Cartography as a Science to Simplify Complexity of the Earth’s Surface
The Earth’s surface is not a static entity but a complex system comprising interconnected substructures at different levels of scale or hierarchy. Cartography is the science dedicated to the representing and mapping of these entities on the Earth’s surface to smaller surface like paper, facing the challenge of distilling the vast complexity onto manageable scales. This complexity is ubiquitous in the way geographic features are organized: from the vast expanses of lands and oceans to the detailed intricacies of seven continents, over 200 countries, and beyond 10,000 cities. Such hierarchical structure follows a long-tailed relationship, revealing a natural scaling order where there are far more small geographic entities than large ones. Due to the impossibility of replicating every detail of the Earth’s surface on a smaller scale, cartography relies on a process of simplification or generalization. This process entails prioritizing ‘head’ elements—key features meaningful for the map’s purpose—while eliminating ‘tail’ elements which are less essential (Jiang 2015a). Therefore, a map is not the territory but a head part of the territory.
Cartography aims to distill the complex Earth’s surface onto maps, while fractals offer a way to understand complexity of geographic features across different scales and discover the inherent hierarchy behind a map. A prime example of this is Mandelbrot’s exploration of the British coastline (Mandelbrot 2967), illustrating how its measured length changes depending on the scale of measurement. This phenomenon reveals that smaller measuring units capture more details of the coastline’s intricate patterns, leading to an increase in the perceived length. The relationship between scale and the resulting length of the British coastline displays a long-tailed distribution and an L-shaped non-linear curve aligning well with the power law on a rank-size plot. Therefore, the fractal can be defined as a pattern or a set where the detail observed (y) and the scale of measurement (x) adhere to a power-law relationship, that is y=xα, with α representing the power-law exponent or the fractal dimension. This discovery not only enhances our understanding of the complexity in nature but also provides cartography with a methodological approach to depict the non-linear and long-tailed nature of complex geographic patterns, emphasizing the importance of maintaining the geographic information across different scales.
The Cantor set and the Koch curve are two iconic fractals under the classic definition. The Cantor set begins as a simple line segment and divided into three equivalent parts, from which the middle one is removed iteratively. Throughout the process of iterations, there are 1, 2, 4…2N segments at the size with respect of 1, 1/3, 1/9, … 1/3N, and the fitted power-law exponent α=-0.63. In contrast, the Koch curve is generated by replacing the middle part with two segments that form an equilateral triangle without the base. There are 1, 4, 8…4N at the size with respect of 1, 1/3, 1/9, … 1/3N, the fitted power-law exponent α=-1.26, which shows the Koch curve is more complex than the Cantor set in terms of their patterns or shapes.
The essence of long-tailed distributions and complexity lies in the non-linearity that there are far more small things than large things. However, the power-law serves merely as a subset of mathematical representations for long-tailed distributions, not its entirety (Jiang 2015b). The classic definition of fractal is overly restrictive, considering only distributions that follow a power-law as true fractals. Both exponential and logarithmic functions also exhibit non-linear, long-tailed characteristics, suggesting a broader spectrum of long-tailed phenomena. In order to encompass this wider range of long-tailed distributions, the head/tail breaks method offers a way to quantify how the structure that far more smalls than larges recurs within a fractal, trying to break down data recursively into head part and tail part by the average number and do it again and again within the head part (Jiang 2015a). This method introduces the ht-index, which counts the iterations of this recurring structure. A more inclusive third definition emerges: a pattern is considered fractal if the “far more smalls than larges” motif repeats more than twice, or if the ht-index exceeds two. Under this expanded definition, numerous geographic features qualify as fractal, which would not be under the strict classical criteria. This broader interpretation seeks to acknowledge the ubiquitous complexity in the natural world (Simon 1962) and affirm the pervasive presence of fractal geometry (Mandelbrot 1982). By doing head/tail breaks, head elements that are essential to the map could be classified and the irrelated tail elements could be eliminated.
Scale and scaling offer distinct perspectives on measurement within the realm of cartography. Scale is the dimensional ratio indicating the relationship between distances on a map or model and their counterparts in the real world. In contrast, scaling explores how scale interacts with the level of detail captured in the representation, revealing the dynamic interplay between magnification and the granularity of information. For instance, human heights typically range between certain values, demonstrating a concept of magnitude. This distribution of human heights, with exceptionally tall or short individuals being rare, follows a normal distribution. However, in the natural world, there exists a profound scaling hierarchy: far more small animals like ants than large ones like elephants. This discrepancy highlights not just a difference in sizes but a scaling pattern where smaller entities vastly outnumber larger ones, illustrating the complex and hierarchical nature of biological and geographical phenomena.
Cartography revolves around the efficient simplification and depiction of the Earth’s surface within constrained spaces and across diverse scales. This complexity is evident not just in the variety and distribution of geographic entities but also in their fractal nature and the associated scaling hierarchy that there are far more small things than large ones. In this process, cartographers face the challenge of choice, determining which elements are crucial and which can be omitted to ensure the map conveys necessary information without exactly covering every detail of the Earth’s surface (Jiang 2019). Fractal geometry offers an approach to quantifying the intricate complexity of geographical phenomena at different scales, uncovering the ubiquitous nature of complexity across all geographical features from the grandest to the minutest. The expansive third definition of fractal sheds light on their commonality and introduces a strategy for mapping the head elements within this complexity, aiding significantly in the process of map generalization. This approach not only navigates the challenges of representing the Earth’s complexity on maps but also enriches our understanding of geographical features through the lens of fractal geometry, marking a crucial advancement in the science of cartography.
References:
- Jiang B. (2015a), Geospatial analysis requires a different way of thinking: The problem of spatial heterogeneity, GeoJournal, 80(1), 1–13.
- Jiang B. (2015b), The fractal nature of maps and mapping, International Journal of Geographical Information Science, 29(1), 159–174.
- Jiang B. (2019), New paradigm in mapping: A critique on Cartography and GIS, Cartographica, 54(3), 183–205. Reprinted as the cover story in the magazine Coordinates, October issue, 9–21.
- Mandelbrot B. B. (1967), How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, 156(3775), 636–638.
- Mandelbrot B. B. (1982), The Fractal Geometry of Nature, W. H. Freeman and Co.: New York.
- Simon H. A. (1962), The architecture of complexity, Proceedings of the American Philosophical Society, 106(6), 467–482.