Fractal dimension and topological invariants as methods to quantify complexity in Yayoi Kusama’s paintings

The symmetry, complexity or periodicity of irregular spatial-temporal patterns could be characterized as beautiful. For many years, numerous investigations have tried to characterize the dynamical processes that lead to the creation of such patterns. Structural properties in many complex systems are the final manifestations of the internal dynamics. Through measures such as connectivity, porosity, texture, roughness, symmetry or temporal correlations to name a few, the complexity of systems can be characterized Ball (2016); Barnsley (1993); Mandelbrot (1982); Etzenhouser et al. (1998); Daccord et al. (1986); Goodchild and Mark (1987). Nature produces many examples of complex patterns Caballero et al. (2012); Barnsley (1993); Andrienko (2018). Some investigations have tried to described the dynamics behind natural complexities, many of which are visual: for example Turing patterns Caballero et al. (2012), symmetries, textures and patterns of marine invertebrates Ball (2016), etc.

Art, of course, is full of complexity but its objective quantification is not trivial. Some studies have quantified particular characteristics of abstract paintings. For instance, Naumis et al. Aragón et al. (2008) evaluated the turbulent luminance of van Gogh paintings, Mekler et al. Martínez-Mekler et al. (2009) measured size-ordered distributions in Vasili Kandinsky. The physical process leading to the emergence of complex patterns in the David A. Siqueiros accidental painting technique Palacios et al. (2019) and in Jackson Pollock’s drip paintings Zetina et al. (2015) have been studied by our group. The fractal dimension of artworks has been widely used to assess the degree of complexity Mureika and Taylor (2013); Taylor et al. (1999); de la Calleja et al. (2016); Redies et al. (2007); Palacios et al. (2019); Zetina et al. (2015). This attempt has created a renewed interest in the subject and opens a new area of study Abbot (2006); Coddington et al. (2008), even if their validity and extend of applicability have been questioned Jones-Smith and Mathur (2006); Jones-Smith et al. (2009). Despite these efforts, a general consensus on the appropriate mathematical tools to quantify the complexity of abstract art has not been reached.

In this report, the complexity of an image is evaluated using those two different mathematical approaches: fractal dimension and Betti numbers. Both measures are used to characterize structural properties as geometry, symmetry, non-regularity, dimensionality and connectivity of a set. Felix Hausdorff developed the mathematical description for Mandelbrot’s sets better known as fractals Mandelbrot (1982). It is well accepted that the Hausdorff-Besicovitch fractal dimension is the usual parameter to characterize irregular spatial-temporal patterns as a result of temporal or spatial evolution of the elements of the system Mecke (1996); Ball (2016); Forsythe et al. (2011). On the other hand, the modern formulation of the theory of algebraic topology Mac Lane (1986) is also a powerful tool to analyze complexity in many systems by using the value of topological invariants, the so-called Betti numbers Pranav et al. (2017); Gerarld (1990); de la Calleja and Zenit (2017); de la Calleja et al. (2016). Despite being widely used in many other areas to quantify structural properties Kosterlitz and Thouless (1973); Gottsche (1990); Spånslätt et al. (2005); Kondic et al. (2012); Xia and Wei (2014), the use of topological invariants as a measure of complexity has not been used much for the case of art. Recently, the Betti numbers of abstract expressionist artworks were computed and used as a measure of complexity de la Calleja and Zenit (2017), with promising results to be applied in other cases.

To validate our proposed methodology to quantify complexity we first apply it to sets of black and white synthetic abstract images formed by randomly adding adding dots (or lines) in a two-dimensional empty area. The advantage of using black-white two-dimensional images allows applying the technique without any previous image processing. Then, the same methodology was used to measure the complexity in Yayoi Kusama’s artworks. Her work is characterized by the use of regular dots patterns. Some critics have cataloged such periodic distribution of dots and nets as ‘simple’. The analysis of Kussama’s works is of particular importance because we found that their fractal dimension hold similar values as other abstract paintings, such as Pollock’s; however, the value of the Betti numbers indicate a lack of connectivity and not necessarily complexity. Hence, we show that the fractality of an image cannot be used alone to quantify complexity. Furthermore, our results indicate that complexity is not related to the visual appreciation of the image.

