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Submission. Direct PNAS Board. a is article This interest.y competing no declare authors The ...... n ...aaye aa n ...... n S.K.H. and H.J., M.S., D.K., B.L., and data; and tools; analyzed B.L. reagents/analytic S.K.H. paper. new research; and the contributed wrote designed H.J., S.K.H. M.S., S.K.H. and M.K.S., and I.-s.S. B.L., H.J., research; M.S., performed M.K.S., M.K.S. B.L., contributions: Author characterizing reproduce authenticate to to (26–28); dates 22–25), evolution creation (3–7, the estimate representation and quantify and recent to fre- style applied artistic Moreover, and of been (17–21). spectrum, have space analyses power color statistical as Fourier in such distributions the paintings to quency dimension, of contributed fractal far properties the so statistical art (16). diverse visual insights characterize of previous disproving assessments qualitative statisti- complementing and Computational and arts, validating quantitative visual both the apply research study and to develop methods to cal researchers able (13–15), paintings been of have scans digital large-scale of bers time?” over evolve principles organizing such painting?,” do in cultur- “How principles and there design “Are transcendent temporally 12): and (11, to ally questions framework long-standing quantitative an a two Addressing develops answer far. study few com- so this a presented spatial challenge, been open the only have on paintings Meanwhile, studies of fashion. position macroscopic of and such quantitative out before large particularly fell aesthetics analyses and history formalist art of fields the in owo orsodnemyb drse.Eal jogkiteuor [email protected] Email: addressed. be may correspondence [email protected]. whom work.y To this to equally contributed M.K.S. and B.L. eands a oitiiet rae audience. broader a which to suggest- nonintuitive concepts, far historical periods, so art remained style of supergroups conventional meaningful and ing periods, time abso- artworks, reveals lute artist’s individual distributions of similarity structures clustering of over clear analysis evolves histori- network systematically The the paintings time. within landscape of proportion ography compositional that uncovers by preferred art Western paintings the represent- of paintings historiography landscape canonical landscape the 14,912 ing dissecting of analysis for coherent The artists. used yet simple as a prin- proportion compositional propose these captures We that how framework time. yes, information-theoretic over if and evolve are art ciples the there within in if principles characteristics is organizing transcendent aesthetics temporally and and art culturally in question foundational A Significance hnst h eetpoieaino nrcdne num- unprecedented of proliferation recent the to Thanks y b aiiinSchich Maximilian , y y b eateto hsc,Cugu National Chungbuk Physics, of Department . y raieCmosAttribution-NonCommercial- Commons Creative d,e aon Jeong Hawoong , . y https://www.pnas.org/lookup/suppl/ NSLts Articles Latest PNAS d utrlData Cultural a,f,2 | , f11 of 1

APPLIED PHYSICAL SCIENCES styles of specific artists (29), and to classify the systematic nov- consensus of the rhizomatic metanarrative of landscape painting elty of artists (30). Moreover, beyond the characterization of that, however, remains so far invisible except to the connoisseur artworks, art historical metadata including exhibition trajectories who is familiar with the corpus as a whole and who by coin- and auction price history shed new light on the dynamics behind cidence has been trained using the identical corpus, which of the careers of artists (31–33). course, is highly unlikely. As such, our study reveals the metanar- Some researchers devised quantitative measures for visual rative inherent in the chosen dataset to a broad multidisciplinary characteristics of artworks using concepts of information theory, audience while offering a benchmark or null model for further including an area aptly titled critical and creative aesthetics as research, qualitative and quantitative, including research dealing information processing (34). with originals and reception aesthetics. A notable departure of study in computational aesthetics dates We choose landscape paintings as the scope of our study for back to 1933 when the American mathematician Birkhoff (35) two reasons. First, landscape paintings more often consist of conceptualized a quantitative aesthetic measure to understand clear horizontal or vertical components compared with other the order and complexity of artworks (36, 37). He regarded paintings such as portrait, still-life, or abstract paintings. For beauty as a mathematical phenomenon and introduced an aes- instance, landscape paintings often have a horizontal boundary thetic measure M , defined as the ratio between “order” O and between a foreground and a background (or possibly, a middle “complexity” C , where O and C were measured based on the ground) or vertical frames composed of trees, cliffs, or build- number of structural regularities and elements of an artwork, ings. Thus, analyzing landscape paintings using the currently respectively. It is this sense of aesthetics aiming to formalize available informational partitioning algorithm has advantages of “orderliness” in artworks that has since driven further develop- directional simplicity of elements in the composition, which also ments