1 Introduction

The interface between art and science has grown over the past three decades with the advent of statistical analysis of the visual characteristics of art works. Although such studies now encompass a broad range of artistic styles, substantial research has been devoted to paintings generated by pouring paint onto the canvas rather than by using traditional brush contact. A number of Twentieth Century artists pursued this technique, including the European Surrealists [1], the Canadian Les Automatists [2], and the American Abstract Expressionists [3]. The latter featured the most famous proponent of the ‘pouring’ technique, Jackson Pollock [4].

Celebrated as Action Painting, these poured works serve as records of the artists’ encounters with their canvases. In Pollock’s case, this encounter involved him painting in the three-dimensional space above the canvas and then letting gravity condense the fluid paint onto the two-dimensional plane of the canvas laid out across the floor. This dynamic process often unfolded at frantic painting speeds, inviting speculation from art critics and the public alike as to whether it is possible to control the pouring technique. Perhaps all artists are instead destined to generate haphazard records of their encounters with the canvas. This debate has been fueled by the lack of traditional compositional strategies displayed in typical poured works - no center of focus, no left or right, and no up or down [3, 4]. Furthermore, respected scholars have misidentified imitations as masterworks in high profile controversies [5–7], suggesting that Pollock’s specific application of the pouring technique might not be unique.

Statistical analysis offers the possibility of resolving this debate by distinguishing between poured patterns generated by different artists. Fractal analysis, which assesses patterns at multiple size scales, has been proposed as a useful tool for quantifying Pollock’s patterns [8–12]. This choice of analysis was motivated by the possibility that the pouring technique might avoid the size restrictions imposed by hand-brush motions and so free up a broader range of body motions. These could then translate into multi-scaled paint trajectories on the canvas that might not be possible within the constraints of traditional painting techniques. A second motivation derived from the fact that natural scenery is composed of fractal patterns [13]. Many observations (made both by Pollock and subsequent art theorists) highlighted the similarity between his patterns and those found in nature [4].

Initial fractal studies calculated the paintings’ ‘covering’ dimension D**0, which quantifies the amount of canvas surface covered by paint at different scales [8–12]. These were quickly followed by multi-fractal studies which employed a spectrum of dimensions, D**0 to D**50, to assess increasingly sophisticated statistical qualities at different scales [14, 15]. Various scaling measures have since been applied to different visual qualities of Pollock’s paintings [16–29]. Significantly, these fractal approaches received support across the disciplines, ranging from the mathematician Benoit Mandelbrot who introduced the term “fractal,” to chief Pollock scholar Francis O’Connor [30, 31].

Whereas many of these studies aimed to gain new insights into the subtle visual characteristics of poured paintings, fractal analysis was also proposed as a potential authenticity tool to distinguish Pollock’s paintings from replicas [5, 6, 11]. Although its application to a high-profile authenticity dispute initially stirred controversy [5, 6], a subsequent artificial intelligence (AI) approach demonstrated a 96% classification accuracy by combining fractal parameters with other pattern-analysis parameters [32]. Of these parameters, the fractal character was found to be the dominant predictor [32]. A similar technique was found to be more accurate than adults and children in distinguishing paintings of abstract expressionists [33]. A more recent AI technique also exploited fractal qualities in poured paintings to distinguish Pollocks with a 99% accuracy [34]. Although our current study is not aimed at developing authenticity tools, it is nevertheless inspired by the fact that different artists have the potential to generate different poured signatures. To investigate the physical origin of such differences, we aim to go beyond the spectrum of fractal dimensions previously used to capture the multi-scaled complexity of poured works.

Previous pattern investigations of art [35, 36] and design strategies such as urban planning [37] have benefitted from a multi-scaled technique known as lacunarity analysis [38]. While fractal analysis quantifies the scaling behavior of the spatial distribution of the paint, lacunarity focuses instead on variations in the voids between the clusters. Introduced by Mandelbrot, the term is derived from “lake” [13] to reflect the geographic analogy with gaps in land distributions. Morphologically, lacunarity probes the textural “clumpiness” originating from translational variations in void sizes at different locations across the canvas. Previously, lacunarity techniques have been applied successfully to a range of physical systems. Examples include categorizing agricultural processes [39], distinguishing geophysical data from different landscapes [40], identifying bone defect patterns in patients afflicted with osteoporosis [41], and investigating the physical distribution of galaxies in the universe [42, 43]. Given this diverse application to natural patterns along with the similarity of Pollock’s work to natural images, it is an obvious extension to translate this statistical tool to the analysis of poured paintings. Lacunarity has the potential to identify new structures and ‘fingerprints’ within poured masterworks. In doing so, this approach might add valuable information for authenticity studies and might also provide a new ‘eye’ on a much debated, revolutionary art technique.

