We used four previously published and two novel datasets of images in our analysis. For an overview of the datasets, their origins, and a reference to exemplary images, see Table 1.

The following four dataset have been published previously: (i) The Jenaesthetics dataset (Amirshahi et al., 2015; Hayn-Leichsenring et al., 2017) contains 1,629 high-quality images of oil paintings of Western provenance that were made available by art museums on the Wikimedia Commons webpages (Google Art Project; set no. 1 in Table 1). The dataset comprises traditional works from art periods that extend from the Renaissance to Expressionism, but no (post-)modern or contemporary art. The paintings depict various subject matters (for example, urban scenes, landscapes, seascapes, architecture, portraits, still lives and nudes). (ii) A similarly diverse dataset of 185 graphic artworks (monochrome works on paper; set no. 2). This dataset corresponds to a previously published dataset of 200 graphic artworks of Western provenance (Redies et al., 2007), but without the 15 examples of (post-)modern and contemporary art that were contained in the original dataset. To extend our study to traditional artworks from other cultural backgrounds, a dataset of (iii) 238 images of Islamic book illustrations (set no. 3) and (iv) a dataset of 215 images of traditional Chinese color paintings (set no. 4), both from different centuries, were downloaded from the Wikimedia Commons webpages and included in the analysis (Redies et al., 2017). Note that the four datasets of traditional artworks represent a large variety of artistic techniques (oil paintings, prints, drawings, watercolors etc.).

The Bad Art dataset (set no. 5) contained 244 digitized reproductions from the MOBA collection, kindly provided by its curator, Mr. Michael Frank. In addition, we downloaded 44 images of artworks from the official web collection of the OBAMA³. Only high-resolution images that did not show obvious artifacts (blurring, JPEG artifacts, reflections, etc.) were included in the analysis. If present, frames around the images were removed.

From the website of the Tate Gallery⁴, 474 images of twentieth century abstract art of Western provenance were downloaded between October 22 and November 13, 2016 (set no. 6). Images of similar abstract art were also obtained from the Kunstsammlung Nordrhein-Westfalen (set no. 7). Fifty-four images were downloaded from the collection's website⁵ between November 6 and December 21, 2016, and another 44 images were photographed by one of the authors previously (Redies and Gross, 2013). We used only high-resolution images and restricted the number of artworks by individual artists to avoid overrepresentation of particular types of abstract art. A preliminary analysis showed that the datasets of abstract art images from the two museums did not differ in the statistical values calculated in the present study. We therefore merged the images of abstract art into a single dataset of 572 images for further analysis.

We determined four statistical image properties by following previously published procedures (Amirshahi et al., 2012; Braun et al., 2013; Redies et al., 2017). Briefly, the properties were calculated as follows:

Non-parametric tests were used throughout the analysis because the values for most measures were not normally distributed. Means for different image categories were compared by a one-way ANOVA (Kruskal-Wallis test, followed by Dunn's post-test). A level of p < 0.05 was considered significant. Effect sizes (η²) of differences between pairs of image categories were calculated by the Wilcoxon rank-sum test. In the box plots, the whiskers bracket 5–95% of the data.

Figure 1 shows box plots for the four image properties for all datasets of artworks. Mean values and standard deviations are listed in Table 2. We observed significant differences between the six image categories (Kruskal-Wallis test) for the fractal dimension (H = 423, df = 5, p < 0.0001), self-similarity (H = 471, df = 5, p < 0.0001), 1st-order entropy (H = 278, df = 5, p < 0.0001) and 2nd-order entropy of edge orientations (H = 544, df = 5, p < 0.0001). Effect sizes (η²; Wilcoxon rank-sum test) for the differences between the individual image categories for the fractal dimension and self-similarity are listed in Table 3, and for 1st-order and 2nd-order edge entropy in Table 4. In Figures 2, 3, the values for 185 randomly selected images from each dataset are plotted in 2d scatter diagrams.

The median values of the fractal dimension (Figure 1A and Table 3) for Bad Art and abstract art are lower than for all of the four traditional art styles. Differences are observed also between the traditional art styles (Kruskal-Wallis test; H = 195, df = 3, p < 0.0001). The highest median value is observed for graphic art. Bad Art does not differ from abstract art in this measure.