in the general theory of computational aesthetics, rather facilitates the interpretation of results. Second, the colors used than the difficult or even evasive concept of “beauty,” which is in the subregions of landscape paintings are often more dis- rooted in older literature (Baumgarten) and is more familiar to tinctly separated than in other genres of painting, resulting in a broader audience. the extraction of larger mutual information. To prove this point, Starting from Birkhoff’s original idea, Bense (37) and Moles we provide comparative results throughout this paper, looking at developed informational aesthetics and formulated the order landscapes and abstract paintings, using an additional auxiliary and complexity of an artwork in terms of redundancy and large-scale dataset (SI Appendix, section I). Shannon’s notion of information (38). Bense (37) considered a Analyzing 14,912 landscape paintings, we specifically address creative action in painting a transition process from an initial the following questions. Arising from the fact that the historiog- state (palette) to a final color distribution on a physical support raphy of landscape painting is often subdivided by nationality, (canvas). Inspired by Bense’s idea, Rigau et al. (39) considered are there nationally specific or transcendent characteristics in Bense’s creative process an information channel between a color the composition of landscape paintings, such as Dutch flatland palette and an image’s spatial regions. They presented a greedy vs. Swiss mountains? What are the characteristic frequencies algorithm that progressively partitions an image into quasihomo- of compositional proportion employed by individual landscape geneous regions by extracting the mutual information between painters in general? How do the characteristic, perhaps domi- color and regions in each step of partition until the painting nantly used compositional proportions change over time? Can is completely revealed. The procedure takes a full image as a we use the frequency distributions of compositional proportion unique initial partition and progressively subdivides it into a ver- as a signature to characterize diverse conceptual groups in art tical or horizontal direction by a line that gives the maximum history such as plain time spans of creation dates, conventional mutual information. Therefore, during the partitioning process, style periods, or individual artists’ oeuvres? information acquisition increases, and the uncertainty of color In the following sections, we show that the distribution of over regions reduces. The information acquisition rate over each compositional proportion successfully captures distinguishing partition step was used to define the painting’s degree of order compositional properties of artists and conventional style peri- (39). More recently, Shin et al. (40) developed a faster algo- ods in art history as recorded in our dataset (i.e., arguably in line rithm for the partitioning procedure (SI Appendix, section II with public consensus). We also demonstrate that there exists has a detailed explanation of the algorithm) so that analyzing a systematically changing trend in the compositional propor- a massive set of paintings is now feasible in a much shorter tion distribution of paintings that allows us to unveil temporally timescale. clustered structures above the level of the life of artists and tran- In this study, we employ Rigau’s dissection algorithm, rooted scending the duration of conventional style periods in landscape in Bense’s information theoretic concept regarding the process painting. of painting, to present a systematic framework assessing the compositional proportion of landscape paintings as an essen- Materials and Methods tial organizing principle. Analyzing a large set of digital scans Datasets. Digital scans of landscape paintings were collected from two of landscape paintings, we reveal the emerging macroscopic major online sources: WikiArt (WA) (13) and the Web Gallery of Art (WGA) trend of frequency distributions of compositional proportions as (14). WA landscape paintings were collected in October 2018 with a total of 12,431 paintings by 1,071 artists assigned to 61 nationalities. WGA data implied in a dataset of 14,912 paintings by 1,476 painters, cov- were collected in May 2016 with a total of 3,610 paintings by 816 artists ering a period from the Western renaissance to contemporary assigned to 20 nationalities. We note that both websites continuously art (mostly from 1500 to 2000 CE). We are aware that our study update the dataset, so a larger volume of landscape paintings is available works with digital surrogates of the original palette (red, green, on the date of publication of this research. and blue [RGB] values in the RGB color space) and the orig- While the overall number of paintings from WGA is relatively smaller inal paintings (digital scans). We are also aware that our study than from WA, the WGA dataset has a larger volume of paintings produced cuts out considerations of reception aesthetics (i.e., how different before 1800 CE. Therefore, we utilize both datasets in a complementary audiences perceive individual artworks in a variety of situations). way. We carefully constructed a unified dataset by filtering out the duplicate Nevertheless, our study makes an important contribution: our paintings from both datasets. During the data preprocessing procedure, we also manually removed unsuitable borders and photo backgrounds study characterizes a dataset that essentially summarizes a col- from the entire digital reproductions in painting dataset. The final dataset lective community consensus regarding the historiography of includes 14,912 paintings by 1,476 painters from 61 nationalities (Fig. 1A)(SI landscape painting, as it crystallizes from the feedback of qualita- Appendix, section I has a detailed description and the data curation process). tive research and visual resource librarianship. We do not claim To build a consistent nationality standard, the nationality of WGA artists that our data reflect art historical fact. Instead, they reflect a was assigned based on Wikipedia’s nationality information. We provide

