1 Introduction

Traditional villages are important carriers of Chinese history and culture and the birthplace of local traditions, possessing rich tangible and intangible cultural heritage (Hu, 2021). Since 2012, the Ministry of Housing and Construction of China has released a list of traditional villages in China, which reflects the unique local customs, social organization, and ecological environment of various regions and ethnic groups in China over a long period of time (Cui et al., 2020; Bian et al., 2022). Sichuan Province, located in southwestern China, is now home to 1,383 traditional villages, of which about 20% are Tibetan traditional villages in the highland mountainous areas of western Sichuan. The Gyaltsen, Amdo, and Baima Tibetans are three clans of the Tibetan ethnic group living in different villages in the western Sichuan plateau. Their villages present diversified landscape features with important spatial, cultural, and social values due to the differences in geographic features, religious beliefs, folk cultures, and production and living styles. However, many traditional villages face challenges such as landscape homogenization, weakening of vernacular, and spatial fragmentation in the process of urbanization, affected by the impact of external culture or capital and slow internal development (Xiaohua et al., 2020; Fang et al., 2022). Therefore, how to protect and develop traditional villages has become a topic of urgent concern in contemporary society (Feng, 2021).

In nature and human society, there are many complex structures and forms that appear to be irregular and fragmented, but in fact have a certain degree of self-similarity and repetition (He & Zhao, 1996). These structures and forms are called fractals, a mathematical concept that can be used to describe and quantify complex phenomena in nature (Mandelbrot, 1967). The fractal theory provides us with a new perspective to think about and analyze the functionality, diversity, and connectivity of landscape space (Martinez et al., 2023; Zhu et al., 2022). Landscape space refers to the spatial structure composed of natural and anthropogenic elements, which is the result of the interaction between humans and nature, as well as the embodiment of human culture and society (Hao et al., 2021; Forman, 2006).

The spatial study of traditional village landscapes with fractal theory has addressed the fractal characteristics of village landscapes, the relationship between ethnicization and village landscapes (Huang et al., 2020), the complexity of village landscapes (Li, 2022), the spatial distribution pattern of landscapes (François, 2022), the prediction of structural evolution (Liang et al., 2017; Yu & Zhao, 2022), and sustainability of landscapes. The research methods include fractal geometry for segmentation and classification of spatial images (Huang et al., 2020; Mahdi, 2016), counting box dimensions for land use and architectural complexity (Florin, 2020; Massimo, 2017), and different scales, perspectives, and dimensions for village spatial morphology (Mahdi & Hassan, 2016; Zhang, 2019; Thanh et al., 2022), analysis on spatial aggregation and planar form (Luo, 2020; Zhu et al., 2022), etc., so as to reveal landscape space’s self-organization, self-similarity (Özdemir, 2021; Bian et al., 2022), and influencing factors including the natural environment, historical context and social development (Asimakopoulos et al., 2016; Dharmendra, 2022). In existing studies show that landscape space has fractal characteristics in nature, and this fractality is the coordination and balance of the structure and function of the landscape with internal and external factors (Wang et al., 2019; Li et al., 2022).

Tibetan traditional villages are a typical landscape space in the highland mountainous areas of western China, and they have unique natural ecological landscapes and diverse cultural landscapes, reflecting the coordination and adaptation of the Tibetan people with the natural and humanistic environments during their long history of development (Liao, 2019). At present, the research on traditional Tibetan rural landscape mainly includes the following aspects: The first is the study of landscape distribution characteristics and influencing factors, using spatial analysis to compare the differences in the morphological patterns of Tibetan villages and to analyze the influence of the natural environment in terms of topography and geomorphology, the structure of the water system and road network, and land use, as well as the socio-economic level of urbanization, transportation conditions, and regional culture on the spatial characteristics of the landscape. (Sun, 2018; Wang & Liao, 2021; Zhang et al., 2022). The second is the study of cultural landscape and conservation, which mainly involves the aspect of diversified cultural forms in the countryside, explores the relationship between historical changes, religious beliefs, ecological and philosophical thoughts, etc., and the cultural landscape, and proposes the strategy for the conservation and development of Tibetan landscapes. Third, the application of theories, mainly using the theory of landscape security pattern (Wang & Liao, 2021), landscape morphology (Hu et al., 2021), and the theory of landscape genetics (Chen & Xie, 2016), etc., to analyze the relationship between natural and cultural factors and the spatial structure, morphology, and pattern of Tibetan landscapes (Liu et al., 2019; Ma & Chen, 2020; Zhang et al., 2022).

