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This paper describes the complexity of gravitational fractals in terms of global and local dimensions. They are presented in Table 1.Table 1 Global and local dimensions of gravitational fractals and attraction basins.Full size tableThe fractal in hexagonal CPT space, shown in Fig. 1, has a very rich structure, and therefore its characterization by means of fractal dimensions requires two approaches: (1) a global approach treating the fractal as a complex whole and (2) a local approach which allows us to determine the dimension of its fragments which are particularly interesting from a research perspective (see also Table 1). In the subsequent part of the paper, the results obtained are presented and interpreted according to the division in the table.Global dimension of boundaries of gravity attraction basinsTwo types of fractal dimensions have been thus far used in this analysis, i.e., the box and ruler dimensions. Figure 3 shows the distribution of the values of these dimensions determined for the boundaries of attraction as a function of space friction μ.Figure 3Comparison of the variability of the global ruler and box dimensions. Legend: The edge of all attraction basins is a function of the μ coefficient; 1–edges of all basins, 2–entire basins.Full size imageFigure 3 empirically confirms a fact known from chaos theory that whenever a fractal represents full chaos, the ruler dimension may be greater than 2 (Peitgen et al.33, 192–209), whereas the box dimension never exceeds this extreme value. Clearly, for a certain value of μ (in this case μ = 0.19), the numerical values of both types of dimensions are identical.In the bottom part of Fig. 3, line 1 illustrates the variability of the shapes of the attraction basins of individual cities depending on the value of μ, i.e., space resistance. The initially extremely complex shapes of the boundaries are smoothed to take the form of straight lines in the case of a large value of μ (μ = 0.52).In turn, line 2 illustrates not only the boundaries of the attraction basins, but also their internal structure. Clearly, the initially chaotic impacts of individual cities on the agent (μ = 0.005) are gradually smoothed out, so that in the final stage of the process they fully stabilize. This means that each city has a geometrically identical basin of attraction. Hence, if the agent is in the attraction basin of city 1 (purple color), it will always be attracted only by that city. This rule also applies to the other cities. It is obvious that the random process occurring at μ = 0.09 is then replaced by a strictly deterministic one. When chaos becomes complete order (Banaszak et al.15, the numerical values of both types of dimensions appear to stabilize at the level of 1.Global dimension of the boundary of each separate attraction basinFigure 1 also shows the geometric image of the attraction basins of individual cities. They were almost identical, and therefore also the fractal dimensions of the boundaries of these basins must match. The validity of this proposition is confirmed by Fig. 4. Six lines representing the distribution of the fractal dimension of the boundaries of the six basins coincide with almost full accuracy. Further analysis of Fig. 4 allows us to infer the conclusion that there is almost total chaos at the value db = 1.9021 (μ = 0.005). On the other hand, as space resistance increases to the value of μ = 0.22, there is a rapid decrease in the value of the fractal dimension of the boundary of each basin to the level of 1.2628; when μ = 0.34, then db = 1.2382. In that case, the value of the fractal dimension stabilizes, and at μ = 0.46, db = 1.2444 and finally for μ = 0.52, db reaches the value of 1.0412. The icons presented in Fig. 4 in lines 1 and 2 have slightly different structures than the icons in Fig. 3, due to different values of μ in certain cases.Figure 4The box dimension of the edges of the attraction basins depending on the μ coefficient (separately for each attractor). Legend: 1–boundaries of single attraction basins, 2–entire basins.Full size imageThe global dimension of the attraction basin of each city as an irregular geometric figureThe full symmetry of the basins of attraction of individual cities can be disturbed by the shape of the geometric figure on which the deterministic fractal is modeled. Such a situation occurs in the present case. Due to the fact that the fractal in Fig. 1 is formed on the surface of a square, the final basins of attraction of cities 1, 3, 4 and 6 are obviously larger than those of cities 2 and 5. Of course, these differences do not occur when considering the surface inside the hexagon.In Fig. 5, the line marked in black color represents the average value of the fractal dimension of the basins of attraction of individual cities, the value of which is (overline{{d }_{b}}=1.77). It can be seen that at very high values of the fractal dimension in the range (1.750, 1.775), there are db oscillations around this line. This is precisely the effect of modeling the fractal on the surface of the square, rather than the properties of this fractal. Therefore, (overline{{d }_{b}}=1.77) should be regarded as the global dimension of the basin of attraction (of each city) treated as an irregular figure.Figure 5Box dimension of the attraction basins as a geometric irregular figure in the gravitational fractal. Legend: 1-basins of the first city, 2-basins of the second city, 7-basins of all cities.Full size imageLocal dimensions of the boundary of the selected characteristic fragmentsFigure 6 presents fractal dimensions, with the Box and Ruler as functions of μ, and the boundaries of the attraction basins of individual cities occurring in all fragments A, …, E.Figure 6Distribution of the values of fractal dimensions of the boundaries of the attraction basins identified in selected fragments of a fractal; Legend: (A, D)-fragments marked in Fig. 1.Full size imageIt is evident that the structures of Fig. 6 (Box and Ruler) are almost identical. This means that, as has been stated earlier, when describing complex fractal objects, it does not really matter which type of dimension is used.Of interest here is the variability of the structure of both figures along with the increase in the value of the parameter μ. Fragments A, …, E (see Fig. 1) are characterized by high complexity, i.e. the intertwining attraction basins of the individual attractors (cities). This observation is confirmed by the numerical results of both fractal dimensions whose values are in the range (1.68–1.82). To illustrate the spatial complexity of these fragments, and thus their dimensions, by way of example, two fractal fragments are considered below: fragments A and D (see also Fig. 7).Figure 7Box dimension of the edge of each gravitation basin in A and D. Legend: The icons show the variability of the fragments A and D due to the share of the attraction basins of individual cities (3, 4 and 6).Full size imageFigure 6 offers important conclusions concerning the organization of social and economic life in the geographical area surrounding individual cities (attractors).