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APPLIED PHYSICAL SCIENCES a effect of color depth on the partitioning process and a comparison between rc = . [6] a + b results obtained from different color-depth systems. Here, a is the height (width) of the first subregion from the top (left), b is the Image Partitioning Using Compositional Information. We apply the image height (width) of the lower (right) subregion for a horizontally (vertically) partitioning algorithm of Rigau et al. (39) based on mutual information of partitioned image, and the proportion is normalized to (0, 1). The closer the color palette and subregions in an image. Each image in the dataset was r is to one, the more the painting is divided at the bottom (right). first converted into a matrix representation whose dimensions correspond c to image width and height. Results The full image is considered an initial partition, and the algorithm progressively subdivides the image according to the partitions that pro- Fig. 1 B and C shows an example of how the partitioning pro- vide maximum information gain at each step. For a random variable C cess takes place in a sample painting by Claude Lorrain (1620 to taken from the set of discrete colors in RGB color space, the color palette 1650) (the three-bit and other color-depth images in which the information of an image is defined as Shannon entropy H(C): actual algorithm works are shown in SI Appendix, Fig. S7). The X original image is considered an initial region, and the algorithm ( ) = − ( ) log ( ). H C P c 2 P c [1] progressively subdivides the image according to the partitions ∈ c C that provide maximum information gain at each step as described

The probability P(c) is given by P(c) = Sc/S, where Sc denotes the number of in Materials and Methods. In the partitioning process, a paint- pixels taking color c and S is the size of the image. ing image is split into quasihomogeneous regions via either a Considering a painting process as a mapping from the color set C to the horizontal or a vertical line. Lorrain’s painting is found to have set R composed of a finite number of regions in an image, the conditional larger compositional information in horizontal partitions than entropy H(C|R) is defined as vertical partitions (Fig. 1C). Therefore, the painting is first par- X titioned in the horizontal direction at the position that gives the H(C|R) = − P(c, r) log P(c|r), [2] 2 maximum information. Then, the compositional proportion rc is c∈C,r∈R determined by the normalized partitioning position given by Eq. where the joint distribution P(c, r) = Pr[C = c, R = r] and the conditional 6. In case of the sample painting, the partitioning takes place probability P(c|r) = Pr[C = c|R = r]. The compositional information gained at the 227th pixel from the top, while the height of the original from introducing a set of partition R is provided by the compositional image is 373. Therefore, rc is calculated as 227/373 ≈ 0.608. information defined as the mutual information: Landscape Paintings Are Well Characterized by Dominant Partitions. I(C, R) = H(C) − H(C|R). [3] In principle, an image can be partitioned repeatedly until the image is fully divided into subblocks of completely homogeneous If the image is decomposed into n regions, R = r1, r2, ... , rn, the com- positional information is given by the generalized Jensen–Shannon colors. As illustrated in Fig. 1D, a subblock of an image can be divergence (JSD): partitioned either vertically or horizontally at any partition pro- portion rc in each step. However, we find that early partitions in

I(C, R) = JSD(C1, C2, ... , Cn) landscape painting show a distinct characteristic compared with n n later steps. X X , [4] ≡ H( πiCi) − πiH(Ci, ri) First, the partition direction of landscape paintings in the i=1 i=1 early partition steps is mostly found to be horizontal in direction where π is the size of region r normalized by the full size S and the regional (Fig. 1E), while there was no directional preference in abstract i i paintings in the first step (Fig. 1F); 86.8% of landscape paintings Shannon entropy H(Ci, ri) for the color set Ci in the region ri is given by were horizontally partitioned at the first partition step with larger X H(Ci, ri) = − P(c|ri) log2 P(c|ri). [5] compositional information than vertically partitioned cases on c∈Ci average (Fig. 1G). The frequent observation of a dominant hori- zontal partition is mainly due to the fact that landscape paintings Therefore, given a partition that dissects an image into two regions, usually include a horizon, where constituting colors begin to I(C, R) gained from the partitioning is maximized if the dissected subre- vary significantly. After around 10 