However, there are fewer studies on the landscape space of Tibetan villages in western Sichuan under the perspective of fractal theory, and even fewer studies on the characteristics and mechanisms of their fractals. This may be due to the complexity and diversity of Tibetan landscapes, and the fact that it is more difficult to obtain the basic data with higher accuracy in the location and geography of villages. Fractal theory can quantitatively describe the spatial structure and study the irregularity, complexity, and diversity of landscape space (Yu et al., 2019; Zhang et al., 2020). Nevertheless, it also has limitations because it is a mathematical model that cannot explain the social, cultural, and historical factors behind it. In this paper, we take the traditional villages inhabited by the Tibetans in Gyatong, Amdo, and Baima in western Sichuan as the research samples. We combine the regional and cultural background of Tibetan villages and use spatial analysis methods and fractal dimensions to measure and evaluate (Zhang et al., 2019). We compare the landscape pattern, morphological structure, and influencing factors of the villages to dissect the fractal characteristics of their landscapes. This helps to strengthen the original authenticity, integrity, and heterogeneity of the village landscape and provides a reference for the conservation and sustainable development of cultural heritage in the study area. The research questions of the article are as follows:

  • RQ1: What are the spatial fractal characteristics of traditional Tibetan village landscapes in western Sichuan and their influencing factors?

  • RQ2: What are the inspirations for the conservation and development of traditional Tibetan village landscapes in western Sichuan based on fractal theory?

2 Research area and method

2.1 Research area

Western Sichuan is located at the intersection of the Tibetan Plateau and the Hengduan Mountains and is the main area where Tibetans live. The Tibetans are divided into several branches, of which Gyatso, Amdo, and Baima are three, and this paper selects representative ethnic traditional villages as the research samples, namely: Xisuo Village, Shenzuo Village, and Minzu Village (as seen in Fig. 1).

Fig. 1
figure 1

Study area map

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Xisuo Village is a Jajong Tibetan village located in Zhuokeji Township, Malcolm City, Aba Tibetan Autonomous Prefecture (102°17′41.148″E,31°52′9.688″N). It has a population of 349, a village area of about 0.15km2, and an average elevation of 2,660 m. It is included in the second batch of Chinese traditional villages and is a key urban area in the “Twelfth Five-Year Plan” of Malcolm City. Surrounded by mountains and water, the village preserves the cultural heritage of traditional Tibetan-style houses, Tusi official fortresses, and the Nyingmapa Dandurand Buddhist Monastery of Tibetan Buddhism, which is more than 700 years old.

Shenzuo Village is a settlement village of the Amdo Tibetan people located in Charlie Township (101°57′59.916″E, 32°43′24.799″N) in Aba Prefecture. It has been included in the sixth batch of Chinese traditional villages list. It is a semi-developed village hidden in the most beautiful colorful forest in Northwest Sichuan, far away from the hustle and bustle of the city. The village has only fifty households, with a village area of about 0.29km2 and an average altitude of 3,100 m. It has 300-year-old Tibetan-style houses, the Golden Monastery of the Gelugpa School of Buddhism and the Longyi Monastery of the Yungtsek Monastery. The original appearance and intact ecological communities of the village were selected as a model village for rural revitalization in Sichuan Province from 2020 to 2021.

Minzu Village is located in Muzuo Tibetan Township, Pingwu County (104°27′54.358″E, 32°39′34.986″N), in the northeastern part of the "Tibet-Yi Corridor" proposed by Mr. Fei Xiaotong and is an important area for the cultural integration of Tibetans, Yi, and Chinese. The village is one of the villages inhabited by Baima Tibetans, with a population of 553 and a village area of about 0.27km2. The village is situated in the valley where the Duobu River and its tributaries converge, and the terrain gradually decreases from east to west, with an average elevation of about 2,000 m above sea level. The village retains a large number of ancient residential complexes and traditional folklore inherited from ancestors, all of which are precious intangible cultural heritages. The village is listed in the fourth batch of Chinese traditional villages and is also an "Excellent Village for Rural Tourism in Sichuan Province".

These three villages are located in different geographical locations with different climatic conditions, geomorphological features, and hydrological characteristics, as well as their own religious cultures and beliefs, folklore, and historical traditions. Therefore, choosing these three villages as research samples can provide us with useful research information, and also help us understand the landscape characteristics and differences of traditional Tibetan villages of these three tribes, so as to provide certain references for conservation and development.

2.2 Research data and method

2.2.1 Data type and resource

The data used in the study are mainly geographic information data such as administrative boundaries and river and water systems. Information about villages was accessed through the Chinese Traditional Villages website (http://www.chuantongcunluo.com) and the Ministry of Housing and Urban–Rural Development website (https://www.mohurd.gov.cn/); Collecting DEM data (30 m resolution) and POI data of the villages on the Geospatial Data Cloud website (https://www.gscloud.cn/) and Gaode Open Platform (https://lbs.amap.com/); Obtaining the ethnic composition of the study area through field survey visits, thus forming a diversified data source.

2.2.2 Research path

This paper adopts a combination of qualitative and quantitative methods to help us answer the research questions. through field visits, surveys, and observations to collect relevant data on the natural environment and cultural characteristics of villages, and combines spatial analyses and fractal dimension calculations to explore the constituent elements of the landscapes of the sample villages, the landscape pattern, and spatial morphology structure, etc., and then validates the model and the analytical results by using the leave-one-out method cross-validation method, and applies multiple regression to find out the factors that affect fractal characteristics. We verified the feasibility of the research path from several perspectives.

