while that of the global pattern increases. If observers prefer the global pattern then ratings will increase with shading. This might be expected based on global precedence. If the visual system evaluates global low spatial frequency information first, it might not need to perform any additional processing. On the other hand if local patterns are preferred then reducing their discriminability should lower beauty ratings. Opacity also has the effect of reducing the pattern's hierarchical character. It diminishes how noticeable the two levels are from each other. If viewers like the presence of both levels then we can predict a drop in ratings with increased opacity.
Examples of Shaded Fractal Symmetry Patterns Used in Experiment 2 Shown for Each Level of Opacity.
Method
Participants
Fifty undergraduates participated to fulfill a class requirement. There were 25 males and 25 females. No subject that participated in the first study was allowed to participate in the second, so they would remain naive regarding its nature and aims. Vision was normal or corrected to normal. Average age of the students was 19.3 years. All participants voluntarily agreed to participate and signed a consent form. American Psychological Association standards and data confidentiality were followed. Datasets, stimuli, and other experimental materials are publicly accessible.
Stimuli
The same patterns from the first study were used. Size, configuration and location of each pattern was identical. There were four types of symmetry or regularity (random, translation, reflection, rotation with eight variations each), presented again at a vertical and horizontal orientation. The only difference was that the unfilled squares in the smaller local pattern were progressively made less visible by shading them in to make them darker. In the zero condition (0.0, or 0%) these squares were white and thus exactly the same as the first experiment stimuli. In the 25% opaque condition (0.25) they were shaded by applying a one-quarter gray scale fill. In the 50% case (0.50) they were filled with a one-half gray scale fill. For the 75% (0.75) condition it was a three-quarter fill and in the 100% opaque condition the squares were completely black. In this latter case the local patterns were completely invisible and all that remained was the global pattern. The four symmetries, two orientations, eight versions, and five shadings crossed to yield 320 total unique stimuli (4 × 2 × 8 × 5 = 320). These constituted a single block of trials. Each participant viewed one block of randomized trials.
We next calculated the fractal dimension of the patterns using the bin counting method, utilizing the FracLac plugin for the ImagJ program set to grayscale. All other details for calculating the dimension were the same as in experiment 1. Adding shading effectively reduces dimensionality by removing the amount of contour in the pattern. The absence of any shading (0% condition) preserves all of the contour at the local level. These patterns are the same as those used in experiment 1 and serve as controls. They contain more edge contour but less black-filled surface area. For comparison, in the 100% shading condition only the global outline of the pattern is visible, the amount of total edge contour is reduced but the amount of black-filled surface regions is greater.
Procedure
Due to restrictions imposed by the global coronavirus epidemic it was not possible to test participants in person in a physical lab setting. This study was therefore conducted online, using Qualtrics software. Participants logged into the Sona system subject pool site and clicked on a link that started the program. They viewed and completed a consent form. Then they read an instructions screen. The instruction enforced a set of viewing conditions. Subjects were told they must perform the study using a laptop and not any other device. They had to be in an upright sitting posture in a chair and viewing the patterns at a distance of 60 cm, to preserve the same stimulus visual angle as before. Participants were instructed to adjust their computer monitors so that they were at the vertical upright and not tilted or rotated. They had to complete responding in a single session. They could not get up or take a rest break until the session was completed.
Participants were given as much time as they needed to respond on any given trial. Beauty judgments were made using the number scale that ran across the top of their computer keyboard. The scale ran from 1 – 7 with verbal labels for each numerical item the same as the instruction set in experiment 1. All other response conditions were also the same. Due to program restrictions reaction time measures were not obtained and participants were unable to complete a follow up questionnaire. On average, the experiment took about 15 minutes to finish.
Results and Discussion
All data were included in the analysis. Beauty ratings were transformed using the formula (Score−Minimum)/(Max−Min) × 100, where Max was seven and Min was one. A three-way analysis of variance (ANOVA) was performed. There was a significant main effect of Symmetry F(3, 147) = 15.51, p < 0.01, ηp2 = 0.24. The main effect for Shading was also significant F(4, 245) = 4.34, p = 0.01, ηp2 = 0.06. As in Experiment 1, Orientation was not significant, F(1, 98)) = 2.70, p = 0.10, ηp2 = 0.01. None of the two-way or three-way interactions were statistically significant. Figure 4 shows the bar plot of the main effect of Symmetry. The general ordering of the means from most to least preferred was rotation, reflection, random, then translation. The two most preferred types, rotation and reflection, did not differ from each other. Neither did the two least preferred types, random and reflection. Table 2 shows the means and standard errors along with the differences based on a Tukey Least Square Means Test (Q = 2.56, alpha = 0.05).
Mean Normalized Beauty Ratings for Each Symmetry Type in Experiment 2. Error bars show ±1 standard error of the mean.
Tukey Least Squared Means Differences HSD Test With Means and Standard Errors for Symmetry in Experiment 2.
Note. Levels not connected by the same letter are significantly different.