partition steps, the ratio of gions are composed of a completely distinct color to other subregions. On the other hand, if the dissected subregions both have same color frequency landscape paintings that are partitioned in the horizontal direc- distribution with the original image, the partitioning offers no meaningful tion saturated at the point slightly below 0.5. However, abstract information [I(C, R) = 0]. paintings did not show any directional preference in composition The procedure to find partitioning positions of an image is as follows. over all partition steps including particularly the first step. It is During the first partition process, one should calculate the compositional interesting that the ratio of horizontal vs. vertical partitions sat- information gain over all possible partitions in both horizontal and vertical urates slightly below 0.5 in both painting genres (Fig. 1 E and directions on the entire image resulting in w − 1 × h − 1 trials, where w and F). We speculate that this saturation effect is mainly caused by h are the numbers of pixels of weight and height, respectively, of the image. two reasons. First, more images exist in landscape aspect ratio Then, the algorithm selects a partition that gives the maximum information. as opposed to portrait aspect ratio, notwithstanding the genre of From the second partitioning process, the algorithm is repeatedly applied to remaining subblocks, and the subblocks are partitioned with lines that landscapes or abstract subjects. This means that simply by chance offer maximum information for each step. In principle, the partitioning pro- more should be partitioned in a vertical direction. Another rea- cess can be continued until the image is fully decomposed into regions of son, especially for landscape paintings, is that whereas large-scale homogeneous colors. Because the conventional partitioning process takes objects in landscape paintings such as sky, earth, and ocean are considerable time to find partitioning positions for a large dataset, we used horizontally placed, there are typically smaller vertical objects the Shin et al. (40) line updating bipartitioning (LUB) algorithm to calcu- often with edges left and right such as trees, plants, and build- late the mutual information for each partition step for faster calculation ings. Consequently, the frequent existence of vertically oriented (detailed descriptions of the LUB algorithm are described in SI Appendix, objects, particularly in the foreground, could cause more fre- section II). quent partition in the vertical direction in the later partition In this study, we focus on the first two partitions of painting images to analyze compositional proportion of a painting. This allows for meaningful steps. We provide more detailed verification regarding the two interpretation as the first two sections classify the images in relation to the explanations in SI Appendix, section III. most dominant compositional features of the painting. After an image or Landscape painting further exhibits distinct characteristics in a subblock of the image is partitioned into two subregions with ratios a : b, the amount of compositional information. The compositional we define the compositional proportion rc of the bipartitioned image as information gained from the first partitions in landscape painting

4 of 11 | www.pnas.org/cgi/doi/10.1073/pnas.2011927117 Lee et al. 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V–H-type could split (Fig. vertical paintings middle. second H–V top the the in from paintings, the H–H-type down in slightly from unlike cuts However, position division first 0.55 the most Therefore, the found ele- was around vertical partition first and frequently the paintings, horizontal H–V-type In both and com- ments. balancing the H–V- in and therefore perspective. features ground, vertical position represent dominant middle to have horizontal paintings a suitable V–H-type three as foreground, corre- with background, a H–H-type paintings a depicting the in of usually such, shown composition as As layers, the subregion 2D). to lower (Fig. the sponds painting in sample top the the horizontal from second position the 0.47 step, partition next .I h –-yepitns h rprino rthori- first 2 of (Fig. proportion distributions the the paintings, of zontal H–H-type peaks the the joint In of near D–G). types found four paintings the ratio ple plot partition we of direction, distributions partition of pairs four nationalities). in across rooted (SI S3 simply selection data, Fig. historical Appendix, the are art and/or of production nationality nationality localized structure of in across changes temporal differences composition global apparent of the while in use rooted the possibly in is difference while most evolution the the that suggests temporal of on also This gradual nationality. based a of concepts significantly exhibit transcending and differ types painting partition landscape in sitions okn ttepsto ftehrznoe ie(.. h physical the (i.e., time over horizon the of are position we the exceptions, at correspond- looking with Essentially, their understanding. and intuitive CE and 2000 the to ratio in dissection 1500 direction ing from horizontal of (86.8%) the groupings landscape step in Western priori first partitioned of a are majority that great as schol- the paintings them ongoing on focus treat to We simply subject paintings. we often conventions, discussion, on arly agreed more are less