Our research goal is to apply lacunarity analysis to controlled poured-painting experiments with systematic data collection and reproducible analysis. In particular, we will determine whether the lacunarity’s probing of spatial ‘clumpiness’ of poured paint trajectories is sensitive to differences in the body motions of artists of different ages. We will consider two distinct groups of artists - children and adults. These two populations are at different developmental stages of their biomechanical balance physiology. Our investigation is motivated by experiments showing that balancing motion displays non-linear characteristics such as fractal patterns [43–61]. In addition to postural sway during standing, these studies extend to dynamical balance actions during walking [57], sports [58, 59], dancing [60, 61], and martial arts [55]. These biomechanical balance studies build on a growing body of work showing the prevalence of fractals in human physiology [62]. If balance physiology plays a dominant role in the painting process, we hypothesize that distinctly different fractal and lacunarity characteristics will be detected in the final paintings created by the children and adults.

Because our preliminary study does not include a direct measurement of the biomechanical balance of the two artist groups, we discuss additional inter-group factors and how these might contribute to the pattern variations. As a motivation for future experiments, we also make a preliminary comparison with two famous poured works by Jackson Pollock (Number 14, 1948) and Max Ernst (Young Man Intrigued by the Flight of a Non-Euclidean Fly, 1942) [4]. These choices are informed by the potentially differing roles played by balance motion in their specific applications of the pouring technique [4, 30].

Finally, we build on our investigations of generating art by considering the act of observation. Previous studies have shown that observers can distinguish between abstract art created by children and adults [63, 64]. Here, we focus on the adult poured paintings and ask a group of observers to rate them in terms of their perceived pleasantness and interest. Given that the fractal repetition of patterns naturally generates high visual complexity, we also consider the perceived complexity of the poured patterns. The goal is to investigate whether there are correlations between these ratings and the fractal and lacunarity characteristics of the paintings.

2 Methods

2.1 Image generation and acquisition

To examine the potential of lacunarity analysis, we employed ‘Dripfests’ which invited participants to paint in the pouring style of Pollock. The pouring technique is influenced both by artists’ physical motions and their choice of materials. The Dripfests provide a controlled environment which standardizes the materials used - for example, the type of canvas, paint, and painting implement. By doing so, our investigations focus on the impact of the artists’ motions. Our work builds on recent investigations which interpret the pouring process through the scientific lens of fluid dynamics [65, 66]. Whereas these studies considered the role of the painting implement’s speed and height relative to the canvas, our experiments also incorporate the patterns traced out by the changing directions of the artists’ motions.

We recruited 18 children (four to six years old) and 34 adults (18–25 years old) for the Dripfest experiments. No inclusion nor exclusion criteria were applied to applicants. We supplied them with identical horizontal surfaces (sheets of paper with dimensions of 61.6 cm by 95.5 cm) and identical vinyl paint diluted to a common fluidity suitable for pouring. They were each supplied with a container positioned on the ground close to the paper. Because the container was not lifted by the participants, container size was not adjusted across the two age groups in any attempt to adjust for weight. Participants were supplied with identical mixing sticks for them to transfer the paint to the paper through the air. Participants were shown the same examples of Pollock’s poured paints and were issued identical verbal requests “to create a painting like the ones shown.” Clarifications were provided when asked. Each participant was allowed to create just one painting. Participants were allocated 40 min, which exceeded the time needed to perform their task in all cases.

For simplicity of analysis and interpretation, the study considers monochromatic paintings on a white background. The paintings were scanned at a pixel size of 0.3 mm to ensure resolution of the finest paint features (∼1 mm). Pollock’s Number 14, 1948 (57.8 × 78.8 cm) [4] and Ernst’s Young Man Intrigued by the Flight of a Non-Euclidean Fly (75 × 60 cm) [4] were scanned from high-quality prints. The images were thresholded at 50% when converting the grayscale images to binary (black and white) images. Visual inspection confirmed that small variations in the threshold value did not change the visual appearance of the binary image. We also confirmed that the variations did not change the count of the number of the binary image’s black pixels. We note that Ernst planned to tap into his subconscious by using the resulting paint trajectories as a springboard for free association. Accordingly, he perceived the image of a face in the paint trajectories of Young Man Intrigued by the Flight of a non-Euclidean Fly [4] and developed the painting further by sketching in a colored face. For the purposes of our study, we are interested in the underlying poured trajectories, and we therefore electronically extracted these trajectories for our analysis using color-separation techniques previously developed for analysis of the painted layers of Jackson Pollock paintings [11]. Figure 1 shows representative examples of the Dripfest paintings.

Figure 1. Representative examples of 23 poured paintings generated during the Dripfest experiments. The first 19 are by adults (Adult19, 13, 32, 34, 7, 3, 18, 27, 23, 12, 9, 26, 16, 20, 29, 31, 4, 5, and 2) and the last four are by children (Child16, 5, 11, and 4).

