Fractal Analyses Reveal the Origin of Aesthetics in Chinese Calligraphy

Di Baofeng

Sichuan University

Luo Maoting

Sichuan University

Shi Kai

Jishou University

Liu Chunqiong

Jishou University

Jiao Yang＊

Sichuan University

Abstract: Chinese calligraphy is a thousand-year-old writing art. The question of how Chinese calligraphy artworks convey emotion has cast its spell over people for millennia. Calligraphers’ joys and sorrows were expressed in the complexity of the character strokes, style variations and general layouts. Determining how Chinese calligraphy aesthetic patterns emerged from the general layout of artworks is a challenging objective for researchers. Here we investigate the statistical fluctuation structure of Chinese calligraphy characters sizes using characters obtained from the calligraphy artwork “Preface to the Poems Collected from the Orchid Pavilion” which was praised as the best running script under heaven. We found that the character size distribution is a stretched exponential distribution. Moreover, the variations in the local correlation features in character size fluctuations can accurately reflect expressions of the calligrapher’s complex feelings. The fractal dimensions of character size fluctuations are close to the Fibonacci sequence. The Fibonacci number is first discovered in the Chinese calligraphy artworks, which inspires the aesthetics of Chinese calligraphy artworks and maybe also provides an approach to creating Chinese calligraphy artworks in multiple genres.

Keywords: Chinese calligraphy, Preface to the Poems Collected from the Orchid Pavilion, fractal analysis, Fibonacci number, aesthetics

Chinese calligraphy, with its artistic and linguistic values, has been admired by people for thousands of years (Gaur, 1994). During this time, people have been contemplating the question: How Chinese calligraphy, as the main representative of China’s ancient civilization, conveys art aesthetics, thoughts and feeling (Burckhardt, 2009, pp. 124-126).

The understanding and appreciation of Chinese calligraphy is focused not only on Chinese character recognition but also on its forms of aesthetic expression, such as the character topology styles, general layout and spatial arrangements (Ouyang & Fong, 2008). The different styles of each calligrapher gave birth to a variety of multiple genres (Burckhardt, 2009, pp. 124-126; Ouyang& Fong, 2008). In the Artificial Intelligence (AI) field, it is valuable to program a computer to appreciate and generate the beautiful calligraphic artworks, and it will enhance the optical character recognition, artistic creation, restoration of ancient calligraphic artworks and more (Li, Song &Zhou, 2014, pp. 299-305). The quantitative analyses of aesthetic expressions of artworks is the cornerstone of this AI field. In recent years, achievements in analyzing the manifestations of artworks, such as music and paintings have been realized by using mathematical analyses (Serrà,Corral, Bogu?á, Haro & Arcos, 2012, pp. 485-487; Hsu & Hsu, 1990, pp. 938-941; Mureika, Dyer &Cupchik, 2005; Lyu, Rockmore & Farid, 2004; Taylor, Micolich & Jonas, 1999). The art in Chinese calligraphy is presented thoroughly and vividly so that its structure can be seen instantly, generating an impression in our minds. However, compared with other art forms such as music and paintings,the analysis of Chinese calligraphy is difficult owing to the complex Chinese character topology and the large number of Chinese characters (Li, Song & Zhou, 2014, pp. 299-305; Zhang & Zhuang,2012). Attempts to imbue computers with the ability to grade and create calligraphic artworks have been rare because of the complexity of Chinese characters (Li, Song & Zhou, 2014, pp. 299-305).Little research has been done on the basis of mathematical analyses to quantitatively discuss its aesthetic expressions, especially the general layouts and spatial arrangements.