Self-similarity (Figure 1B and Table 3) is also lower for Bad Art and abstract art than for the traditional art categories. The traditional art styles differ from each other (H = 186, df = 3, p < 0.0001), except for Western oil paintings and Chinese paintings. Self-similarity values are highest for the graphic artworks and the Islamic artworks.

For 1st-order and 2nd-order entropy of edge orientations (Figures 1C,D and Table 4), values of abstract art are widely scattered, ranging from very low to near maximal values. Mean values of 1st-order edge entropy for Bad Art are slightly lower than those for the Islamic and Chinese art but similar to the mean values for the oil paintings and graphic artworks. Mean values of 2nd-order edge entropy are higher for Bad Art than for abstract art but lower than the mean values for all categories of traditional art. Some of the traditional art categories also differ from each other (1st-order entropy, H = 28, df = 3, p < 0.0001; 2nd-order entropy, H = 182, df = 3, p < 0.0001).

The scatter diagrams visualize these findings in more detail. For traditional artworks, data for the fractal dimension and self-similarity are concentrated toward the upper right corner of the plots (Figures 2A–D); values are high for both measures and the data points overlap extensively (Figure 2G). Bad Art and abstract art (Figures 2E,F,H) also overlap with traditional art, but a larger proportion of data points scatters toward the lower left corner of the plots, i.e., they show lower values for both measures.

Similar observations can be made for the scatter plots of 1st-order and 2nd-order edge entropy (Figure 3). Close to maximal entropy values are found for all four categories of traditional art (Figures 3A–D,G). For 1st-order edge entropy, the data points of Bad Art (Figures 3E,H) overlap with the traditional art categories, especially with Western oil paintings and graphic art. For 2nd-order edge entropy, there is also a considerable overlap, but a larger proportion of Bad Art data points are low compared to the other art styles (Figures 3E,H). For abstract art (Figures 3F,H), we again observe some overlap, but even more data points are found at lower values for both 1st-order entropy and 2nd-order entropy (Figures 3F,H).

In conclusion, there is a considerable overlap between all six art categories. However, Bad Art and, even more so, abstract art differ in the measured average image properties from most of the traditional art styles. Moreover, the dataset of abstract artworks contains images that scatter widely in their values for 1st-order entropy and 2nd-order entropy of edge orientations (Figures 1C,D; see also S.D. values in Table 2).

To quantify the overall differences between the artworks of the different categories, we calculated their Mahalanobis distances to reference datasets in the 4d space that is spanned by the image properties studied in the present experiment. Specifically, we calculate the distances of each image in a given category to the median of one of the other categories (here called the reference dataset) by a pairwise comparison between image categories. Note that we can calculate the Mahalanobis distance, as a baseline, also between images of one category and the median of that same category. Because the values for none of the image categories passed the D'Agostino and Person omnibus normality test, we calculated median values and median absolute deviations (M.A.D.s) for the Mahalanobis distances (Table 5).

Figure 4 shows box plots of the results. For the left-hand part of the figure, the reference dataset are the Western oil paintings. Distances of all four traditional art datasets to the Western oil paintings are shown. Western graphic art has distances larger than the median distance of oil paintings to themselves, whereas the distances of Islamic and Chinese art to the Western oil paintings are of similar magnitude. For the right-hand part of the Figure 4, a dataset of 185 images that were randomly selected from each of the four traditional art categories (740 images in total; dataset no. 8 in Table 1) served as the reference dataset (red box plot in Figure 4). On average, distances are larger for Bad Art and, in particular, for the abstract artworks than the distances of images in the traditional artwork dataset to itself. Mean distances for all possible pairwise comparisons between the six original datasets are summarized in Table 5.

As a measure of the dispersion of the distances, we calculated the median absolute deviation of each distance to the median distance of the reference dataset for each comparison. Results in Table 5 confirm that the deviations are larger for the comparisons of Bad Art and abstract art to the traditional artwork categories, respectively, than for the comparisons between the traditional artworks (Figure 4).

In conclusion, as described previously (Redies et al., 2017), we find that, to some extent, artworks of Western, Islamic and Chinese origin share a specific pattern of statistical image properties. The image properties of Bad Art and abstract art considerably overlap with this pattern. At the same time, a large number of Bad Art and abstract artworks deviate considerably from the traditional art categories; these deviations are more pronounced for abstract art than for Bad Art.