authorship and or attribution, style date, of concepts these oepoetpclcmoiinlpooto o aho the of each for proportion compositional typical explore To ofr ehv xlrdcaatrsiso h rttwo first the of characteristics explored have we far, So r c a ealdsaitc) hl eaeaaeta l of all that aware are we While statistics). detailed has a ags erte04 oiinfo h o.I the In top. the from position 0.40 the near largest was F n ,oragrtmde o atr h exact the capture not does algorithm our G, and hw h eprldsrbto ftedatasets the of distribution temporal the shows r c ntefloigaaye oalwfrclear for allow to analyses following the in r c norfl ieageae e fland- of set time-aggregated full our in r c .Fnly –-yepaintings V–V-type Finally, ). r copne ytpclsam- typical by accompanied c ntredfeettpsof types different three in NSLts Articles Latest PNAS r c sfudna the near found is | f11 of 5 ,

APPLIED PHYSICAL SCIENCES Fig. 2. Landscape painting partition preference in horizontal vs. vertical direction up to the second most dominant dissection. (A) For the time-aggregated dataset, the fraction of H–H-type paintings is highest, followed by H–V-, V–H-, and V–V-type paintings: H–H > H–V > V–H > V–V. Colored and gray bars indicate the proportions of partition types for landscape paintings and abstract paintings, respectively. (B) Gradual temporal evolution in the use of the four composition types. A moving time average was applied, where each time bin was set to include the same number of paintings (500). The fraction of H–V-type paintings gradually decreases, whereas the fraction of H–H type gradually increases to become dominant since the midnineteenth century. The fractions of V–H- and V–V-type paintings are stable over time. (C) The box plots summarize the distribution of fractions of each partition type across different nationalities in five consecutive time periods. The five time bins were set to include equal numbers of paintings. Behaviors of individual nationalities follow a similar temporal trend to the aggregate result (A and B). Distributions for individual nationalities are shown in SI Appendix, Fig. S14 in detail. (D–G) Four types of joint distributions of partition ratio rc emerge from the dataset, here accompanied by typical paintings sampled near the peaks of the distributions for (D) H–H-, (E) H–V-, (F) V–H-, and (G) V–V-type paintings. The first and second partitions of sample paintings are represented by solid and dashed lines, respectively. Painting images credit: WikiArt.

horizon between earth and sky) or the most dominant chromatic tion bias toward the modern era (SI Appendix, section V has the difference in the associated color gradient. statistics). In Fig. 3C, we exhibit a random sample of 66 (50%) Fig. 3 exhibits the change of the distribution of rc measured individual painters from the 131 representative painters further and encapsulated in three different conceptual categories: 1) dis- restricted for the purpose of visualization. The distributions in all crete creation date time spans (20-y bins from 1500 to 2000 CE), three conceptual categories are sorted and colored by the median 2) conventional artistic style periods, and 3) the oeuvre of indi- year in each group. vidual artists. We analyze the top 25 and arguably least contested In all three cases, the distributions of rc are mostly unimodal conventional style periods with the largest number of artworks in and varying smoothly over art historical time, indicating that the dataset for the rest of our study. These conventional style there are characteristically dominant compositional proportions periods compose 92.8% of all artworks in the dataset. Regard- used by landscape artists (equivalent to the peak of the respec- ing individual oeuvres, we focus on 131 painters, all within the tive distribution), while the distribution gradually changes over top 10% based on the number of paintings in our dataset, in time instead of showing a singular or a random preference in two groups before and after 1800 CE, resulting in 30 and 101 peak shift. The change of peaks over time periods in Fig. 3A painters from the two periods, respectively. We introduce this shows three stages. Initially, in the midsixteenth century, a small separate selection criterion as the number of recorded paintings rc around one-third was dominant. However, the peak rc grad- by painter largely increases from 1800 CE. Applying a uniform ually increased from the late sixteenth century and reached its selection criterion across all time would possibly cause a selec- highest value from 1640 to 1680 CE. This high peak rc was

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APPLIED PHYSICAL SCIENCES construction. We further study the distributions of rc of indi- SD (σ = 0.15). Community 3 has smaller peak position (peak vidual painters. Fig. 3C shows the distributions of compositional rc = 0.303), and the distribution is much wider (σ = 0.21). Com- proportions for the random 50% sample of individual artists munity 2 has intermediate characteristics between communities 1 with the largest numbers of paintings in our dataset. While natu- and 3 (peak rc = 0.393, σ = 0.17). Another remarkable feature is rally subject to larger fluctuations due to the smaller number of that without including any explicit metadata regarding informa- images per artist, the peaks for individual artists still demonstrate tion on the activity time of artists or their stylistic classification, a similar trend in