Firstly, GIS spatial analysis is utilized to understand land use and landscape element composition. Through spatial analysis, data such as land type, vegetation distribution, and water system distribution can be obtained (Hu et al., 2021). Land use data, like a mirror reflecting human-land relations or social relations, can objectively explain the spatial pattern and constituent elements, avoid the uncertainty of land use classification, and guarantee the accuracy of the spatial location as well as the area of the study area, so as to improve the accuracy and effectiveness of the analysis of the landscape pattern (Ji & Zhong, 2021). Next. Landscape elements were extracted based on land use, which can help us understand i.e. the individual components of the landscape, including abiotic, biotic, and cultural landscape elements. Abiotic landscape elements are mainly natural geography, such as topography, climate, and hydrology of villages; biotic landscape elements are mainly plants and animals and habitats, such as arable land, water bodies, grasslands, woodlands, etc., which have the functions of production and environmental services; and cultural landscape elements include towns and rural settlements, which are superimposed on natural landscapes under the role of human activities (Wang & Liao, 2021), such as the study landscape, which is the most important component of the landscape, and the most important component of the landscape is the natural landscape. 2021), such as ancient watchtowers, Buddhist temples, official fortresses, and Tibetan-style dwellings in the study area. Then, based on the extraction of landscape elements, FRAGSTATS was used to calculate the landscape spatial indices of the sample villages, which reflected the landscape pattern information in terms of the number of landscape patches, aggregation, and density to obtain quantitative characteristics of the landscape, and thus provide more comprehensive data for the measurement of fractal features (Chen, et al., 2019).

Fractal features refer to the similar structure and form of objects or systems at different scales, which can be described by a number of indicators, among which the fractal dimension is a scale-dependent indicator of fractal features, which can be used to characterize the spatial morphological complexity of the landscape, and quantitatively reflect the self-similarity and irregularity of spatial structure. Therefore, we used the counting-box dimension to measure the fractal dimensions of village architectural clusters, vegetation areas, public spaces, village boundaries, road networks, and water systems (Li et al., 2020) from a "point-line-plane" perspective to analyze their morphological features and structures (As shown in Table 1).

Table 1 Counting-box dimension measurement objects and characterizing meaning
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The Counting-box dimension is a widely used method to calculate fractal dimension, which is based on the principle of covering the target image with a box whose side length is \({\delta }_{i}\); the square box side length scale is \({\delta }_{i}\)(usually \({\delta }_{i}\) is set to a power of 2, That is \({\delta }_{i}\)= \({2}^{i},i=\) 0,1,2,……,n); The number of images covered by boxes is N(\({\delta }_{i}\)), If it is consitent with N(\({\delta }_{i}\))∝\({{\delta }_{i}}^{-dim},\) Then it indicates that the study area has fractal characteristics. The fractal dimension is calculated based on the logarithmic relationship between the number of squares and the size of the squares with the following equation:

$$\mathrm{Dim}_{box}=-\frac{\mathrm{logN}\left(\updelta\right)}{\mathrm{log}\left(1/\updelta\right)}$$
(1)

where \({\mathrm{Dim}}_{box}\) denotes the fractal dimension, N denotes the number of boxes used, and \(\delta\) denotes the box size. The principle of measurement can be understood as a coarse visualization of the fitting of the study object with boxes (He & Zhao, 1996).

Meanwhile, in order to verify the validity of the research results, leave-one-out cross-validation is an available method to effectively assess the robustness and generalization ability of the research results and models with small samples, avoiding the problem of overfitting or underfitting (Wild et al., 2022). The principle is to use one sample at a time as the test set and the remaining samples as the training set, and repeat the process N times, obtaining one prediction error each time; repeat the process until all the samples have been tested. The average of the N prediction errors is the final prediction error. Then compare the error and goodness of fit between the test set and the prediction set, if the MSE value is smaller and the R is closer to 1, it means the model validity is better. The equation is as follows:

$${CV}_{\left(n\right)}=\frac{1}{n} \sum_{i=1}^{n}{MSE}_{i}$$
(2)

where n is the number of samples in the dataset and \({MSE}_{i}\) is the mean square error at the i th time when the I th sample is left as the test set. In general, the smaller the prediction error, the better the model. This method can help to improve the reliability and validity of the research results, as well as provide more references and lessons for subsequent research.

$$\mathrm{R}=\frac{\sum_{i=1}^{n}({x}_{i}-\overline{x })({y}_{i}-\overline{y })}{\sqrt{\sum_{i=1}^{n}{({x}_{i}-\overline{x })}^{2}}\sqrt{\sum_{i=1}^{n}{({y}_{i}-\overline{y })}^{2}}}$$
(3)

where R is the coefficient of determination, which indicates the correlation between the independent variable and the dependent variable. the absolute value of R ranges from 0 to 1, and the closer it is to 1, the stronger the correlation and the better the fit. \({x}_{i}\) and \({y}_{i}\) represent the ith observation, \(\overline{x }\) and \(\overline{y }\) are the mean values of x and y, respectively.