1.

Out of all the separated fragments, only in fragment A do we find the attraction basins of all the cities intertwined across the entire range of variation μ, i.e. (0.00–0.48). Hence, the graph of fractal dimension (db) (blue line) as a function of μ is continuous, and when the resistance of space is the greatest (μ = 0.48), the fractal dimension d = 1.00. This means that chaos has given way to total order, and fragment A has been symmetrically divided between cities 1 and 6. Hence, there are two colors left, namely red and purple.

2.

A similar situation occurs in the case of fragment D (yellow line), where the attraction basins of individual cities intertwine continuously within the range: 0.00 ≤ μ ≤ 0.46. Beyond the value of 0.46, the entire fragment D is filled with purple: the closest city 1 dominates it.

The research conducted here also confirms the conclusions presented in previous works by Banaszak et al.15,16 concerning the transformation of chaos into spatial order, which means the stabilization of permanent dominance, usually of one attractor (city). Thus, with regard to fragments A and D, in fragment A there is a constant dominance (in half of the area) of cities 1 and 6, from the limit value of μ = 0.24 onward. In the case of fragment D, beginning with the value of μ = 0.36, only city 1 dominates (purple). That is, in the final phase of establishing the order in spatial interactions in the arrangement of areas A and D, the role of the dominant attractor (city) is played by city 1 (purple).Due to the symmetry of Fig. 1, similar effects can be observed in other parts of this fractal, located symmetrically in relation to A, …, E (see Supplementary Material).Figures 1 and 6 confirm the findings, known in the theory of city development, that urban (and other) centers rise in the hierarchy (or their rank decreases), depending on the external and internal factors conditioning their development. In the model used in this study, the parameter μ represents external factors (space resistance). If μ values are low, all cities are attractive from the point of view of spatial interactions and create their own but symmetrical basins of attraction. When the resistance of space increases, one city becomes the dominant center, and its basin of attraction is a uniform compact isotropic surface.However, this is not a simple mechanism, since, as has been demonstrated by simulation experiments described in this paper, within a certain range of μ values, another city (attractor) may dominate the others during chaotic interactions. The dynamic history of urban development confirms this observation, for example, in relation to historical capitals of some countries that have lost their functions as administrative capitals.Local dimension of the boundary of each attraction basin in a selected fragment of a fractalFragments A, …, E (Fig. 1 and the Supplementary Material) consist of mutually intertwined basins of attraction (six cities) whose boundaries with complicated courses have a fractal dimension, e.g. a box dimension.Figure 7(fragment A) shows the distribution of db as a function of μ in this fragment. In the case of total internal chaos, the fractal dimension of the boundaries of the attraction basins of all cities is identical and amounts to 1.9152. A clear differentiation of db is noticeable from μ = 0.1 onward. It should also be noted that orange and blue, red and purple, yellow and green lines mutually coincide. The red–purple line tend towards db = 1 as μ increases. However, orange, blue, yellow and green lines reach a value of db = 0.The fractal dimension db = 1.0 is most closely represented by the blue line (city 2), then the red line (city 6) and the purple line (city 1). Since these lines almost coincide, and the red and purple lines are the last to reach