1 
 
 
A Fractal Perspective on Scale in Geography 
 
Bin Jiang and S. Anders Brandt 
 
Faculty of Engineering and Sustainable Development, Division of GIScience 
University of Gävle, SE-801 76 Gävle, Sweden 
Email: bin.jiang@hig.se, sab@hig.se 
 
(Draft: September 2015, Revision: June 2016) 
 
 
 
Abstract 
Scale is a fundamental concept that has attracted persistent attention in geography literature over the 
past several decades. However, it creates enormous confusion and frustration, particularly in the 
context of geographic information science, because of scale-related issues such as image resolution, 
and the modifiable areal unit problem (MAUP). This paper argues that the confusion and frustration 
arise from traditional Euclidean geometric thinking, with which locations, directions, and sizes are 
considered absolute, and it is now time to revise this conventional thinking. Hence, we review fractal 
geometry, together with its underlying way of thinking, and compare it to Euclidean geometry. Under 
the paradigm of Euclidean geometry, everything is measurable, no matter how big or small. However, 
most geographic features, due to their fractal nature, are essentially unmeasurable or their sizes 
depend on scale. For example, the length of a coastline, the area of a lake, and the slope of a 
topographic surface are all scale-dependent. Seen from the perspective of fractal geometry, many 
scale issues, such as the MAUP, are inevitable. They appear unsolvable, but can be dealt with. To 
effectively deal with scale-related issues, we present topological and scaling analyses illustrated by 
street-related concepts such as natural streets, street blocks, and natural cities. We further contend that 
one of the two spatial properties, spatial heterogeneity is de facto the fractal nature of geographic 
features, and it should be considered to the first effect among the two, because it is global and 
universal across all scales, which among the practitioners of geography should receive more attention.  
 
Keywords: Scaling, spatial heterogeneity, conundrum of length, MAUP, topological analysis 
 
 
1. Introduction 
Scale is an important, fundamental concept in geography, yet it has multiple definitions or meanings, 
some of which seem to be contradictory. Among the various definitions (Lam 2004), map scale is the 
most commonly used, referring to the ratio of distance on a map to the corresponding distance on the 
ground. Scale is also closely related to map generalization for selectively representing things on the 
Earth’s surface on a map and it can refer to the pixel size of an image, i.e. resolution. An image with 
small pixels has high resolution, while one with big pixels has low resolution. In this regard, scale is 
synonymous with the level of detail of an image, which is closely related to the notions of scaling up 
and down (Wu et al. 2006, Kim and Barros 2002) for translating statistical inference and reasoning 
from one scale to another. Scale is also commonly used to refer to the scope or extent of a study area. 
A large scale of study area (such as a country), if mapped, implies a small-scale map, whereas a small 
scale of study area (such as a city), if mapped, implies a large scale map. Obviously, confusion and 
frustration arise from multiple, seemingly contradictory meanings, and how to translate statistical 
inferences across scales. On the other hand, the confusion and frustration make scale even more 
interesting and challenging. In addition to the quantitatively defined scales, there are other 
qualitatively defined scales, such as micro-, meso- and macro-scales, and local, regional, and global 
scales. 
 
The concept of scale has generated extensive literature over the past two decades (e.g., Sheppard and 
McMaster 2004), along with emerging geospatial technologies including geographic information 

2 
 
science and remote sensing (e.g., Tate and Atkinson 2001, Weng 2014). Between 1997 and 2014, 
Goodchild and his colleagues produced eight publications with scale in the titles, including two books 
(Quattrochi and Goodchild 1997, Zhang et al. 2014). There are of course numerous other writings in 
the literature where scale and scale related issues such as the modifiable areal unit problem (MAUP) 
have been of persistent interest and challenge in geography and in geographic information science in 
particular. However, previous discussions are usually constrained to Euclidean geometry because all 
the meanings of scale in geography are about sizes in a ratio, or an absolute value related to 
geographic features or their representations. As a consequence, the major concern surrounding scale is 
how it affects geospatial data collection and analysis results with respect to accuracy and reliability. 
This is understandable because maps are initially produced for depicting and measuring things on the 
Earth’s surface. Unfortunately, most geographic features are not measurable, or the measurement is 
scale-dependent because of their fractal nature (Goodchild and Mark 1987, Batty and Longley 1994, 
Frankhauser 1994, Chen 2011b). For example, the length of a coastline, the area of a lake, and the 
slope for a topographic surface are all scale-dependent, so they should not be considered absolute. 
Unfortunately, to a large extent our fundamental thinking on scale issues so far has been based on 
Euclidean geometry. 
 