2.2 Image analysis

2.2.1 Lacunarity analysis

The traditional algorithm for determining the lacunarity of a pattern is the gliding box technique [38]. Consider a painting image of height H and length L**T containing occupied pixels (i.e., sites featuring the painted pattern) and unoccupied pixels (sites featuring the painting’s background). A square box of side-length L < L**T is placed at one corner of the painting and the density of occupied pixels (s) within the box is determined. The box is then advanced along the painting by a small increment ε = 3 mm, and the density is calculated for each advancement over the painting’s entire length and height. By plotting the number of boxes featuring pixel density s measured for the box size L, the frequency distribution of densities, n(s, L), is then generated. This can be converted to a probability density, p(s, L), by normalizing the density with respect to the total number of boxes NT(L) of box size L. A box is considered to be occupied if it contains one or more occupied pixels. The number of occupied boxes, N(L), is obtained by summing over the boxes of a given size L. The value of L is reduced in equal increments of 3 mm. The first and second moments of this distribution of occupied pixels are calculated as

Z1=∑i=1NLsipsi,L(1)

Z2=∑i=1NLsi2psi,L(2)

These first and second moments (Equations 1, 2) relate to the mean and variance of the distribution, respectively. The lacunarity of the pattern at the scale L is then defined by the ratio

ΛL=Z2Z12(3)

Given that Λ(L) is proportional to the distribution’s variance, patterns characterized by large variations in void size will be quantified by large lacunarity, while tightly clustered patterns with little void variation will show low lacunarity. A scaling plot can be generated by repeating this calculation of Λ(L) in Equation 3 for decreasing box sizes and plotting log Λ(L) vs. log(L). The shape of the resulting lacunarity scaling curve yields rich information about the statistical nature of the clustering. For example, pure fractal distributions result in a linear curve on the log-log scaling plot [38].

In addition to the curve’s shape, additional parameters are useful for quantifying the scaling behavior of the pattern. For example, the zeros of the lacunarity curve serve to uniquely identify fundamental scale sizes of the pattern. The difference between the maximum and minimum values of log(Λ(L)) across a given scaling range is defined as the lacunarity depth

ΔΛ=logΛmaxΛmin(4)

and the lacunarity slope (the average rate at which the lacunarity changes with scale) is given by

ς=−Δ⁡logΛΔ⁡logL(5)

A pattern with little change in clumpiness as the size scale L is reduced is said to show little “depth” ΔΛ (Equation 4) and is characterized by a small ς value. In contrast, patterns with rapidly increasing clumpiness as the size scale L is reduced have a larger depth ΔΛ and are characterized by a large ς (Equation 5).

These concepts are demonstrated in Figure 2 for two poured paintings featuring unusual painted features. In each case, the lacunarity scaling curve (Λ(L) vs. log(L)) shows a gradual increase in Λ(L) with decreasing scale L. Measured over identical scaling ranges, the overall change in Λ(L) is larger for the top painting and so it shows greater lacunarity depth and slope. In both cases, a painted feature (a large blob and a dense band) dominates the pattern density at around the 10 cm scale. This can be seen in the scaling plots as a rise in Λ(L). This rise is bigger for the top image because the blob has a higher density.

Figure 2. Demonstrations of the lacunarity analysis. Images of the paintings (61.6 × 95.5 cm) are shown on the left and their lacunarity scaling curves are shown on the right. The top painting is Child1 and the bottom painting is Adult15.

For all Dripfest paintings we employ box sizes ranging from L = 4 pixels (∼1mm, corresponding to the finest observed paint feature) up to L = 2048 pixels (∼61 cm, corresponding to the width of the sheet of paper). The smallest L value examined (the fine-scale “cut-off”) is set at 2.5 mm to avoid resolution issues and the largest (the coarse-scale ‘cut-off’) is set to L = 31 cm to match the scaling range used in the fractal analysis (see below). A linear slope is fitted to the scaling region that generates the largest R2. Although the L range of this linear region typically spans from the smallest analyzed L value up to a “transition” value of approximately 5 cm, the precise range varies between paintings. Accordingly, we do not measure ς based on the slope of this linear fit. Instead, we adopt a standardized approach of calculating the slope based on the lacunarity depth ΔΛ observed across the full size range (L = 2.5mm to 31 cm).

2.2.2 Fractal dimension

Fractal dimension D**0 describes how the patterns occurring at different magnifications combine to build the resulting fractal shape of the painting [8, 13]. For a painting consisting simply of a smooth line (i.e., containing no fractal structure), D**0 has a value of 1, while for a completely filled canvas (again containing no fractal structure), its value is 2. However, the repeating patterns of a fractal line cause the line to begin to occupy space at different size scales. As a consequence, its D**0 value lies between 1 and 2. For a poured painting featuring many fractal lines, its D**0 value will similarly lie between 1 and 2.