In addition, the correlation between fractal dimensions and landscape elements (natural landscape, humanistic landscape) was analyzed to find the factors affecting the spatial fractal characteristics of the landscape (Tao et al., 2022). The multiple regression method is a commonly used statistical model that describes the relationship between multiple independent variables and a dependent variable with the following equation:

$$\mathrm{D}={\beta }_{\mathrm{O}}+{\beta }_{1}{x}_{1}+{\beta }_{2}{x}_{2} +...+ {\beta }_{m}{x}_{m }+\epsilon$$
(4)

where D denotes the dependent variable, representing the fractal dimension of various landscape elements; \({x}_{1}\), \({x}_{2}\), ……\({x}_{m}\) denote the independent variables, representing the landscape factors in the study area. \({\beta }_{0}\), \({\beta }_{1}\), \({\beta }_{2}\)……\({\beta }_{m}\) denotes the regression coefficient, which represents the degree of influence of the independent variable on the dependent variable. The \(\epsilon\) denotes the error term. In model analysis, the degree of correlation between the independent and dependent variables is mainly based on the F-value, P-value, and the goodness of fit R2 (Li et al., 2021). For example, the F value, that is, the ratio of variance to residuals, the larger the F value, the better the model fit, the residuals are small and the difference is significant; the size of the P value, indicating the significant level of the F value, the P value is less than 0.05 means that the F value is significant, the selected variables are valid; the goodness of fit R2 test, the closer the value of R2 is to 1, it means that the better the fit, and the opposite is a poor fit.

3 Analysis and results

3.1 Landscape elements and spatial pattern characteristics

Traditional Tibetan villages in western Sichuan located in high-altitude mountainous areas or canyon zones, such as the Kamba, Gyatso, and Amdo Tibetan villages in Ganzi and Aba, are mostly distributed in the transition zone between the Sichuan Basin and the Tibetan Plateau, in a scattered pattern, interspersed with Han Chinese villages; such as the Baima, Ershu, and Moya Tibetan villages in Mianyang, Ya'an, and Ganluo, which are either distributed along the valleys, in a linear pattern along the river, or clustered in the alpine canyons and neighboring Yi villages. This information reflects the regional differences and diversity of traditional Tibetan villages in western Sichuan and provides the background and basis for the subsequent spatial analysis of the landscape. The natural and cultural environments have nurtured the unique residential landscapes of Tibetan settlements. Residential houses and monasteries are the main building types in the villages, and the residential spaces are built according to the topography, roads or water systems, and distributed around the monasteries; the monasteries and sacrificial places are the public spaces and the centers of religious activities of the villages, and they occupy an important position in the settlements; the residential spaces and the public spaces, due to the influences of the natural environments and the religions and cultures, have formed a "boundary-center" pattern. The residential space and public space form a "boundary-center" pattern due to the influence of the natural environment and religious culture (Zhang et al., 2022;). The landscape of Tibetan villages has distinctive spatial organization and pattern characteristics, reflecting the spatial and cultural characteristics of Tibetan settlements (Sang, 2018).

This paper selects three branches of traditional Tibetan villages as research samples, namely Jiarong, Amdo, and Baima, which represent the landscape features and differences of different regions, beliefs, and customs of Tibetan settlements in the western Sichuan Plateau. In this section, we use spatial analysis to extract the landscape elements of the three villages, and according to the calculation and analysis of landscape indices, we quantitatively describe the spatial characteristics of landscape elements such as buildings, road networks, vegetation, etc., and explore their interrelationships, so as to reveal the spatial pattern of sample village landscapes and better understand the formation mechanism of landscape spatial fractal characteristics.

  1. (1)

    Landscape elements and distribution characteristics.

Landscape elements refer to elements of rural landscapes that have unique functions or characteristics, such as architectural settlements, roads, rivers, woodlands, and mountains. In short, landscape elements are the composition of natural, artificial, or mixed elements of a landscape (Xu, 2022). We mainly use high-resolution remote sensing image data as the basic data, combined with computer vision and image processing algorithms, to automatically or semi-automatically extract and segment landscape elements such as architectural clusters, road networks, water systems, vegetation, and other landscape elements in the villages, and to obtain a preliminary understanding of their contours, distributions, and location information. As shown in Fig. 2, based on the extraction of landscape elements from the sample villages, we observe both similarities and differences in landscape distribution characteristics.

Fig. 2
figure 2

Sample village landscape spatial pattern map

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On the one hand, the villages are generally located at high altitudes, with settlements scattered at the foot of the mountains or on the slopes, and consistent with the topography. The architectural layout follows the mountains and is adjacent to water sources, forming a coherent spatial sequence. The spacing of the buildings is large and there are no continuous streets or walls, showing a state of large dispersion and small settlements. Next, as an important landscape element, the road is the supporting skeleton of the village surface area, which relies on the mountain and topography and guides the spatial movement; the main road is generally close to the water system and parallel to the contour line, while the secondary ones are mostly perpendicular to the main road, presenting irregular forms such as herring-bone and dendrite, and meandering and staggeringly distributed in the village space.