Scale in fractal geometry (Mandelbrot 1982), as well as in biology and physics (Bonner 2006, Jungers 
1984, Bak 1996), is primarily defined in a manner in which a series of scales are related to each other 
in a scaling hierarchy. For example, a coastline is a set of recursively defined bends, forming the 
scaling hierarchy of far more small bends than large ones (Jiang et al. 2013). Therefore, a new 
definition of fractal could explicitly be based on the notion of far more small things than large ones 
(Jiang and Yin 2014, Jiang 2015a), in analogy with Christaller’s (1933) central place theory (cf. Chen, 
2011a) where there are many small villages but few large cities. Another definition of scale is simply 
the measuring scale, ranging from smallest to largest, to measure both Euclidean and fractal shapes. 
This measuring scale, from an individual rather than a series point of view, is equivalent to image 
resolution or map scale. This measuring scale makes many geographers believe fractal geometry can 
be a useful technique for dealing with scale issues. However, this view of fractal geometry is dubious. 
Fractal geometry is not just a technique but could also offer a new paradigm or new worldview that 
enables us to see surrounding things differently. Fractal geometry is a science of scale because it 
involves the universal scaling pattern across all scales from smallest to largest. On the contrary, 
geography dominated by traditional Euclidean geometry focuses on a few scales for measuring 
individual sizes. 
 
This paper aims to advocate fractal thinking as a way to effectively deal with scale issues in 
geography and geospatial analysis. We think that mainstream views on scale of practitioners in 
geography, as briefly reviewed above, are not in line with the same concept in other sciences such as 
physics, biology and mathematics. In spite of the fact that fractal geometry has been intensively 
studied in geography (Goodchild and Mark 1987, Frankhauser 1994, Batty and Longley 1994, Chen 
2011b), the fundamental way of thinking of most geographers while dealing with scale issues is still 
Euclidean. This situation still exists more than forty years after fractal geometry was established. On 
the other hand, this situation is understandable, because, with the development of geospatial 
technology, measurement with high accuracy and precision has been a major concern. To measure 
things, we need Euclidean geometry, whereas to develop new insights into structure and dynamics of 
geographic features, we need fractal geometry.  
 
Section 2 introduces fractal geometry, in particular the underlying way of thinking, and put it in 
comparison with the Euclidean counterparts. Based on fractal geometry or fractal thinking, Section 3 
presents several fallacies or scale related issues in geography, such as the conundrum of length and 
MAUP. To avoid these scale related issues, Section 4 illustrates street-based topological and scaling 
analyses that enable us to see the underlying scaling patterns. Finally Section 5 discusses two spatial 
properties that are closely related to the notion of scale, and summaries our major points towards the 
fractal perspective on scale. 
 
 

 
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4 
 
In the course of generating the Koch curve, the initial segment of one unit is a simple regular shape. 
The Koch curve, on the other hand, looks complex, although the generation involves the simple 
repetition of division and replacement. First, the Koch curve is irregular and much more complex than 
the initial segment. Second, the Koch curve has an infinite length. Under the Euclidean geometry 
framework, anything is measurable, no matter how big or small. Why the Koch curve has an infinite 
length puzzled mathematicians for more than 100 years, until Mandelbrot (1967) solved the mystery. 
Geographic features such as coastlines bear the same property of the Koch curve. The length of a 
coastline is unmeasurable, or specifically, it is scale-dependent. In this way, the Koch curve and a 
coastline are essentially the same in terms of scale dependence. However, a coastline belongs to a 
statistical fractal with a limited scaling range, while the Koch curve is a strict fractal with an infinite 
scaling range. Therefore, fractal geometry offers a new worldview for viewing surrounding things 
such as trees, coastlines, and mountains.  
 