To investigate the complexity in abstract images, two methods were considered to assess the structural properties of images: the fractal dimension and the value of topological invariants. Both procedures have been used in many cases from spatial distribution of galaxies to music Pranav et al. (2017); Barnsley (1993); Bigerelle and Iost (2000). In this study these mathematical parameters were calculated for a set of synthetic abstract images composed with dots or lines. As a result, two different measures were obtain for a given image. This allows us to compare and contrast the structural properties of the images. Evidently, in the art world the painting composition contains multiple components, such as dense lines, layers of paint overlay, cluster nodes, voids, or shapes and shadows. And many of those objects live in a three dimensional space.

In this work we evaluate the complexity of synthetic abstract images considering the values of the fractal dimension and the Betti numbers. The procedure to calculate complexity was the following: series of synthetic abstract images were produced using a script written with a Matlab$\copyright$. Two kind of images were tested: dots and lines. In both cases a predetermined number of black objects (dots or lines) were placed randomly within a rectangular blank space. To ensure repeatability, at least three images were produced for each condition. By progressively adding dots or lines on the given area, the black area increases as dots or lines overlap. The area size was kept fixed for all cases. For the case of dots, three series were constructed with dots of three mean diameters: $D_{1}=0.01$, $D_{2}=0.02$ and $D_{3}=0.03$. The dot size distribution has a standard deviation of 10% of the mean diameter. For the case of lines objects of fixed length, $L=0.1$, were placed randomly in the space with a random orientation. Three series were generated by changing the thickness of the lines, obtained series of $T_{1}=0.1,T_{2}=0.3$, and $T_{3}=0.5$. The mass distribution of dots or lines on the blank space did not have any distribution preference, hierarchical aggregation or imposed self-similarity. Typical images generated with dots are shown in Fig.1.

As mentioned previously, the aim of this report is to analyze the complexity of synthetic abstract images by measuring their fractal dimension and the value of their two Betti numbers. The case of synthetic images composed by dots is presented and analysed here. The case for lines shows qualitatively similar results; hence, it is presented in Appendix A. By observing the changes in fractal dimension and Betti numbers as the number of dots increases we can evaluate how the complexity evolves. We use ‘simple’ distributions of dots motivated by our interest to evaluate the complexity of paintings by Japanese avant-garde artist Yayoi Kusama. In 1959, she created Infinity Nets, a 10 meters paint with meticulously inscribed patterns suggestive of a white net, over a slightly darker grayish-white background. Her work has been compared to Jackson Pollock’s Nakajima (2006); Sydney (1958). Visually, it is evident that the two artists’ abstract expressionists works are significantly different. However, as shown below, our measurements of their fractal properties suggest that their paintings have similar structural properties.

Yayoi Kusama is an international avant-garde Japanese artist. Her ‘net painting’ style has been deemed as her most prototypical obsessional art Pollock (2006). Some critics have interpreted her style as ‘culturally specific’, ‘historically feminine’ and ‘subjective painting without image’ Pollock (2006). The art of the so-called queen of Polka Dots is characterized by constructions of by consecutive dots. The world-wide success of her recent Infinity Rooms, has made Yayoi Kusama one of the most recognizable and admired contemporary artists.

Kusama’s paintings have evolved significantly from simple stylistic observations in the 1950s to pure and personal expression of her mental illness in the 1990s Nakajima (2006). Technically, Kusama’s brushstrokes are very thick and do not seem to finish.

Considering the technique describe above, we analyse Kusama’s artworks in a more objective manner. In order to quantify the complexity of Kusama’s infinity net style, $30$ paintings were selected. A complete list of these works is included in Appendix B. Most of the artworks used here are from the following collections: Dot’s obsession Kusama (2019, accessed 10/27/2020a), Infinity Nets Kusama (2013), My eternal soul Shimbun et al. (2017) and Silkscreens from 2009 Kusama (2019, accessed 10/27/2020b). The specific images were selected by considering the appearance of the characteristic repeated dot pattern and the absence, as much as possible, of figurative expressions or objects. The analysis of complexity in pieces of art is possible because of the two dimensional nature of the paintings. Although it is, in principle, possible to extend our proposal to 3-dimensional spaces it was not attempted.