On the other hand, the differences in the layout of residential buildings. Jiarong and Amdo Tibetan residence is called "watchtower or walled house", the plane layout is hexagonal, octagonal, or twelve shape, and the building's internal space is often in the form of "three rooms, two halls, and one room", divided into dimensional Buddha Hall area, living area, and livestock enclosure area. The spatial layout of the dwellings of the Baima Tibetans incorporates the layout characteristics of more western Sichuan dwellings, with the "L"-type or "concave"-type compound as the main feature. The spatial arrangement of Tibetan houses is often built in family units, forming a small community (Chen & Leung 2021). Therefore, the spatial organization of residential architecture is not only influenced by the natural environment, and architectural form but also by cultural and customary factors.

Furthermore, the location of monasteries has strong spatial differences due to factors such as topography, village area, and religious sect. The location of Tibetan Buddhist monastery buildings is generally centered on mountains, often built on mountain tops or slopes with higher elevation (As shown in Fig. 2), which reflects the ideas and teachings of the Manjushri-style Buddhist cosmology and the three realms (Wu & Shi, 2003; Wang, 2014). The Gyaltsen and Amdo Tibetans practice Tibetan Buddhism, and the monasteries and villages have formed various spatial distribution patterns, mainly adjacent, central, and separated. For example, the Dandarun Monastery, Zhaqu Monastery, and Coronation Mountain Baodu Monastery in Xisuo Village are in the center or separate from the village, while the Zhaga Monastery and Raiwaji Monastery in Shenzuo Village are adjacent to the village, and the Langyi Monastery and Charlie Monastery have a layout of “the upper village and the lower monastery” with the village area (as seen in Fig. 3b-c). The Baima Tibetans are different from the other two ethnic groups in that they believe in the Bon religion, worship all things in nature, and fear the gods, often building smaller temples in their homes and in the fields and mountains, dedicated to the gods of the mountains, the gods of the white stones, and the gods of the land. In ethnic villages, for example, temples that enshrine mountain gods and white stone gods are usually located in the center of the village, where folk beliefs and religious activities take place (as seen in Fig. 3a). Monasteries are places where religious culture is disseminated and faith spaces where villagers rely on spirituality.

Fig. 3
figure 3

Spatial Distribution of Monasteries and Village

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In addition, the water system and vegetation landscape are important natural geographic features in the village, which enrich the sense of spatial hierarchy and also reflect the ecological diversity (Li et al., 2017). They also play the functionality of the landscape, i.e., the rivers and ponds provide the necessary water supply for the local villagers’ life and production; the woodlands and grasslands provide the villagers with sufficient resources for animal husbandry and cultivation. As can be seen from the figure, the distribution of architecture, water system, and vegetation landscape presents an interlaced and complex organic space.

To summarize, the residential houses, monasteries, roads, and streets are built according to the topographic environment and embedded in the water system and vegetation, forming the spatial interface of the village. The unique natural and humanistic environmental factors are integrated into the spatial structure and layout of the village, playing a role in the spatial organization and functionality of the landscape. This also reflects the development logic of traditional Tibetan villages in adapting to complex environmental influences and shows the interactive process between humans and nature (Dai et al., 2023). (2) Landscape index and spatial pattern analysis.

The landscape index is a comprehensive system for assessing, analyzing, and describing the spatial quality and characteristics of the landscape, and it is an important method for studying the spatial organization and layout of village landscapes (Chen et al., 2019). In the paper, from the actual situation of the sample village, some landscape indices were selected to explore the spatial pattern characteristics of each landscape element in the village, such as the number of patches (NP) and the density of patches (PD), which can reflect the scale and distribution characteristics of the landscape; the maximum percentage of patches (LPI), the landscape shape index (LSI), the average area of the patches (AREA_MN), the average dimensioning index (FRAC_MN) and the (FRAC_AN), which can reflect the morphological characteristics of the landscape, and the aggregation index (AI), which can reflect the distribution characteristics of the landscape (As shown in Table 2). According to the description of the landscape index, we can understand the differences in landscape spatial patterns of different Tibetan villages (As seen in Fig. 4). Spatial pattern is the spatial relationship between elements in the landscape, which can objectively reflect the relative location and organization of the elements (Hu et al., 2021). This analysis process helps to identify the diversity and complexity of landscape space, as well as its function and value.

Table 2 Landscape index analysis of the villages
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Fig. 4
figure 4

Characteristics of landscape spatial pattern in sample villages (Self-drawn)

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According to the graphs, from the analysis of landscape indices such as the number of patches, density, aggregation degree, and adjacency degree of buildings, roads, water system, and vegetation in three villages, namely Xisuo Village, Shenzuo Village, and Minzu Village, we can objectively compare the characteristics of the scale, form and distribution of the landscape. After identifying the indices by dimensionality reduction and dimensionality division, we quantitatively describe the relationship between different samples or different variables in the form of cluster analysis and point and line diagrams. From the landscape indices NP and PD, Xisuo village has the highest number of patches and patch density, and the landscape scale is higher than that of Shenzuo village and Minzu village; from the LPI and LSI, the ratio of the largest patch to the landscape area is the highest in Shenzuo village, which indicates that the landscape morphology of this area is relatively simple, while that of Xisuo village is more complex; from the values of AREA_MN and FRAC_MN (a range of values of 1–2), it indicates that the landscape space of the three villages has irregularity and complexity in landscape space; the patch area and shape of Xisuo Village are more scattered, while the landscape of Shenzuo Village and Minzu Village is more concentrated; from the AI value, the landscape distribution of all three villages is more aggregated.