rc . Consequently, we are able to investigate the artists and styles are meaningfully grouped in terms of time temporal clusters of individuals in detail based on similar use of period (Fig. 4 A and B). At a minimum, this means the metadata compositional proportions in the following section. are coherent with the visual features of the images, pointing to a coherent classification by individual qualitative art historians. Of The Similarity Network of Individual Artists and Conventional Styles course, critical eyes may surmise that the emerging metanarra- Is Subject to Meaningful Clustering and Confusion. The fact that tive seems almost too coherent to be true. This would mean the each compositional proportion distribution for time periods, classification is coherent yet leaves out everything that does not artists, and style concepts has distinguishing characteristics nat- fit into the all to smooth story of Western art. urally suggests consideration of clustering between the distri- Looking into details, community 1 in Fig. 4A is characterized butions to characterize separation and confusion. Which artists by high rc paintings and contains most of the artists before 1850 and styles are similarly characterized with regard to their typi- CE. Conventional styles up to the midnineteenth centuries in cal rc distribution? Which are far apart from each other? Can the dataset fall into this group. Community 2 exhibits a peak we find communities of artists and styles sharing a similar use of at rc = 0.393, which is above the middle of the canvas and con- dominant compositional proportion? To answer these questions, tains many artists who worked during the late nineteenth century we compare the distributions of rc within and between individ- and some in the early twentieth century. Realism, impression- ual artist oeuvres and styles periods. Doing so, we effectively ism, and related painters characterize this group in an exemplary treat both “artists” and “style attributions” as groups of artworks, way. As we have seen in Fig. 3, from the periods of realism much like they may form latently in the mind of an art historian and impressionism onward, the dominant partitioning propor- or a trained neural network. We define a proportion similarity tion began to decrease. As such, this period bridges the classically measure J between two distributions of rc between Pi and Pj preferred composition to the newly preferred composition in using an information-theoretic distance, the JSD: the modern period. This result is in line with the established studies that realism and impressionism served as the transition J = 1 − JSD(Pi , Pj ) toward the modern era. Lastly, artists in community 3 show a Pi + Pj 1 [7] distribution of rc with the smallest modal value among the three = 1 − [H ( ) − (H (Pi ) + H (Pj ))], 2 2 groups (Fig. 4B) with a less pronounced peak rc found at the near–one-third position (rc = 0.303). Looking at temporal dis- where H is the Shannon entropy with base 2 and the proportion tribution, the paintings in community 3 are mainly concentrated similarity J is bounded between zero and one. in the early twentieth century, with some in the late nineteenth We investigate how individual artists cross-correlate with mul- century. The characteristic shape of the rc distribution is broader tiple styles in terms of horizon choice. We first create a bipartite than in the other two communities, representing a more diver- or two-mode network composed of landscape artist oeuvres sified use of proportional dissection. However, contrary to the and conventional style periods, respectively, to detect groups common notion that the styles or isms of the modern era are of artists and style periods with similar habits of composition distinct stylistic expressions, their distributions of compositional in paintings. We measure the distributions of rc of the same proportions are similar and clustered into a same group: the 131 representative individual painters and 25 conventional style peak rc of aggregated paintings (Fig. 4B, third column) and each periods as used in the previous subsection. Then, we set the style’s peaks (Fig. 3B) were all found near rc = 1/3. In other weight of links between pairs of artists and styles by the propor- words, “anything goes” not only across modern isms but also, tion similarity between their rc distributions. From the originally within each single ism, while “anything” is bounded with a bias fully connected bipartite weighted network, we then compute to rc = 1/3. As such, it is not individual isms that correspond a significance criterion for each link following the procedure to conventional style periods such as renaissance or baroque. described in ref. 42. We only keep significant links at the level Instead, one could argue that the sum of isms is the actual style of α = 0.366, which maximizes Nb /N0 − Lb /L0, where Nb (Lb ) period of isms, which all behave the same with respect to pro- and N0 (L0) are the numbers of nodes (links) in the filtered net- portional dissection while differing in more limited ways only, work and the original network, respectively. Then, we apply the particularly in their textual phenotype. One could say all isms Louvain community detection algorithm (43, 44) on the filtered share the same generating function regarding the dissection of network, which is a fast hierarchical community detection algo- landscapes. rithm widely used in network science, to find hidden