Whereas the traditional method for lacunarity studies is the gliding box technique, the traditional algorithm for determining the fractal dimension of a pattern is the box-counting technique [13]. The two methods share various commonalities, and the box counting technique can be seen as a subset of the gliding box technique. Following the same procedure as the lacunarity analysis, the square box of side-length L < L**T is placed at one end of the painting and the box is advanced along the painting by increment ε. Instead of calculating the density s of occupied pixels within each box, the ‘box counting’ technique simply determines if the box contains any occupied pixels. The number of occupied boxes, N(L), is summed for the given box size L. The scale invariance of fractals is generated by the power law N(L) ∼ L−D0 [13] and so D**0 can be extracted by fitting the slope of the linear region of the scaling plot log(N) vs. log(L).

The box-counting method therefore assesses the coverage of the pattern at different size scales and the extracted fractal dimension is referred to as the covering dimension. The steeper slopes for high D**0 fractals are indicative of the higher number of occupied boxes at fine scales when compared to an equivalent low D**0 fractal. As such, the D**0 value has an important impact on the visual appearance of fractal patterns. For fractals described by low D**0 values, the small content of fine structure builds a very smooth, sparse shape. However, for fractals with D**0 values closer to 2, the larger amount of fine structure builds a shape full of intricate, detailed structure [12, 67, 68]. More specifically, because the D**0 value charts the relative contributions of fine to coarse structure, D**0 serves as a convenient measure of the visual complexity generated by the repeating patterns.

2.2.3 Multifractal theory

Covering dimension belongs to a “multifractal” spectrum of dimensions. Accordingly, D**0 joins an infinite number of dimensions Dq, -∞ < q < + ∞ [69]. Whereas covering dimension asks simply whether a box is occupied, higher dimensions in the spectrum probe increasingly refined measures of the pixel occupation with the boxes. In particular, the multifractal spectrum performs a closer examination of visual appearance by employing the dimensions to yield key information about the degree to which complexity is manifest in a pattern. The multifractal spectrum of dimensions is readily determined by using the gliding box technique to calculate

Zq,L=∑i=1NLpiLq,(6)

where pi(L) is the relative density of the pattern contained in the box discussed earlier.

The logarithmic slope τ(q) of the partition function Z(q,L) is related to the multifractal dimension D**q according to the relationship

Dq=τqq−1,(7)

τq=d logZq,Ld logL(8)

In the limit q = 0, the expression for Z(q, L) in Equation 6 reduces to N and we obtain our standard relationship N(L) ∼ L−D in which D = D**0. When q→1 in Equations 7, 8, D**q is known as the information dimension (and serves as a measure of the Shannon information entropy [70]). The q = 2 multifractal dimension is of particular interest to lacunarity studies. Referred to as the correlation dimension, this examines the “mass” of the pattern within the boxes by employing

M=∑i=1NLpiL2,(9)

It has been shown [38] that in the case of linearity, the slope ς of the lacunarity spectrum Λ(r) yields the correlation dimension according to the relationship

D2=E−ς,(10)

where E is the dimension of the space in which the fractal pattern is embedded (E = 2 for our two-dimensional canvases). Therefore, measuring correlation dimension becomes a second approach to probing lacunarity. A convenient measure of the depth of a multifractal is defined as ΔD = D**0 − D∞ [15]. Mono-fractals are defined as having no depth (i.e., D**q = D**0). However, multi-fractal patterns with large depth will be characterized by low D**2 values. Through Equation 10, large depth corresponds to large ς and therefore a large lacunarity depth ΔΛ. Therefore, the clumpiness measured by lacunarity is a fundamental quality of multifractal character.

A multifractal analysis is performed on each painting by plotting log(N) vs. log(L) to extract D**0 and log(M) as calculated in Equation 9 vs. log(L) to extract D**2. Some of the scaling plots feature a distinct ‘knee’ at around the 5 cm transition reported in the lacunarity analysis. This knee is indicative of a bi-fractal behavior, whereby the scaling behavior switches between different slopes and therefore different fractal dimensions below and above the transition. Previous Pollock studies have discussed the pouring technique in terms of two interactive pattern-generation processes [8–11]. Pollock’s motions across the canvas are expected to dominate patterns above an approximate transition of 5 cm, while patterns below this scale are expected to be dominated by the paint dynamics of the fluid paint. This model will be considered in more detail in the Discussion section. Previous Pollock investigations employed a bi-fractal fit to the data to quantify the “knee” feature in the scaling plots [11]. These included the transition L value at which the “knee” occurs and the associated change in slopes. Because the knee feature is not the focus of the current study, here we report qualitative observations of their occurrence.

Because of the limited size scales associated with the two scaling behaviors, here we fit the data across the combined size range in part to gain a larger fitting range, but also to emphasize the interactive character of the two fractal generation processes (see Discussion). This approach also ensures that the fractal and lacunarity parameters are analyzed across the same sizes. Both dimensions (D**0 and D**2) are calculated by linear regression performed on the scaling plot data within the combined range. Using this approach, the linear range spans from 8 pixels to 1024 pixels, corresponding to spanning from 2.5 mm to 31 cm.