Figure 4 show four of the most iconic Kusama’s artworks: $(a)$ No. Red B (1960), $(b)$ Pumpkin Yellow T (1992), $(c)$ Infinity Nets Yellow (1960) and, $(d)$ Untitled (1967) Abrams (2018, accessed 10/27/2020). The pieces of art of Kusama are full of the symmetric and irregular distribution of dots which apparently are a result of simple compositions. However, our results provide compelling evidence that this fact is questionable.

The fractal dimension $f(\alpha)$ for the selected set of paintings is shown in Figure 5. As can be observed, the fractality of the Kusama’s artworks is between $1.7<f(\alpha)<1.9$, with a mean value of $f(\alpha)=1.8285$ (dashed black line). These range of fractal dimension is in good correspondence with other abstract images with a high density of dots with the smallest diameter as we observed in Figure 2. Also, Kusama’s artworks fractal dimensions can be compared with those for other abstract expressionisms de la Calleja et al. (2016). Interenstingly. the fractality of the art pieces of Yayoi Kusama evaluated in this report have values similar to those calculated for Jackson Pollock’s paintings Taylor et al. (1999); de la Calleja et al. (2016). This result suggests that the despite the obvious differences in style and composition the complexity of Kusama’s paintings, using this metric, is similar to that of Pollock’s.

Now, to complete the complexity characterization of Kusama’s paintings we proceed to calculate the values of the Betti numbers. To do this, the images are binarized choosing a threshold level following ordinary image processing techniquesSchneider et al. (2012). The value of $\beta_{0}$ number quantifies the connectivity of a set and, while the value of $\beta_{1}$ number counts the holes. Figure 6 (a) show the Betti numbers for Kusama’s paintings listed in Appendix B. Note that both $\beta_{0}$ and $\beta_{1}$ are normalized with the area of each painting (A), to allow comparison between works of different sizes. The mean value of $\beta_{0}$ (brown dashed line) and the mean value of $\beta_{1}$ number (green dashed line) are also shown. In general, we find that $\beta_{0}>\beta_{1}$ There is expected since the Kusama’s dots are mostly disconnected, each one of them represents a single connected domain with would lead to a high $\beta_{0}$ value. Since there are few overlaps, only a few holes appear, therefore one would expect the images to have low $\beta_{1}$ values. This, of course, is not always the case. In many instances the high density of dots are in balance with the number of holes.

We have conjectured that a high image complexity corresponds to cases when the number of connected regions is roughly the same as the number of holes, $\beta_{0}\sim\beta_{1}$ de la Calleja et al. (2016). Fig.6$(b)$ shows the ratio $\beta_{0}/\beta_{1}$ of all the works analyzed here. Overall, it is clearly observed that $\beta_{0}>\beta_{1}$. This is good agreement with can be deduced from the visual inspection of the paitings: dispersed non-overlapping points do not lead to the formation of many holes. Of all the works, only 1/3 of them a value of $\beta_{0}/\beta_{1}$ close to unity. Hence, from this metric the complexity of Kusama’s paintings seems to be smaller. This observation is in sharp contrast with what was concluded from the fractal dimension analysis. It is also interesting to note that for the case of Pollock’s paintings, the ratio $\beta_{0}/\beta_{1}$ was closed to unity, at least for works close to the dripping period.

The fractal dimension and Betti numbers can both be used to characterize complexity with a non-predictable structure. Fractal properties are present in a wide and extend the number of objects. While the fractal dimension measures certain spectral properties of an image, the Betti numbers quantify the topological shape. As demonstrated above, they can both provide a metric for complexity of abstract images. One interesting aspect conclusion drawn from this study was that, although Kusama’s art works have fractal dimensions comparable to other abstract paintings, the value of the Betti numbers are strikingly different.

The analysis of image complexity is a useful non-invasive technique to characterize artworks. With the analysis presented here we demonstrate that it is possible to capture essential information about painting styles. Yayoi Kusama has painted many pieces with dots, and her influence and impact have been significant. Kusama’s paintings are created with consecutive and order distribution of dots. From our analysis, the fractality of Kusama’s art indicates high complexity while the value of the topological invariants show more homogeneous textures with a high degree of dispersiveness. Hence, we can conclude the complexity of Kusama’s art work has a richer depth than that observed for other abstract painters, including Jackson Pollock.