We investigated the calligraphy work Preface to the Poems Collected from the Orchid Pavilion which was praised as the best running script under heaven. It is the most important handwriting work of a great calligrapher named Wang Xizhi (303-361), created during the Eastern Jin Dynasty (317-420).However, the original copy was lost during the Tang Dynasty (618-907). Among all the facsimile copies, one copy, with the length of 24.5cm and width of 69.9cm, called the Shenlong Version, created by a calligrapher named Feng Chengsu (617-672) during the Tang Dynasty, is the most renowned.This artwork consists of 28 columns with 324 characters. For over one thousand years, many research findings have been achieved with respect to its history, calligraphy, literary ideology, etc. Its overall layout structure, as a key component of the work, remains a mystery to be explored in multiple genres.Producing an outstanding calligraphy artwork, with various stroke shapes, requires many skills.However, the stroke shapes are not the only sources for the aesthetic perception of calligraphy works.For example, when people are appreciating the calligraphy work of Preface to the Poems Collected from the Orchid Pavilion feelings of softness and fluency are also created. Such calligraphy has been described as: “The dragon jumping over the heavenly gate and the tiger lying in the watchtower of the phoenix” (Ouyang & Fong, 2008; Alsop, 1982; Yee, 1974; Robinson, 1995). These mysterious feelings,mainly subjective, relate to every character in the calligraphy. The question was: Could people manage to describe and capture such mysterious feelings in a quantitative way? The answer seems to be yes. By exploiting the fractal method in a time series, we have unveiled several macroscopic statistical patterns and aesthetics characterizing this calligraphy artwork.

Methods

Materials and Data Extraction

The Chinese characters have some formal square structures. In Chinese calligraphic artworks,in order to pursue the aesthetic perception, calligraphic characters were often produced to show an inclination of being spiritually round with the specified round stroke representing and writing trajectories. Moreover, with the aim of pursuing the overall beauty of character shapes, Chinese calligraphy characters are not of uniform size in some artworks. That is “square in shape, round in inclination; spirit in square, and round in form” (Kang, 2006). Thus, in Chinese calligraphy,handwriting and composition merge in the idea of combination and complementation of squares and circles. Considering the importance of font size in general layout, we ignored the stroke shapes of each character and encoded the Chinese calligraphy code word’s dataset.

Based on units of squares and circles, the areas of the 324 characters were measured in the facsimile copy by the actual length scale of Preface to the Poems Collected from the Orchid Pavilion based on spatial georeferencing respectively. The traditional Chinese characters are written from right to left and from top to bottom. By the corresponding square and circle units, these characters area data with word order are shown in Figure 1 respectively.

Detrended Fluctuation Analysis

To demonstrate the existence of a correlation, the detrended fluctuation analysis (DFA) (Penget al., 1994, pp. 1685-1689), a benchmark method to quantify long-term correlations, was used. The fluctuation function F(l) at the special scaling l of a time series follows a power-law relation,where F(l)∝lα. When α=0.5, corresponding to the uncorrelated time series, the larger (or smaller)α indicates long-term correlation (or anti-correlation).

Box-Counting Technique

The fractal dimension (DB) can quantify the scaling relationship between the patterns observed at different magnifications (Lyu, Rockmore & Farid, 2004; Taylor, Micolich & Jonas, 1999). For Euclidean shapes, DBis the integer value. For example, for a smooth line, DB=1 ; while for a filled area, DB=2. However, fractal shapes, DBhave a non-integer value. DBvalue can be determined by the well-established “box-counting” technique. When one covers an object with a grid of square boxes of size L, the number of occupied squares isN(L)∝L-DB. In fractal geometry, the power law generates the scale invariant properties.

Results and Discussions

Statistical Patterns of the Data

We found that the cumulative distribution of these area series could be matched to a stretched exponential distribution,P(S)=A·exp[-B·(S/RS)C]. Wherein P(S) is the cumulative probability distribution of the circle, S is the area, RSis the mean area, and A, B and C are the fitting parameters. For the circle and square areas series, CC=3.605 and CS=3.517 respectively. These results are shown in the Figure 1. This statistical distribution has been found both in natural and social phenomena, such as climate records (Bunde, Eichner, Kantelhardt &Havlin, 2005), Internet traffic (Cai, Fu,Zhou, Gu & Zhou, 2009) and financial markets (Wang, Yamasaki, Havlin &Stanley, 2006). The stretched exponential distributions of the series suggest the existence of correlation in this calligraphy.

Figure 1 Schematic diagram showing the cumulative distribution of characters area data in the calligraphy artwork Preface to the Poems Collected from the Orchid Pavilion.

The calligraphy artwork Preface to the Poems Collected from the Orchid Pavilion(The Palace Museum, 24.5cm×69.9cm) was showing in the upper inset. In the bottom left corner,we found that the fluctuations in the characters area data corresponding square and circle units.In the bottom right corner, the cumulative distribution of these area series could be fitted by a stretched exponential distribution.