Although extended, we emphasize that the fitting range is nevertheless limited compared to infinitely repeating mathematical fractals. Instead, the paintings bear a closer resemblance to so-called ‘limited-range fractals’. These are typical of physical objects due to restrictions on their smallest and largest possible patterns [71]. In our case, the smallest patterns are limited by the finest paint lines (∼1 mm) and the largest are limited by the reduced counting statistics for the larger boxes (the limit of the linear regression fitting procedure is 31 cm, corresponding to only 6 boxes covering the canvas area). Because of their limited range, and also because they quantify two scaling behaviors (below and above the 5 cm transition), we adopt the term ‘effective dimension’ for D**0 and D**2.

2.3 Image perceptual ratings experiments

To investigate perceived complexity, visual interest, and pleasantness of poured paintings varying in lacunarity, we recruited 91 participants (23 females, 68 males, 1 nonbinary) via the online recruitment platform, Amazon MTurk. The inclusion criterion was normal or corrected to normal vision. No exclusion criteria were applied to applicants. The average age of participants was 33.85 years (SD = 10.70 years). Most participants were from the USA (68 participants or 74.7%), followed by India (11 participants or 13.2%) and the UK (8 participants or 8.8%). Prior to beginning the experiment, all participants were provided information before indicating their consent. Participants were reimbursed with US$2.50 for their participation.

Because our research focus is on responses to the visual complexity of the painted patterns, 15 of the paintings generated by adults were excluded from this study. These excluded paintings contained features that resembled recognizable patterns such as crosses, diagonal lines, or straight lines. Accordingly, participants rated a total of 19 poured paintings by adults, shown in Figure 1. None of the children’s paintings were included because of the significant difference between adult and children poured paintings’ properties in both fractal dimension and visual appearance.

Each painting was rated on the three bipolar scales to assess their perceived complexity (1-simple to 7-complex), pleasantness (1-unpleasant to 7-pleasant), and interest (1-boring to 7-interesting). Pleasantness was used as the term to assess aesthetic appeal because it was considered to be a broader, more general visual quality than beauty. The paintings were positioned in the middle of the screen and were scaled to subtend 55% of the screen size vertically and horizontally. On each trial, the sliders were positioned below the painting image with the “unpleasant-pleasant” slider on the top, “simple-complex” slider in the middle, and “boring-interesting” slider at the bottom. Before the ratings commenced, participants previewed all paintings to familiarize themselves with their appearance. During the preview stage, paintings were shown without sliders and in a quick succession with each painting displayed for 1.5 s. During the rating phase, the exposure to each painting was unlimited. Participants spent on average 8.6 s (SD = 9.1 s) viewing each painting per trial.

2.4 Human subjects protocols

Adult image generation was granted a protocol exemption from the UO Institutional Review Board. Subject recruitment was conducted starting 01/05/2004, and the paintings were generated during the first two weeks of June 2004 (01/06/2004–15/06/2004). Written informed consent was obtained for all participants.

Children’s image generation was performed under the UO protocol “Language, Cognition, and Social Cognitive Development” and was approved by the UO Institutional Review Board on 09/11/2001 under an expedited review, given that the data was collected during standard preschool activities and required no experimental intervention beyond consent. The paintings were generated starting on 25/03/2002 and ending on 13/12/2002. The children’s parents gave their written informed consent and their children gave verbal informed consent.

All experimental procedures for the perception experiments were approved by the UNSW Human Research Ethics (UNSW HREAP-C) Advisory Panel for the period starting 20/11/2017 and ending 20/11/2022. Data was collected during a 24-hour period starting on 31/03/2018. All participants provided their written informed consent for the study.

3 Results

3.1 Painting analysis

Figure 3 shows the results of the analysis applied to four paintings: example adult and child paintings, and also to Pollock’s Number 14, 1948 and Ernst’s Young Man Intrigued by the Flight of a Non-Euclidean Fly. The two Dripfest images are representative of the general visual appearances of adult and children’s paintings (see also Figure 1). The adult paintings have higher paint densities than those of the children and they also typically display a larger degree of ‘splatter’ (i.e., larger variation in the width of the paint trajectories).

Figure 3. Representative examples of analysis performed on Child8 (row (a)), Adult9 (row (b)), Pollock’s Number 14, 1948 (row (c)) and Ernst’s Young Man Intrigued by the Flight of a Non-Euclidean Fly (row (d)). The left column shows the painting image, the middle image shows fractal scaling plots (log(M) and log(N) vs. log(L)) and the right column shows the lacunarity analysis (log(Λ) vs. log(L)). See text for details.