To get a better visual grasp of some of the differences described above, example images are shown for traditional art in Figure 5, for Bad Art in Figure 6, and for abstract art in Figure 7. Proceeding from the upper left part to the lower right part of the figures, images are shown for Mahalanobis distances to the center of traditional artworks (red in Figure 4) above 5, then around 5, to around 2 and 1, which is close to the median distance of traditional artwork to themselves.

Of the most deviant traditional artworks, the Manet painting (Figure 5A), shows a prominent railing in the background with bars that are oriented in parallel and result in low entropy values. The Degas painting (Figure 5B) and the Chinese portrait painting (Figure 5E) display large homogeneous areas that are delineated by long and relatively straight edges, resulting in a low fractal dimension and low 2nd-order edge entropy, respectively. In the van Gogh drawing (Figure 5C), lines are predominantly oriented in parallel (low 2nd-order edge entropy). In the Islamic book illustration (Figure 5D), cardinal orientations prevail (low entropy values). These characteristics set the images apart from the traditional artworks with more typical values (Figures 5F–N).

Images of Bad Art (Figure 6) can differ from traditional artworks with respect to several statistical properties. For example, a low fractal dimension, which corresponds to the impression of low image complexity, is observed in the images shown in Figures 6A,D,E,G. In some other images, the prominence of particular orientations manifests itself in low values for edge entropy (Figures 6A,B,F,G). Examples of artworks with values similar to traditional artworks are shown in Figures 6I–O.

For some examples of twentieth century abstract art of Western provenance (Figure 7), deviating values can be directly related to their style and visual appearance. For example, the painting by Kenneth Noland (Figure 7A) shows close to maximal 1st-order entropy of edge orientations because all orientations are present at equal strength due to the strictly circular structure of the painting. At the same time, 2nd-order edge entropy is very low because of its high co-circularity (Sigman et al., 2001). In the paintings depicted in Figures 7B–E, both 1st-order entropy and 2nd-order entropy are low because the range of orientations is rather restricted. This deviation from traditional artworks is particularly evident in the works by Piet Mondrian (Figure 7C) and Josef Albers (Figure 7D), which consists of horizontal and vertical orientations only. Self-similarity is low in paintings that exhibit large homogeneous areas (Figures 7A–C,E,F). There are also many abstract artworks with values well within the traditional art category (Figures 7I–M), including works by Joan Miró (Figure 7J), Paul Klee (Figure 7K), and Wassily Kandinsky (Figure 7L). Interestingly, the drip paintings by Jackson Pollock (Figure 7H) resemble traditional paintings, except for a fractal dimension and self-similarity that are higher than for most other abstract and traditional artworks.

The present work confirms previous results that large sets of traditional artworks share specific statistical image properties on average. Examples for common properties are the Fourier spectral slope (Graham and Field, 2007, 2008; Redies et al., 2007; Melmer et al., 2013; Mather, 2014), self-similarity (Hayn-Leichsenring et al., 2017), edge orientation entropy (Redies et al., 2017) and variance of filter responses (Brachmann et al., 2017). Similarities have been found both across cultures (Western, Islamic and Chinese) (Graham and Field, 2008; Melmer et al., 2013; Redies et al., 2017), for different genres of Western Art and for different content depicted in artworks (Redies et al., 2007, 2017; Hayn-Leichsenring et al., 2017). A comparison of traditional artworks with various non-art images (natural or artificial objects, patterns and scenes) confirmed that the image properties are characteristic for traditional artworks. These findings are in line with the general suggestion that visual artworks represent stimuli with specific perceptual properties that are neither arbitrary nor random (Fechner, 1876; Bell, 1914; Dowling, 2014; Redies, 2015; Renoult et al., 2016). These properties might contribute to the “significant form” that has been postulated by Bell (1914). However, there are varying degrees of overlap with the properties of natural patterns and scenes (see Introduction). Such similarities provided the basis of ideas on how the perception of artworks is linked to the processing of natural scenes and objects, to which the visual system is adapted in evolution and development (Orians, 1986; Hodgson, 2006; Redies, 2007; Graham and Meng, 2011; Taylor et al., 2011). These ideas should be viewed as complementary, not as contradictory (Hodgson and Verpooten, 2015; Redies, 2015) to concepts that view visual art as a cultural phenomenon wherein contextual and conceptual factors play a central role (Dickie, 1974; Danto, 1981). Despite these overall similarities across cultures and art genres, there is a considerable inter-individual variability in the preference for statistical image properties within groups of people of similar cultural background, both for art and non-art stimuli (for example, see Jacobsen, 2004; Mallon et al., 2014; Bies et al., 2016; Güclütürk et al., 2016; Lyssenko et al., 2016).