2.2 Fractal and Euclidean thinking 
Besides that Euclidean geometry considers regular simple shapes, and fractal geometry irregular 
complex shapes, there are more profound facets in how the two geometries differ (Mandelbrot and 
Hudson 2004) (Table 1). Euclidean geometry focuses on pieces or parts, while fractal geometry 
focuses on the whole. Euclidean geometry looks at individuals, while fractal geometry looks at 
patterns. This holistic or pattern view of fractal geometry implies a recursive view of seeing 
surrounding things. The Koch curve at iteration 3 (Figure 1) is just a Euclidean shape that consists of 
64 segments of all the same scale of 1/27. Seen from the recursive or fractal geometric perspective, it 
becomes a fractal shape, involving 85 segments of four different scales with far more short scales than 
long ones. 
 
Table 1: Comparison of Euclidean and fractal thinking 
 
Euclidean thinking 
Fractal thinking 
Regular shapes 
Irregular shapes 
Simple 
Complex 
Individuals 
Pattern 
Parts 
Whole 
Non‐recursive 
Recursive 
Measurement ( = scale) 
Scaling ( = scale free) 
 
Hence, fractal geometry offers a way of seeing our surrounding geography differently. Euclidean 
geometry mainly measures shapes (Euclidean shapes), directions, and sizes. Fractal geometry aims to 
see underlying scaling. Simply put, Euclidean geometry is used for one particular scale or a few scales, 
while fractal geometry aims for scale-free or scaling that involves all scales. The term scale-free is 
synonymous with scaling, literally meaning no characteristic mean for all sizes. This difference is 
very much like that between Gaussian and Paretian thinking (Jiang 2015b), which refer to more or 
less similar things (with a characteristic mean), and far more small things than large ones (without a 
characteristic mean, or scale-free), respectively. For example, a tree is better characterized by all sizes 
of its branches, or how the branches (scales) form a scaling hierarchy of far more small branches than 
large ones, rather than only by its height. It is fair to say that both Euclidean and fractal geometries 
aim to characterize things, but with different means; the former through measurement (at one scale), 
and the latter through scaling (across all scales). However, without individual Euclidean shapes, there 
would be no fractal pattern. It is scale that bridges individual Euclidean shapes and a fractal pattern. 
Without scale, there would be no fractal geometry. Scale plays the same important role in geography 
as in fractal geometry as a science of scale. 
 
Fractal geometry is not just limited to patterns. It can also be applied to a set. For example, the set of 
numbers, 1, 1/2, 1/3, 1/4, …, 1/1000, constitutes a fractal because there are far more small numbers 
than large ones within the set, based on the definition of fractal using head/tail breaks classification 
method (Jiang and Yin 2014, Jiang 2015a). The 1,000 numbers are created by following Zipf’s Law 

 
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7 
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4. Topo
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Figure 5: (
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8 
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with the 
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s into the 
terms of 
ometrical 
19 street 
Panel (b). 
et blocks 
converted 
egrees of 
Panel (b) 
cate how 

9 
 
So called ‘natural streets’ are created from adjacent street segments with good continuity or with the 
least deflection angles or with the same names (Jiang et al. 2008, Jiang and Claramunt 2004). These 
streets are naturally defined and can be a basic unit for spatial analysis through a graph representation, 
or a connectivity graph, in which the nodes and links represent individual natural streets and their 
intersections (see Panels (a, c) in Figure 5). This topological representation provides an interesting 
structure, involving all kinds of streets in terms of both length and degree of connectivity. In other 
words, interconnected streets behave as fractal when seen from the topological perspective, as there 
are far more less-connected streets than well-connected ones. Thus, natural streets can be a more 
meaningful unit than arbitrarily imposed areal units. Natural streets, or their topological 
representations, suffer less from scale effect because geometric details such as accuracy and precision 
play a less important role. What matters is relationship. The topological view enables us to see the 
underlying scaling pattern, where we can assign point-based data into individual natural streets, rather 
than to any modifiable areal units for spatial analysis. 
 
4.2 Street blocks and natural cities 
Street blocks also demonstrate the scaling property of far more small things than large ones. The street 
blocks refer to the minimum rings or cycles, each of which consists of a set of adjacent street 
segments. Obviously, a country’s street network usually comprises a large amount of street blocks 
(Jiang and Liu 2012). The street blocks are the smallest unit and are defined from the bottom up, 
rather than imposed from the top down by authorities. The street blocks are smaller than any 
administratively or legally imposed geographic units. They can be automatically extracted from all 
kinds of streets, including pedestrian and cycling paths. It is understandable that the street or city 
blocks defined by authorities are just a subset of the automatically extracted street blocks. For 
example, the number of London census output areas, which is the smallest census unit in the UK, is 
just half that of the street blocks that can be extracted from the OpenStreetMap databases. 
 