Considering the scaling plots shown in the second column in Figure 3, the covering dimension analysis (the blue data line plotted relative to the right y axis) and correlation dimension analysis (the green data line plotted relative to the left y axis) are shown for the full range of box sizes - from 1 mm up to 61 cm. All paintings reveal a mild knee feature at approximately 5 cm caused by an increase in slope at the larger scales (the top and bottom paintings are examples of distinct knees). The dimensions extracted from the linear regression method applied across the combined range are listed in Tables 1, 2. We note that the D**2 values are typically slightly smaller than the corresponding D**0 values. The values of both dimensions are typically larger for the adult paintings than for the children’s. The histograms of the covering and correlation dimensions shown in Figure 4 emphasize these findings.

Table 1. A list of the D**0, D**2, and ς values for the adult paintings (The bolded labels signify paintings that are shown in Figure 1).

Table 2. A list of the D**0, D**2, and ς values for the children’s paintings (the bolded labels signify paintings that are shown in Figure 1).

Figure 4. Histograms comparing the adult and children painting scaling parameters: (a) D**0, (b) D**2 and (c) ς. In each case, blue columns signify the adult paintings and gray columns signify the children’s paintings. The values for Pollock’s Number 14, 1948 are indicated by the black arrows and Ernst’s Young Man Intrigued by the Flight of a Non-Euclidean Fly by the red arrows.

Figure 3 also shows the results of the lacunarity analysis for the four paintings. The lacunarity curves suggest a change in behavior at around the 5 cm transition, similar to that of the dimension curves. In each case, the red dashed line shows a fit to the approximately linear region observed at the smallest scales. While the linear region approximates to the sizes dominated by the paint dynamics, the precise range and shape of the curves vary. The ς values calculated from the lacunarity depth observed across the full size range (Methods) are listed in Tables 1 and 2 and plotted in the histogram in Figure 4c. They are typically larger for the children’s paintings than for those of the adults. In each of the histograms, the values for Pollock’s Number 14, 1948 are indicated by the black arrow and Ernst’s Young Man Intrigued by the Flight of a Non-Euclidean Fly by the red arrow. Whereas the former lies within the adult distribution, the latter lies within the children’s.

Figure 5 depicts the distribution of the three scaling parameters for paintings generated by the adults and children. Detailed descriptive statistics for the scaling parameters of the two samples of poured paintings are provided in Table 3 below.

Figure 5. The violin plots depicting the distribution of D**0 (left panel), D**2 (middle panel) and ς (right panel) scaling parameters for paintings produced by adults (blue) and children (gray).

Table 3. A compilation of statistical analyses of the D**0, D**2, and ς values of the adult and child paintings.

Independent t-tests confirm that there are statistically significant differences in scaling parameters between paintings produced by adults and children for all three scaling parameters. For D**0, the paintings produced by adults (M = 1.907) exhibit higher average values than those produced by children (M = 1.688) and this difference is statistically significant (t(50) = 13.8, p < 0.001, Cohen’s d = 4.01). Similarly, the paintings produced by adults (M = 1.848) compared to those produced by children (M = 1.647) have significantly higher average D**2 values (t(50) = 10.0, p < 0.001, Cohen’s d = 2.92). Finally, the average lacunarity slope ς values are higher for paintings produced by children (M = 0.383) than adults (M = 0.144). In this case as well, the difference is statistically significant with large effect size (t(50) = −12.3, p < 0.001, Cohen’s d = 3.57).

3.2 Perceptual ratings results

The 15 adult paintings excluded from the perception study feature recognizable objects. There are no statistically significant differences between the included and excluded paintings regarding their lacunarity (t(32) = −0.504, p = 0.617, Cohen’s d = 0.174), D0 (t(32) = 0.716, p = 0.479, Cohen’s d = −0.247), or D2 (t(32) = −2.073, p = 0.046, Cohen’s d = −0.716) when adjusted for multiple comparisons. The 19 selected paintings are therefore representative of the whole group in terms of their fractal and lacunarity parameters.

The descriptive statistics for the ratings of complexity, interest and pleasantness are shown in Table 4. The three types of ratings are positively correlated with Pearson coefficients of correlation ranging from 0.270 (p = 0.01) between complexity and pleasantness, to 0.586 (p < 0.001) between complexity and interest. The correlation between ratings of interest and pleasantness is 0.539 (p < 0.01). The average ratings of perceived complexity, interest, and pleasantness as a function of image lacunarity slope ς values are depicted in the top row of Figure 6. We fitted a linear regression model to the complexity, interest, and pleasantness ratings with lacunarity values as a predictor. While there is no statistically significant relationship between lacunarity and perceived complexity (F(1,908) = 0.369, p = 0.529, R**2 = 0.0004), the associations between lacunarity and perceived interest (F(1,908) = 21.97, p < 0.001, R2 = 0.024) and pleasantness (F(1,908) = 91.87, p < 0.001, R2 = 0.092) are modest, but statistically significant.

Table 4. Descriptive statistics for the ratings of complexity, interest and pleasantness.