Although rotation was preferred the most it did not differ statistically from reflection based on the post-hoc means test. This result differs from the first study. It is not clear how to account for this difference. However, we can conclude that rotation and reflection are preferred more than random and translation, since this finding is consistent across both studies. The absence of a shading interaction suggest that this difference is not due to the introduction of opacity.
Figure 5 shows the bar plot for the main effect of Shading. There is a general downward trend, with beauty ratings decreasing with an increase in opaque shading. The effect based on relative mean ordering is monotonic. Table 3 shows the Tukey test results (Q = 2.72, alpha = 0.05). The zero (0.0) and one (1.0) levels of opaqueness are statistically different, the 0.25, 0.5, and 0.75 conditions do not differ. This implies that observers like patterns where the organization at both levels of spatial scale is most visible. There was no support for the global precedence hypothesis. Patterns with global or equivalently just one spatial level of information were liked the least.
Mean Normalized Beauty Ratings for Each Level of Shading in Experiment 2. Error bars show ±1 standard error of the mean.
Tukey Least Squared Means Differences HSD Test With Means and Standard Errors for Shading (Gray Scale Opacity) in Experiment 2.
Note. Levels not connected by the same letter are significantly different.
Fractal dimension decreased with increased opacity. Table 4 shows the mean and standard deviations for the five levels. We then correlated ratings with dimensionality. There was a negative but non-significant correlation of judged beauty with decreasing fractal dimension, r(245) = 0.43, p = 0.06. The trend of this finding at least is in line with previously cited work showing peak preferences for values around 1.3. If we take edge length as a proxy for complexity, then observers are preferring more complex stimuli. However it should be noted that opacity also increases the proportion of the total pattern with filled black areas. This corresponds to a larger pattern in terms of surface area with fewer distinct numbers of parts.
Fractal Dimension Statistics for the Experiment 2 Patterns Based on Shading (Gray Scale Opacity).
General Discussion
Most previous studies of symmetric preference have been for patterns where the symmetry occurs at one level of spatial scale only. In two experiments we studied the perceived beauty of a new class of hierarchical symmetric patterns. These patterns are characterized by the presence of the same symmetry at a local and global level. We found that for these fractal symmetries reflection and rotation are preferred to translation and random, reproducing and generalizing this finding from previous research with patterns at a single scale. In Experiment 1 fractal dimension was controlled for and equivalent across conditions. In Experiment 2 dimensionality drops with increased opacity an indicator of decreased fractal complexity. Ratings generally trended downward with dimensionality suggesting a complexity preference.
Nucci and Wagemans (2007) independently varied the number of local and global axes in reflectional dot symmetries from zero to two. They measured the amount of time it took to perceive regularity in the patterns (symmetry detection). The results showed that the greater the number of global axes, the shorter the reaction time and hence the easier the task. Number of global axes corresponded well to a calculated holographic measure of goodness, roughly equivalent to the Gestalt notion of pragnanz or “good figure”. The effect was found for both vertical and horizontal orientations. Although their patterns were not true fractals, the results have some applicability here. Fractals may be considered a type of “good figure”. They introduce correspondences between local and global structure that can unify and strenthen perceptual salience. Our rotated and reflected patterns create a more holographic organization in the overall figure than do translation and random. These last two types of symmetry produce more fragmented global structure. For translated fractals there is regularity in the form of repetition between sections of the overall figure. Similarly, for random fractals one can map local patterns onto one another but in a more patchwork way without any stronger unification of the whole.
We also examined the orientation of the patterns and found that with the exception of reflection, it did not affect beauty judgments. Reflection may be an exception because it has biological and evolutionary significance. Faces and bodies more often have a vertical bilateral axis and so can signal health, immunocompetence, and attractiveness (Grammer & Thornhill, 1994). Human faces and bodies also have fractal reflective symmetry. All of the major features of a face, including the eyes, nose, and mouth, have bilateral symmetry and are embedded within the larger bilateral symmetry of the head. Breasts, genitals, and other body features are also examples of local features that are reflective inside of the global reflective symmetry of the body. The human visual system may thus have evolved to detect and evaluate the correlations between fractal reflective symmetries for evaluation of sexual fitness. However, this evolutionary explanation applies only in the case of vertical reflection. It does not predict that rotation would be preferred over reflection. Also, preference for symmetry is likely influenced by both individual differences and education. Leder et al. (2019) for instance, found that art experts preferred asymmetric and simple shapes in comparison to non-art experts.
In experiment two, we introduced levels of opacity to reduce the discriminability of local symmetry and hence the pattern's hierarchical or fractal characteristics. Patterns where this discriminability was greatest were rated most beautiful. Global only patterns without any perceivable fractal features were rated the lowest. Shaded patterns fell in-between these two extremes. It should be noted that contrast, or a sharp luminance edge by itself was not a predicter of ratings. If participants had preferred contrast then ratings would have been highest for 0% and 100% shading, and lower for the intermediate 25%, 50%, and 75% conditions. The monotoic ordering of means instead suggests that it is the disruption in salience between local and global levels that matters.