Based on the above landscape index analysis, we found that there are certain similarities and differences in the landscape spatial patterns of Xisuo Village, Shenzuo Village, and Minzu Village. On the one hand, the similarity is mainly reflected in the scale, shape, and distribution, such as the number and density of patches in the three villages are not high, which indicates that the scale of the landscape is small; if the shape index and the dimension index of the patches are higher, the landscape spatial pattern is more complex, irregular and rich in details; if the aggregation index is all high, the landscape distribution is more aggregated, and the same kind of patches are adjacent or close to each other. On the other hand, the variability is mainly reflected in the degree of fragmentation of landscape patches, i.e., the degree to which the landscape space is divided into patches of different types and sizes. This may be related not only to geographic location and natural conditions but also to the human environment (Wei et al., 2022). Because the effects of different types of human activities on the diversity and evenness of landscapes are complex, e.g., construction activities or agricultural activities in the villages may lead to a decrease in the diversity of landscapes, while forestry activities may also increase the diversity of landscapes (Xu et al., 2019). Overall, the three sample villages are characterized by distinct landscape patterns, and these characteristics provide a basis for subsequent fractal measures and analysis of influencing factors.

3.2 Fractal measurement and spatial pattern

Previously, we used methods such as spatial analysis and landscape indices to characterize the landscape elements and spatial patterns of the sample villages and found the differences in the landscape space of different villages. However, these methods are difficult to comprehensively reflect the complexity and diversity of landscape space and fail to reveal its intrinsic formation mechanisms and influencing factors. Therefore, we adopt fractal theory to further analyze it. Landscape space is composed of various landscape elements and subspaces of different sizes and shapes, which can present the heterogeneity of the landscape and is the result of various ecological processes at different scales (Wang et al., 1999). This means landscape space will present similar structures and shapes at different scales. Landscape elements such as buildings, roads, water systems, vegetation, etc. are components of the village landscape space, which are interconnected with each other and together constitute the overall form of the landscape space.

In the study, we will apply the method of counting box dimension to measure the fractal dimensions of architectural clusters, vegetation communities, public spaces, cluster boundaries, and road networks and water systems of the sample villages in three dimensions: points, lines, and surfaces, and compare the fractal characteristics of different villages, so as to discover the differences in landscape spatial characteristics of different villages. Landscape space is a complex system composed of multiple elements with fractal characteristics, with certain morphological features and structural laws (Liang et al., 2017). The fractal dimension is an important index to describe the complexity of fractal structure, which can reflect the fractal characteristics and morphological features of landscape space. By calculating the fractal dimension of different landscape elements, we can gain a deeper understanding of the complexity and diversity of the village landscape space and reveal its intrinsic formation mechanism and influencing factors.

In the landscape index analysis, we used FRAC_AM and FRAC_MN indicators to measure the complexity and irregularity of the patch shapes, which can reveal the fractal status of the landscape space (As seen in Fig. 5). From the data table, we found that the water system had the highest average fractal dimension, indicating that the patch shapes in the water system were the most complex, while the other landscape types had relatively lower fractal dimensions, but with some differences. The complexity and irregularity of the patch shapes in the water system might be related to the natural development process of the water system, which was influenced by factors such as topography, geology, climate, etc., and showed a certain degree of self-similarity and self-organization, thus exhibiting higher fractal characteristics. The other three landscape types were more affected by human activities and planning, and their patch shapes were more regular and simpler, with lower fractal characteristics. These differences reflected the heterogeneity and diversity of different landscape elements in the spatial structure and also provided a basis for understanding the characteristics and patterns of the rural landscape space.

Fig. 5
figure 5

Comparison of FRAC_AM and FRAC_MN indicators for three village landscape elements

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However, it was not enough to evaluate the fractal characteristics of the landscape space only by FRAC_AM and FRAC_MN indicators, because these indicators could only reflect the fractal status of individual patches, but not the fractal status of the whole landscape space. In order to further analyze the fractal characteristics of the landscape space of the three sample villages, we used the counting-box dimension method to measure and compare the differences in the landscape morphology of the villages. The measurement steps are as follows:

First, binary images of the sample village's boundaries, buildings, vegetation, roads, and other elements are extracted using the mapping software (Ai). Then extract the building settlement patches and take the area between the building silhouette and the vegetation silhouette as the public space and fill it with black color, and the building settlement and the public space form a black-and-white map bottom relationship (Zhang, 2019), and process it as a TIF format image file; in the same way, extract the vegetation patches with the area between the colony boundary and the building and public space contour, and carry out the color filling and image processing. Subsequently, set the settlement boundaries, road networks, and other forms as TIF format image files with the same scale. Next, import the processed image files into Fraclab2.2 software, set the upper and lower limits of box sizes and # of box sizes in Box Dimension: Binary Data; and set the value of \({\delta }_{i}\) to a power of two, i.e., 1024,512,256,128,64, count the total number of boxes containing at least one point in the image, and carry out the counting box dimension measurement of each landscape dimension according to Eq. 1 (As seen in Table 3).