Figure 6. Results of the perceptual rating experiment. Top row: The average ratings of visual complexity (left), interest (middle), and pleasantness (right) are plotted against ς for selected Dripfest paintings. Middle and bottom rows: The average ratings of visual complexity (left), interest (middle), and pleasantness (right) are plotted against fractal scaling dimensions D**0 (middle) and D**2 (bottom) for selected Dripfest paintings. Error bars represent 95% Confidence Intervals. Colored lines in each panel represent linear regression fits to the data, with shaded regions representing 95% Confidence intervals.

In the middle and bottom rows of Figure 6, we show the same ratings of complexity, interest and pleasantness as a function of fractal dimensions D**0 and D**2. The linear regression model fitted to the complexity, interest, and pleasantness ratings with D**0 and D**2 as predictors revealed a similar, but inverse relationship to that observed with lacunarity. The association between D**0 and D**2 and perceived complexity is not significant (D**0: F1,908 = 0.378, p = 0.539, R2 = 0.0004; D**2: F1,908 = 0.173, p = 0.677, R2 = 0.0002). The association between D**0 and D**2 and perceived interest (D**0: F1,908 = 11.71, p < 0.001, R2 = 0.0127; D**2: F1,908 = 22.11, p < 0.001, R2 = 0.0238) and pleasantness (D**0: F1,908 = 67.56, p < 0.001, R2 = 0.069; D**2: F1,908 = 90.21, p < 0.001, R2 = 0.092) is again modest, but significant.

4 Discussion

4.1 The interplay between fluid and motion dynamics

We begin with a review of Pollock’s pouring technique to emphasize the relevant size scales over which his patterns were generated. Painting in the air above the canvas, his paint trajectories served as a direct record of his motions. These records capture the multi-scaled movements of his body mechanics, including his hands, arms, torso, and legs. Whereas his motion dynamics defined the direction of the paint trajectories, the shape of their splattered edges was dictated by the second part of the process - the fluid dynamics that evolved under the pull of gravity and subsequent interaction with the canvas. These two interactive patterns can be viewed as the ‘skin’ and ‘bones’ of his poured patterns: the bones represent the trajectories that map out his travels around the canvas (motion dynamics) and the skin represents the splattered variations in the trajectory widths (fluid dynamics).

This interplay between the motion and fluid dynamics is presented schematically in Figure 7. The trajectory patterns (the bones) dominate at scales larger than approximately 5 cm, while the splatter from the fluid dynamics (the skin) dominate below this size transition [9–11]. Both types of patterning feature multi-scaled structure quantified by covering dimensions. We label these dimensions as D**0T (the 0 subscript signifies that it is a covering dimension and the subscript T signifies that it is quantifying the trajectory dynamics) and D**0S (the subscript S signifies that it is quantifying the spatter dynamics). Because distinct physical factors influence these two processes, the difference in the values of D**0T and D**0S is expected to vary from artist to artist [11]. Large differences will result in the appearance of the ‘knee’ in the scaling curves at around 5 cm (Figure 3). As with previous Pollock studies [11], the Dripfest painting knees reflect a steeper slope at larger scales (i.e., D**0T > D**0S). Low splatter paintings potentially create more distinct knees (e.g., the top and bottom paintings of Figure 3) because of the splatter’s lower D**0S. Future quantitative studies will examine knee size and occurrence in detail using bi-fractal fits.

Figure 7. Schematic representation of the ‘bones’ (top) and the “skin” and “bones” (bottom) of the poured patterns.

Defining the two size regimes and their associated dimensions is important for investigating the difference between the adult and children’s paintings. We first focus on the lower size regime dominated by the fluid dynamics. The value of D**0S is influenced by material parameters such as paint density and viscosity, along with the surface texture and porosity of the canvas. However, these factors are standardized across the two groups of painters in our study and so do not factor into the observed inter-group variations. Differences in artistic technique can also influence the dynamics of the falling paint. Previously, two research groups have interpreted Pollock’s poured paintings in terms of the physics of fluid dynamics [65, 66]. Along with the properties of the paint, these investigations incorporated the height and speed of the painting implement (including both linear and rotational actions) into their fluid dynamics equations. The conditions required to achieve fluid “jets” (continuous flows of paint) rather than discrete drips were examined, along with the potential to manipulate the stability of this flow process (including mastering an instability known as “coiling”). Similar to Pollock’s work [65], both the adult and children’s works are typically dominated by stable jets of paint.