community We also investigate the proportion similarity matrices between structure. all pairs of artists and conventional style periods (Fig. 4 C and D). In the final similarity matrix in Fig. 4A, we observe that the To compare the clustering of artists and style periods within each compositional similarity network has a clear community struc- concept with the joint artist–style clusters, we newly grouped ture with three groups of artists and style periods based on their individual artists and style periods into three communities using distribution of using particular compositional proportion. Within the same community detection algorithm. Clustering of artists each community, the link density is high, and the weights of links and styles (Fig. 4 C and D, respectively) correlates meaningfully tend to be much larger than between communities off the diago- with the artist–style clusters, again exhibiting clear clustering nal. Each community has two distinguishing features from the structure in time. At the same time, the pure artist and pure others (Fig. 4B). The first feature is the shape of the aggre- style clusters deviate from the joint artist–style clusters (Fig. 4A) gated rc distribution of paintings in each community. Fig. 4B in detail and consequently, also from the respective one-mode shows the distribution of rc and production year of paintings projections (SI Appendix, Figs. S19 and S20), which means we by all artists in each community. The distribution of dissection can identify exceptions to the respective superera mainstream in proportion rc in each community has clear difference from the individual artist oeuvres and style concepts. others. Community 1 has the largest peak dissection position This effect is in line with Fig. 3 B and C, where the system- (peak rc = 0.531), and the distribution is sharper with smaller atic evolution of rc appears more clear for conventional style

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Community 2 Community Active year Active r D c = 3) n h itiuini hre ihsalrS (σ SD smaller with sharper is distribution the and 0.531), h itiuino iscinproportion dissection of distribution The A. = 1.Tedsrbto fpouto ae o l anig in paintings all for dates production of distribution The 0.21). A, Artist-Artist similarity Artist-Artist and C, Style-style similarity n )Pooto iiaiymti ewe all between matrix similarity Proportion D) and D a efudin found be can NSLts Articles Latest PNAS . V section Appendix , SI r c ifr lal across clearly differs =

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APPLIED PHYSICAL SCIENCES periods than for individual artists due to the existing hetero- structure in the proportion similarity network between groups geneity on the level of individual. This result points to the fact of images representing the oeuvre of individual artists and con- that the grand narrative of Western art history, as expressed ventional style periods. The analyses revealed that artists in the in the coarse-grained conventional style periods, is too smooth same community of the proportion similarity network were actu- and somewhat artificial, failing to capture the rich nonlinear- ally grouped into temporally similar periods even though no time ity of art production throughout time on an individual level. information from the metadata was included in the analysis. In Put in other words, our quantification indicates that a proper other words, the dissection profiles of artist oeuvres and style grand narrative of art history requires a multiplicity of perspec- periods do latently encode date information, again either rooted tives, both qualitative and quantitative, as a great amount of in art history or at least the consensus story of art in which our nonlinear detail gets lost in an overly conventional mainstream dataset is rooted. narrative. Fourth, the distributions of compositional proportions of diverse styles or isms of the modern era were found to be similar Discussion and clustered into the same group, which counters the common For millennia, artists have discussed and developed various com- intuition that the diverse isms function much like style periods. positional techniques to represent their own creative ideas and Possible limitations of our study and future work include the purposes. The results explored in this study provide a macro- following. Although the dataset used in this study includes some scopic picture of how landscape artists have used proportional Japanese and Chinese landscape paintings, our dataset mainly dissection in their compositions according to the latent consensus focuses on paintings by European artists (i.e., the so-called canon of mostly Western art historiography, as expressed in the widely of Western European art). As such, the dataset represents a available and broadly used WA and WGA datasets. Analyzing conventional notion of art history that is biased not only geo- 14,912 landscape paintings covering more than 500 y, we have graphically but also, toward artists of male gender and very likely retraced the evolution of compositional proportion in landscape streamlined in relation to a more complex reality. That said, our paintings. study is nevertheless foundational, as it reveals the streamlined First, our results show that the dominant modes of landscape nature of canonic art history, which is a valuable and necessary composition based on partition direction take gradual changes step preceding all future