The present results confirm our initial expectation that (1) there is a large overlap between all six art genres investigated in the present study, and (2) many images of Bad Art and (post-)modern abstract art of Western provenance differ in their statistical properties from traditional art from prestigious museums. As a quantitative corroboration of these findings, we calculated the Mahalanobis distances between the artworks in the multidimensional space that is spanned by the four image properties. In this space, the distances from the Bad Art and abstract art images to the traditional artworks are larger on average than the distances of the traditional artworks to themselves (Figure 4). Moreover, the scatter of the values for Bad Art and abstract art is large compared to that for traditional art (Table 5).

In the case of Bad Art, the deviating examples (Figures 6A–H) show several stylistic peculiarities not found in most traditional artworks, such as a predominance of particular orientations (resulting in low edge orientation entropy), large homogeneous areas (low self-similarity) and low complexity (low fractal dimension).

For some images of twentieth century abstract art, similar deviations were observed (Figures 7A–G). For example, the strictly circular painting by Kenneth Noland (Figure 7A) exhibits a high degree of co-circularity, which is a regularity that diminishes the 2nd-order entropy of the edge orientations (Geisler, 2008). The paintings by Piet Mondrian (Figure 7C) and Josef Albers (Figure 7D) are deviant because they contain horizontal and vertical lines exclusively (low edge orientation entropy) as well as large homogeneous areas (low complexity and self-similarity).

We speculate that the deviations in some Bad Art images might possibly arise because the artists lack training or talent in traditional art styles. As a consequence, the artists may fail to (or do not intend to) accomplish an image structure that is similar to that of prestigious traditional artworks. For abstract art, the deviating values in some images correspond to stylistic peculiarities (for example, see paintings in Figures 7A,C–F). We thus suggest that some (post-)modern artists transgressed the boundaries of traditional artworks, as defined in the present work by the four statistical image properties. This conclusion is in agreement with common art-historical and philosophical interpretations of (post-)modern art (for example, see Dickie, 1974; Danto, 1981), but it supports this notion with objective and quantitative evidence from a computer-based analysis of a large set of artworks.

However, not all stylistic specializations in abstract art result in deviations from the statistics of traditional artworks. Examples are the drip paintings by Jackson Pollock who created paintings with a highly complex, fractal (self-similar) structure (Taylor, 2002; Taylor et al., 2011). Although the resulting values for self-similarity and fractal dimension are higher than for traditional artworks, the other values for his painting remain within the range of traditional artworks. Importantly, the Mahalanobis distance to traditional artworks is relatively low (Figure 7H). Other artworks that, despite their distinctive abstract styles, have a spectrum of values similar to traditional artworks include paintings by Joan Miró (Figure 7J), Paul Klee (Figure 7K), Wassily Kandinsky (Figure 7L), Ernst Nay (Figure 7M), and others (not shown). We thus conclude that, as far as we can tell from the image properties measured in the present study, not all (post-)modern abstract artists abandoned the characteristic image structure of traditional art. On the contrary, the specific pattern of image properties in traditional art, which has prevailed for centuries and across different cultures, extends well into our modern times (Redies, 2014). Whether this pattern is necessary or sufficient to elicit the perception of visual beauty - or any particular aspects of visual aesthetics, such as liking, harmony, interest, aesthetic appeal, “good Gestalt” or “visual rightness”—remains to be studied by experimental means.

The present results are important because they provide, to our knowledge for the first time, an objective basis for a scientific comparison of traditional artworks that differ in their aesthetic claim, and for a study of the formal similarities and differences between the various types of traditional art and abstract art. Formal image analyses may complement and inform philosophical and art-historical analyses, which focus on contextual and cultural aspects of artworks (see Introduction). Formal image properties, such as 2nd-order edge entropy, are difficult to grasp and hard to describe in every-day terms. We propose that their investigation can provide access to an intuitive artistic knowledge that is based on perceptual mechanism and is largely inaccessible to cognitive introspection. To date, this hidden knowledge may have escaped the intellectual dissection of different art genres by contemporary art historians and art critics. Evidently, more research is needed to relate formal image statistics to art-historical knowledge, especially for (post-)modern and contemporary art.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.