The Relationship Between Long-term Correlations and Human Emotion

The single scaling exponent α calculated over the whole series is not informative regarding the evolution of the correlation structure. Within all 28 columns of characters in the Preface to the Poems Collected from the Orchid Pavilion, we considered a sliding window consisting of six columns of characters and calculated the local α parameters. The number of datasets in each window was chosen to provide a sufficient number of points to perform the estimates. The shift between two successive windows was set 1 column. The reckoning is shown in Figure 2 that the correlation structure in the 16thcolumn changed abruptly. Before the 16thcolumn, there is an anti-correlation feature (α＜0.5), namely when the previous character is bigger, the following character is smaller. After the 16thcolumn, there is a long-term correlation feature (α＞0.5). Here the size of characters presented a continual rule.

The local correlation characteristic was reflected by the single scaling exponent calculated by the detrended fluctuation analysis. The sliding window consists of six columns of characters. The shift between two successive windows was set 1 column.

Figure 2 The evolution of the correlation structure in the Preface to the Poems Collected from the Orchid Pavilion

Fractal Analysis of the Artwork

The left insets of Figure 3 (a) and Figure 3 (b) reflect the box-counting analysis of the whole circle and square areas series respectively. The transition between two power-law processes is almost equal to the mean of each character’s area. At smaller spatial scales, DC2≈1 (circle) and DS2≈1(square), reflecting the linear section of the curve fluctuation diagram. While at larger spatial scales, DC2=1.642 (circle)and DS2=1.652 (square), which reflect the scale invariant properties. Furthermore,we considered a sliding window with six columns of characters and calculated the local DBparameters. The probability density of DBvalues is shown in Figure 3. For circles, the mean value of DBvaluewhile for squares,

Figure 3 Box-counting analyses of the fluctuation curve in characters area data.

In the left insets of (a) and (b), the box-counting analysis of the whole circle and square areas sequences were provided respectively. The breaks all occur at about lg(L)=2.5. The local fractal dimension(DB) parameters with a sliding window with six columns of characters were calculated. The probability density of the local DBvalues of circle and square areas sequences is shown in (a) and (b) respectively.

Discussions

We concluded that the spatial distribution pattern for the size of characters is not random. In the uncorrelated time series, a Poisson distribution, as lg[P(S)]∝－x, is expected (Bunde, Eichner,Kantelhardt & Havlin, 2005). We considered that when the calligrapher was creating this work,his feelings could be flexible, which gave rise to the changing in size of the characters. As a result, the local correlation structure is supposed to change during the creation of this work, and the change revealed that the feelings of the calligrapher changed during the process of creating the art work. At the beginning, the calligrapher might be cautious in order to keep the balance of the whole art work. When the former character was bigger, the following character was smaller (anticorrelation feature). Later in the creating period, the author might enter a free state with less care regarding the size of characters but more focus on the consistency of the whole work. That is the long-term correlation.

The size change of characters was more inclined to reflect expressions of the calligrapher’s complex feelings. Too simple or too complex a change was likely to be irritating. The secret to creating an outstanding calligraphy work is to balance the predictability. Calligraphy work at the same time is literature work. When we read through the work, we understand when the author’s emotion rises and falls but we do not always notice the size change of characters. Our brain tends to predict what comes next. Due to the unique individual perception of the calligraphy, some expectations are matched while others seem to be ignored. An experienced calligrapher rewards part of our expectations to keep us attracted, while delicately violating the other expectations. This is the fundamental reason why a calligraphy work looks smooth and consistent within one breath.

Human emotions are the decisive element for the style of the whole piece, while calligraphers’unique writing skills decide every structure of each character. The means of fractal dimensions of the whole circle and square areas series were close to the Fibonacci sequence. This is an exciting outcome. Fibonacci numbers have been applied to various artistic works in various forms,including the paintings of Leonardo da Vinci. But its application has been never previously found in Chinese calligraphy. The Fibonacci sequence endows Preface to the Poems Collected from the Orchid Pavilion with geometry and aesthetic beauty.

Conclusions

In summary, with the fractal method in a time series, a natural bridge between the quantitative analysis and Chinese calligraphy has now been built. The Fibonacci sequence has been discovered in the Chinese calligraphy artworks, which provides aesthetics to Chinese calligraphy. This study presented an approach to understanding the beauty of Chinese calligraphy, and could provide assistance for the application of AI techniques to create Chinese calligraphy artworks in multiple genres.