Table 3 Measurement of fractal dimension for each landscape dimension in the sample villages
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It is seen in the measurement that the fractal dimension becomes more prominent as the size of the box decreases and the number increases. We select the best fractal dimension from them, which is in the range of 1–2. The landscape spaces of the three sample villages have strong fractal characteristics, but there are also some differences. From the measurement results (As seen in Fig. 6), the \({D}_{VA}\) and \({D}_{PA}\) of the three villages are relatively close to each other, while the \({D}_{BS}\) and \({D}_{RW}\) ones are quite different. Specifically, \({D}_{BS}\) and \({D}_{SB}\) are higher in Xisuo and Shenzuo villages, and relatively lower in Minzu village, indicating that the spatial structure of the first two villages is the most complex and irregular, while the spatial structure of Minzu village is relatively simpler. Secondly, \({D}_{VA}\) and \({D}_{PA}\) of all three villages are relatively close to each other, indicating that these two dimensions of spatial fractal characteristics are similar in the three villages. In addition, \({D}_{RW}\) is stronger in Cecil Village, while that of Ethnic Village is relatively weaker. Through the calculation of fractal dimension, the fractal features of different landscape spaces can be further quantified, compared, and interpreted from different perspectives and scales. From the whole to the local, the village space presents multi-level, multi-scale, and diversified spatial patterns.

Fig. 6
figure 6

Analysis of the comparison of landscape fractal characteristics in the sample villages

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In order to assess the validity of the research results, we utilized the leave-one-out method of cross-validation. Landscape indicator data, spatial analysis data, and measured dimensionality data were organized into a dataset, and the Cvpartition function of MatlabR2022b software was used to divide the dataset, and the data were cross-validated using the Crossvalind Function. The performance metrics of the model on the test set were evaluated by calculating the mean square error (MSE) and the goodness of fit (R) by Eq. 2,3. The results show that the MSE is 1.3591, the model fits the training data well and the prediction error on the test set is small. In addition to MSE, R is also an indicator of model performance (Zeng et al., 2022). According to the results of the goodness-of-fit values, the average value of subset R is about 0.8987, which is close to 1 and has a good fit. Overall, the validation results can be used as a reference to assess the performance of the model, but further analysis and judgment are needed.

In a word, through fractal measurement, we found the differences in the spatial fractals of the landscape of traditional Tibetan villages in western Sichuan, and these features are also closely related to the landscape pattern. Landscape pattern is an important aspect that affects landscape spatial fractal features, and different landscape elements and organization will form the diversity and complexity of spatial patterns. At the same time, the landscape's spatial fractal characteristics also reflect the adaptability and coordination of traditional Tibetan villages in the natural and humanistic environments, reflecting the interactive process and results between humans and nature.

3.3 Analysis of fractal influence factors

The fractal of landscape space can quantitatively characterize the morphology of landscape space but cannot reveal the influencing factors behind it. The degree of fractal of landscape space is related to the natural environment, such as topography and geomorphology, vegetation type and coverage, climatic conditions, etc. It is also influenced by human factors. Generally speaking, the stronger the human activities are, the lower the degree of fractalization of landscape space, the more regular the shape of patches, and the simpler the boundaries are; on the contrary, the more irregular and complex the boundaries are (Song et al., 2022).

Traditional Tibetan villages in Sichuan are culturally important as the center of Tibetan Buddhism spreading in the region. Religious beliefs have had a profound impact on the social structure, village organization, folk culture, and spatial layout of the settlements (Liu et al., 2019; Sun, 2018). In addition, the language, costumes, diet, music and dance, and folk crafts in each village also show diverse folk styles and local characteristics. The geographic environment, religious beliefs, and traditional folklore of traditional Tibetan villages in Sichuan have created many unique cultural landscape phenomena, which together construct spatial dimensions in different contexts (Wang & Liao, 2021).

In this section, we first sort out the material and non-material elements in the natural and humanistic landscapes of the three villages and establish an assessment system (As seen in Table 4) to further explore the factors affecting the spatial fractal characteristics of the landscapes.

Table 4 Landscape features of three sample villages (self-drawing)
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Then, we applied the multiple regression method, taking fractal dimension as the dependent variable, material elements (such as terrain, vegetation, buildings, streets, etc.), and immaterial elements (such as religious beliefs, traditional arts, folk festivals, etc.) as the independent variables; and analyzed the correlation between the natural environment, social and cultural factors, and fractal dimensions according to Eq. 4. Thus, the reasons for the differences in the landscape space of different Tibetan villages are revealed, and some insights are provided for conservation and development.