Changes in direction of the artists’ motions are also expected to influence the fluid dynamics. Visual inspection (see, for example, Figures 1, 3) shows that, at the lower size scales dominated by the fluid dynamics process, the children’s trajectories are not splattered and are instead characterized by smoother edges (i.e., with lower D**0S values approaching one-dimensional lines). In contrast, the adult paintings feature multi-scaled splatter spreading out from their trajectories. This splatter features an abundance of fine scale structure indicative of high dimensional fractals. These differences in splatter and their associated D**0S values could originate from the artists’ motions. Motions characterized by higher D**0T values will feature many fine-scale twists and turns in the artist’s directions across the canvas, and this promotes splatter patterns with high D**0S values. This emphasizes the interactive nature of the patterns in the two scaling regimes. For conditions such as our Dripfests, in which the materials are standardized, we can therefore expect a degree of correlation between D**0T and D**0S. This expectation is supported by the observation that all three of our scaling parameters - D**0, D**2, and ς - are fitted across both scaling regions and, in doing so, successfully distinguish differences between adults and children. In summary, all three parameters are sensitive to the artists’ motions: for scales above 5cm, they directly quantify the fractal character of the artists’ motions, and at scales below 5 cm they map the fractal character of the motion-influenced paint splatter.

4.2 The potential role of the Artist’s biomechanical balance

Having discussed the sensitivity of D**0, D**2, and ς to artist motions, we now consider how these parameters might be influenced by their biomechanical balance mechanisms. Given that artists typically lean over their canvas during the pouring process, their body motions might be influenced by natural biomechanical balance dynamics which then transfer into the paint trajectories appearing in the artwork. It is well documented that because of physical and behavioral differences, children become ‘off-balance’ more readily than adults [72–84] and their postural motions show large, rapidly changing excursions from their stable position [85–88]. Because the transition from child-like to adult-like postural characteristics occurs at six to eight years of age [84, 88–94], we expect our two groups of painters to exhibit distinctly different biomechanical balance motions.

Of particular interest for the poured painting experiment is the way the two groups accommodate their stability limits. Children have narrower limits, are less accurate in judging them, less cautious as they approach them, and approach them more frequently [89, 94]. Due to their lower coordinated muscle responses, they often over-correct and take longer to restabilize after reaching their limits [83, 89]. During the poured painting task, children would therefore be expected to spend more time close to their limits and change direction less fluidly when responding to them. Children use strategies that minimize their degrees of freedom [90], resulting in more pre-planned ballistic responses [89, 95, 96]. They also restrict themselves to completing one response before initiating the next and directional changes become less fluid [95]. Taken together, these results suggest that the children’s group might show simpler, one-dimensional trajectories that change direction less often when compared to the richer, more varied trajectories of the adults.

Significantly, a growing body of research focuses on the non-linear patterns in people’s postural sway [44–61]. When maintaining balance, the body exhibits multi-scaled sways as an effective biomechanical mechanism for positioning the body relative to the feet and these have been quantified by Lyapunov exponents [44] and fractal dimensions [45]. Differences have been measured between healthy and balance-challenged participants [45], between younger and older adults [47], and also across infant development [48]. Intriguingly, balancing with eyes closed suggests that postural instability increases with a loss of visual information [46]. This might have relevance for activities such as the pouring process that unfold at speeds sufficiently high to limit observations. In addition to stationary balance, fractal characteristics also appear in dynamic activities such as walking [57] and running [58]. While the role of postural sway motion during stationary stance is to fine-tune body motions with centimeter-sized variations, dynamic balance mechanisms require larger sized patterns [58]. Thus, the dynamic balance mechanisms associated with the pouring technique can be expected to span across a significant size range of the scaling plots shown in Figure 3.

A potential interpretation of Figure 4a is that the covering dimensions of the poured patterns (i.e., their D**0 values) are being influenced by the fractal characteristics of their biomechanical balance motions. If this is the case, the high D**0 values observed in the adult paintings might be indicative of multi-scaled swaying motions that promote their dynamic stability through an effective positioning of their body’s center of mass. In particular, the large contributions from fine scale correctional sways appear to be important. In contrast, the lower D**0 values of children’s paintings might arise because their immature biomechanical balance lacks these fine scale corrections. Although speculative, this is consistent with the above developmental research highlighting that younger participants suppress their natural motions as a control strategy, resulting in one-dimensional trajectories that turn infrequently.

Figure 8b shows that the paintings’ D**2 values are typically lower than their D**0 values. This fall off in dimension with increasing q is expected for multi-fractals and is a signature of their multi-fractal depth. Although the focus of our experiment is on D**2 because of its relationship to lacunarity, we note that the fall off continues for higher fractal dimensions. For example, the adult mean values of D**0, D**2 and D**50 are 1.86, 1.84 and 1.82, respectively. The inverse relationship between ς and D**0 revealed in Figure 8a follows from the linear relationship between D**2 and D**0 because of the mathematical relationship D2=2−ς (Methods). In terms of the physical interpretation of this result, the higher ς values for the children’s paintings are a signature of the increased clumpiness of their patterns at fine scales.

Figure 8. (a) A plot of lacunarity slope ς against covering dimension D**0, and (b) a plot of correlation dimension D**2 against D**0. In each case, the blue dots represent paintings by adults and the gray dots represent children’s paintings. The black dot represents Pollock’s Number 14, 1948 an