research that is interested in putting over time while transcending nationality. Before the midnine- this biased canon into question. We also believe our study can teenth century, the vast majority of paintings were frequently function as a proper starting point to quantitatively investigate characterized by H–V-type composition followed by H–H, V– principles of composition covering broader conceptual groupings H, and V–V types. However, the use of H–V type continuously (beyond artists and conventional style periods) and a broader set decreased, whereas composition with double horizontal parti- of cultures and regions. Our methodology is readily applicable to tions (H–H type) gradually became the most frequent composi- paintings and any other two-dimensional representational forms tion type. Meanwhile, the fractions of V–H and V–V types were after an appropriate dataset is prepared. In further analyses, kept almost stable over entire period. Because of the fact that this comparing the characteristics of photographs as uploaded into pattern transcends national groups, the long-standing division of online social media, randomly selected from street view panora- landscape art history by nation must be put into question, in favor mas, or procedurally generated could be intriguing avenues of of a broader comparative perspective. Concretely, our result con- study (45). An algorithmic limitation of our current partition- firms the stance of scholarly literature that has not emphasized ing methodology is that it only considers horizontal and vertical national concepts of landscape painting for decades, while sug- divisions. Some artists, for example, might dominantly apply gesting that library classification, metadata in visual resource diagonal composition in painting. Therefore, developing algo- collections, and the categorization of encyclopedia entries should rithms that can detect more complex compositions would be a follow suit. challenging but likely rewarding issue. Our study mainly focused Second, the dominant proportions of paintings characterized on the trend in the first horizontal dissection within paintings. by the distribution of rc gradually varied over time in a suspi- In addition, one could explore patterns of composition based ciously smooth way, which could be rooted in art history itself or on higher-order dissections. Going beyond painting, we believe more likely, artificially in the selection bias of dataset curators or that our framework can also find use in understanding composi- of course, the underlying art historical literature. Reaching the tion and geometric proportions in other art forms, including in highest modal ratio rc during the seventeenth century, includ- particular typography, film, architecture, etc. ing the baroque period, the favored ratio rc began to decrease gradually over time and moved toward the ear one-third posi- Data Availability. All study data are included in the article, SI Appendix, and tion from the top of the painting. As qualitative art historians Dataset S1. in recent decades have progressed over the notion of a smooth ACKNOWLEDGMENTS. B.L. and H.J. acknowledge the support of National consensus story of Western art and have emphasized the anec- Research Foundation of Korea Grant NRF-2017R1A2B3006930. M.S. is dotally evident multiplicity in the evolution of artistic expression, funded within the European Union (EU) Horizon 2020 project CUDAN this study serves as a reminder that the available large-scale (Grant 810961), and acknowledges initial funding through The University datasets might be rooted in older literature, likely perpetuating of Texas at Dallas Arts & Technology Fellowship #1 (anonymous donor). S.K.H. was supported by the Basic Science Research Program through the potentially severe biases. National Research Foundation of Korea funded by Ministry of Education Third, beyond this gradual change in compositional propor- Grant 2017R1D1A1A02019371. For relevant discussions, M.S. thanks Richard tion, we employed network analyses to detect hidden community Brettell, who passed away during the final revision of this paper.

1. H. Wolfflin,¨ Principles of Art History (Courier Corporation, 2012). 6. D. Kim, S. W. Son, H. Jeong, Large-scale quantitative analysis of painting arts. Sci. Rep. 2. A. Riegl, Historical Grammar of the Visual Arts (Zone Book, New York, NY, 4, 7370 (2014). 2004). 7. B. Lee, D. Kim, H. Jeong, S. Sun, J. Park, Heterogeneity in chromatic distance in images 3. A. Elgammal, B. Liu, D. Kim, M. Elhoseiny, M. Mazzone, “The shape of art history in and characterization of massive painting data set. PLoS One 13, e0204430 (2018). the eyes of the machine” in Thirty-Second AAAI Conference on Artificial Intelligence 8. R. Arnheim, Art and Visual Perception (University of California Press, 1974). (AAAI, 2018). 9. E. B. Feldman, Varieties of Visual Experience; Art as Image and Idea (Prentice Hall, 4. H. Y. Sigaki, M. Perc, H. V. Ribeiro, History of art paintings through the lens of entropy New York, NY, 1971). and complexity. Proc. Natl. Acad. Sci. U.S.A. 115, E8585–E8594 (2018). 10. M. Livio, The Story of Phi, the World’s Most Astonishing Number (Broadway Books, 5. M. Yazdani, J. Chow, L. Manovich, Quantifying the development of user-generated 2008). art during 2001–2010. PLoS One 12, e0175350 (2017). 11. M. Andrews, Landscape and Western Art (Oxford University Press, 1999).

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