The findings revealed a strong fit of the model (R = 0.842), indicating a significant correlation between the independent and dependent variables. The variance analysis yielded an F-value of 23.567, with a p-value of 0.000, confirming the significance of the regression equation (As seen in Table 5). The regression coefficients, in terms of their sign and magnitude, were utilized to determine each independent variable's direction and degree of influence on the dependent variable. A positive coefficient indicated a positive correlation, suggesting that an increase in the independent variable would lead to a corresponding increase in the dependent variable. Conversely, a negative coefficient indicated a negative correlation, implying that an increase in the independent variable would decrease the dependent variable. The absolute value of the coefficient reflected the magnitude of the influence. The t-value and p-value were utilized to assess the significance of each independent variable, with a higher t-value and lower p-value indicating greater significance (Li et al., 2021; Qi et al., 2019).

Table 5 Multiple regression analysis results (self-drawing)
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All independent variables passed the significance test, indicating their substantial impact on the fractal dimension. Specifically, religious beliefs, vegetation coverage, streets, and folk festivals positively influenced the fractal dimension, implying that a higher richness of these elements was associated with a greater fractal dimension. This observation can be attributed to the fact that these elements embody the cultural connotations and way of life of Tibetan villages, contributing to the uniqueness and hierarchical structure of the landscape space. In contrast, terrain, architecture, and traditional art had a negative impact on the fractal dimension, suggesting that a greater richness of these elements was associated with a lower fractal dimension. This phenomenon may be attributed to the limitations imposed by the natural environment and construction techniques, resulting in a relatively simpler and more regular landscape space form and structure.

In summary, we employed multiple regression analysis to examine the correlation and influence relationship between the fractal characteristics of traditional Tibetan village spatial patterns and their material and non-material landscape elements in diverse regions and cultural backgrounds. The findings indicate that the landscape space of Tibetan villages is influenced by the natural environment and cultural factors, reflecting complexity and diversity.

4 Discussion

The fractal theory is a mathematical tool for quantitatively describing and explaining complex phenomena in nature, including irregular structures and forms. In this study, we apply it to Tibetan traditional villages in western Sichuan, aiming to analyze and understand the functionality, diversity, and connectivity of the landscape space of these villages (Zhu et al., 2022). In this paper, we selected the traditional Tibetan villages in western Sichuan as the research object and chose the traditional villages inhabited by the Jiarong, Amdo, and Baima Tibetans as the research samples. We mainly used fractal theory, combined with spatial analysis, leave-one-out cross-validation, and multiple regression measurements, to explore the landscape elements, spatial patterns, morphological features, and influencing factors of the villages. The study reveals that Tibetan traditional villages in different regions and cultural backgrounds have landscape fractal characteristics of self-similarity, diversity, and complexity. In addition, the differences in the influence of factors such as natural environment, religious beliefs, and folk culture on these villages were emphasized. These findings provide useful insights for the conservation and sustainable development of traditional villages. Specifically, preserving originality is the basis for maintaining the historical and cultural values of villages, enhancing integrity helps to strengthen their ecological and social values while enhancing heterogeneity can enhance aesthetic and innovative values (Wang et al., 2019; Li, et al., 2023a).

However, we must admit that there are some limiting factors in this study, mainly including the following: first, the sample size of the study is relatively small, covering only three clades of Tibetan traditional villages, and therefore not representative of traditional village characteristics in other regions and ethnic groups. Second, the research data mainly come from remote sensing images and GIS data, while more comprehensive fieldwork and social survey data are lacking. Finally, our research methodology is mainly based on fractal theory and spatial analysis methods, without a comprehensive exploration of other possible methods and techniques. Future research can be expanded in the following directions:

Firstly, we recommend an all-encompassing exploration of landscape spatial attributes and influencing factors within traditional villages across different regions and ethnic groups. This entails a comprehensive examination, both vertically and horizontally, to compare the similarities and differences in their fractal characteristics and patterns, thereby broadening the spectrum of research samples and scopes.

Secondly, a more extensive collection of field surveys and sampling data is warranted, alongside an exploration of alternative methods and techniques routinely employed in traditional village research, such as machine learning, simulation, and other contemporary approaches (Hu et al., 2022; Yang et al., 2016). Additionally, it would be advantageous to incorporate methodologies and theories derived from sociology, anthropology, and other relevant disciplines (Li et al., 2022) to foster a more profound and comprehensive understanding of traditional villages' spatial attributes and cultural significance.

5 Conclusion

This paper preliminarily discusses the fractal characteristics and influencing factors of the landscape space of traditional Tibetan villages in western Sichuan, providing a new perspective for a deeper understanding of the landscape space of Tibetan traditional villages. The fractal method quantitatively describes the morphological complexity of the landscape space and reveals the inherent self-organization and self-similarity of the landscape space. In addition, we observed that natural and human environmental factors not only affect the material form of the landscape space, but also reflect the Tibetan people’s cognition and attitude towards nature and social life. In order to protect the traditional village landscape, we suggest following the principles of originality, integrity and heterogeneity. This means that we should respect the historical context and regional characteristics of the village, avoid destroying its original appearance; balance the protection and development of traditional villages, avoid excessive human intervention, and pay attention to the value of ecology, society, culture and other aspects; plan or design the landscape space from the perspective of fractal theory, enhance its spatial perception and functional multiplicity, promote the spatial integration of urban and rural areas, and improve its adaptability and innovation.