Global precedence predicts that simple patterns without local details might be liked more because they require less computation and are easier to assess. However this was not the case. These patterns were in fact judged the least beautiful. Global precedence it turns out cannot account for a number of the findings in the hierarchical literature (Castillo et al., 2015). It is weakened or absent for displays with a small number of large elements (Kimchi et al., 2005). Also, local elements can be detected more easily when they are part of a global order, suggesting interactions between levels rather than a strict “top-down” approach (Burack et al., 2000). For fractal patterns it seems likely that both levels are processed in parallel or that there are feedback loops between them. This appears to enhance aesthetic judgements but the neural mechanisms behind this effect are as of yet unclear.
Our findings suggest that we prefer more complex fractal patterning where there is structure at multiple levels of spatial scale. This makes sense given that fractal patterns are ubiquitous throughout nature and that humans probably evolved to process this information in landscapes, faces, and bodies. It is well-established that the visual system processes images at different spatial frequencies in parallel and that communication between these levels is used to facilitate shape and pattern recognition (DeValois & DeValois, 1988). It may be that the representation of symmetries at spatial scales creates a state of resonance. This process might occur through neural synchrony and the result could be a pleasurable or positive hedonic sensation.
The results of Experiment 2 demonstrate an effect of symmetric redundancy. Patterns that are more similar to one another at different scales may reduce processing load by facilitating synchrony or correlational activity during spatial frequency analysis. This increased efficiency of neural computation could induce a sense of aesthetic appreciation. One could explore this idea further by varying how similar the symmetries at each level are to one another. On the other hand though, it might be the degree of difficulty in determining this two-part structure that best explains the results. This possibility is explored next.
One way to interpret hierarchical stimuli is according to the predictive coding model (Van de Cruys & Wagemans, 2011). In this view, the visual system generates a prediction or expectation of what it will see. This is then matched against the incoming input. The discrepancy between the two indicates the level of perceptual surprise that is experienced. These predictive errors are interpreted as positive when they are resolved and stability is reinstated. In fact, many artists have been shown to deliberatively introduce incongruities that are then resolved by the viewer to produce a sense of pleasure. The authors determine that too little discrepancy is boring and that too much is exhausting. Therefore, there is an intermediate level of discrepancy that produces an optimal level of surprise which will in turn produce the greatest level of aesthetic appreciation. This level probably differs across individuals and is affected by a number of stimulus factors.
Muth and Carbon (2013) presented observers ambiguous two-tone images. They found that the greatest increase in liking occurred when participants realized a pattern contained a face. They called this an “aha” moment. The study suggests again that resolving an ambiguity causes increased aesthetic appreciation.
In our patterns the surprise or “aha” moment would occur when viewers realize there is emergent organization. We did not formally ask our participants whether they perceived this. However, the shading variable in the second experiment provides some insight into how the predictive coding model might apply to our stimuli. Increased shading disrupts perceptual discriminability of the local patterns making it harder to detect hierarchical structure and experience surprise. If we assume that detecting this global gestalt is inherently difficult then the least amount of shading would provide the greatest opportunity to discover fractal characteristics and experience surprise. This is congruent with the Experiment 2 results. This conjecture however cannot be evaluated in the present experiment because we did not calculate any type of emergence metric for our patterns. The determination of such a metric could be used to measure how easy it is to perceive a pattern's gestalt qualities. In future work, we could use this metric to predict fractal recognition and beauty ratings.
A last set of comments concerns philosophical approaches to the study of beauty using patterns like ours. Makin (2017) argues that the traditional methods of psychophysics are not adequate to the study of empirical aesthetics. One of his assumptions is that stimuli are multi-dimensional and that when changes are added to a pattern they alter the perceived whole. In other words, pattern perception is innately holistic and interdependent. He calls this the “Gestalt nightmare”. Skov and Nadal (2019) counter this argument by noting that perceptual experiences are not entirely driven by object properties but also by top-down factors. In our second experiment we obtained an example that counteracts the so-called “Gestalt nightmare”. When shading was added to our patterns it did not in fact interact with our other variables. The two-way ANOVA showed that shading did not interact with symmetry, orientation or the symmetry by orientation effect.
However, we would argue that even in cases where stimulus properties do interact they can still be studied using the traditional methods of psychophysics and experimental psychology. The interaction terms in statistical tests can be used to assess the degree of independence for two or more variables and also provide clues as to why they vary, based on the interaction plots. These interactions can then be understood further by additional testing. Methodology is also useful. As Skov and Nadal (2019) mention, methods in neuroscience help decode the mechanisms that occur between object properties and preference. For a long time it has been recognized in science that reductionism is only one tool toward understanding natural phenomena, much of which is holistic and interactive. But now there are a wide array of techniques in psychology and other areas that can be used to assess complex dynamical systems. These include stochastic, oscillatory, cellular automata, agent-based modeling, and chaotic computing (Friedenberg, 2009).
Footnotes
Declaration of Conflicting Interests
Funding
ORCID iD