the value db = 1, at μ = 0.48, fragment A is symmetrically covered in red and purple. Therefore, with very high spatial resistance, fragment A is dominated by two cities, namely by 1 and 6.In turn, Fig. 7(fragment D) illustrates the variability of the fractal dimension of boundaries of the attraction basins in this fragment. This dimension depends on the complexity of the mosaic patterns formed in this fragment, with varying μ values. When the values of μ are close to zero, all cities contribute to filling the space of fragment D. When μ = 0.18, city 1 (purple color) falls out of the competition for space, but only up to the value of μ = 0.24, when it starts to compete again with other cities. From the point of view of spatial interactions, in the final phase of this process (μ = 0.44), city 2 (blue) and city 6 (red) dominate to a small extent, because cities 3, 4 and 6, starting from μ = 0.3, do not play any role in fragment D.Figure 7 shows that the value μ = 0.3 is a characteristic point. It is a locus where all the curves representing the attraction basins of individual cities meet. As has already been stated, three of them lose their influence over the space of fragment D.Local dimensions of parts of the attraction basins treated as an irregular geometric figureIn each of the selected fragments A, …, E, some of the boundaries of the attraction basins of individual cities are distributed differently. They create certain holes in the form of irregularly colored mosaic patterns that have a certain fractal dimension. To present its variability, fragments A and D were used again. Figure 8 shows the distribution of db values depending on the value of μ.Figure 8Local dimensions of parts of the attraction basins treated as an irregular geometric figure in (A) and (D). Legend: The icons illustrate the variability of the shape of some of the attraction basins of individual cities in fragment (A) and (D) for cities 3, 4 and 6.Full size imageThe function has several characteristic points. Up to the value of μ = 0.04, attraction basins show a jumble in which no predominant color or shape can be identified. The fractal dimension is then: db = 1.7697. From this value onwards, where μ = 0.042, the interior of fragment A becomes increasingly ordered. With a value of μ = 0.125, the city’s attraction basins 3 and 4 begin to disappear in fragment A. The same happens to the city attraction basins 2 and 5 for the value of μ = 0.24.The final effect of the increase in space resistance (with μ = 0.50) leads to the filling of fragment A with two colors, i.e., purple and red. This means that cities 1 and 6, have won the competition for the space of fragment A. In this case, the fractal dimensions db equal 1.90.Figure 8 presents the variability of the fractal dimension and the effects of the competition for space between cities in fragment D. As is the case in fragment A and all others, i.e. B, C and E (see the Annex with Supplementary Material), the intertwined attraction basins are represented by the area consisting of an endless number of differently colored dots. Hence, up to the value of μ = 0.042, fragment D is dominated by pure spatial chaos that extends over its entire area. It is characterized by the fractal dimension db = 1.7697. This means that with an increase in the value of μ, for the emergence of an irregular shape of a geometric figure, chaos must be accompanied by an increase in the value of the fractal dimension. Its limiting value is number 2. Then, spatial dominance is usually gained by one city and the examined fragment is filled with one color (‘the winner takes it all’).This is precisely the situation in Fig. 8 where city 1 (purple color) has apparently won the competition. Since this color fills area D completely, we find the plausible result db = 2.0. More