Topological analysis of the street blocks begins with defining the border number, which is the 
topological distance far from the outermost border of a country. The border is not a real country 
border, but consists of the outermost street segments of the street network. Those blocks adjacent to 
the border have border number one, and those adjacent to the blocks of border number one have 
border number two, and so on (see Panel (b) in Figure 5). All the blocks are assigned a border number, 
indicating how far they are from the outermost border. Interestingly, the block(s) with the highest 
border number constitutes the topological center of the country (Panel (b) in Figure 5). In the same 
way, we can take all city blocks as a whole to define the topological center as the city center. The 
topological center differs from the geometric center, or the central business district that is commonly 
the city center. 
 
The scaling property of far more small blocks than large ones enables us to define the notion of 
natural cities emerging from a large amount of heterogeneous street blocks. All the street blocks are 
inter-related to form a whole. The whole can be broken into the head for those above the mean, and 
the tail for those below the mean, as shown in the head/tail breaks classification method described in 
Jiang (2013). Those small street blocks in the tail constitute individual patches called natural cities. 
See Panel (b) in Figure 5 for an example of a natural city. Natural cities are defined from the bottom 
up. A large amount of street blocks collectively decides a mean value as a cut off for the city border. 
The head/tail breaks can recursively continue to derive patches within individual natural cities. In 
other words, all city blocks within a natural city are considered a whole, and those below the mean 
value in the tail (high-density clusters) are considered hotspots of the natural city. It is essentially the 
fractal nature of street structure, or the scaling property of far more small blocks than large ones, that 
make the natural cities definable. We can assign point-based data into city blocks, or natural cities, 
rather than any modifiable areal units for spatial analysis. 
 
In summary, as many modifiable areal units are imposed by authorities or images from the top down, 
such as administrate boundaries, census units, and image pixels, it is inevitable that these units or 
boundaries are somehow subjective. They were defined mainly during the small-data era for the 
purpose of administration and management, but are still used in the big-data era. However, objectively 

10 
 
defined units such as natural streets, street blocks, and natural cities should therefore be better 
alternatives for scientific purposes, and they reflect the new ways of thinking about data analytics in 
this big data era.  
 
 
5. Discussion and summary 
The concept of scale is closely related to spatial heterogeneity, one of the two fundamental spatial 
properties. The other property is spatial dependence, or auto-correlation, which has been formulated 
as the first law of geography: Everything is related to everything else, but near things are more 
related than distant things (Tobler 1970). The Tobler’s law implies that near and related things are 
more or less similar. Therefore, spatial variation for near and related things is mild rather than wild in 
terms of Mandelbrot and Hudson (2004). However, spatial heterogeneity is about far more small 
things than large ones, or with wild rather than mild variation. The two spatial properties are closely 
related and can be rephrased as such. There are far more small things than large ones in geographic 
space – spatial heterogeneity, but near and related things are more or less similar – spatial dependence. 
In this regard, spatial heterogeneity appears to be the first-order effect being global, while spatial 
dependence is the second-order effect being local. Therefore spatial heterogeneity provides a larger 
picture across all scales ranging from smallest to largest, while spatial dependence is a more local 
pattern.  
 
As a fundamental concept in geography, scale has been a major concern for geospatial data collection 
and analysis, not only in geography and geographic information science, but also in ecology and 
archaeology. Although Goodchild and Mark (1987, p. 265) concluded that “fractals should be 
regarded as a significant change in conventional ways of thinking about spatial forms and as 
providing new and important norms and standards of spatial phenomena rather than empirically 
verifiable models”, this paper suggests fractal geometry thinking finally should become a new 
paradigm, rather than a technique recognized in the current geographic literature. To effectively tackle 
scale issues in geospatial analysis for better understanding geographic forms and processes, this 
fractal or recursive perspective is essential. Many people tend to think that a cartographic curve is just 
a collection of line segments – Euclidean way of thinking, but actually it consists of far more small 
bends than large ones – fractal way of thinking. After comparing various definitions of scale in 
geography and cartography, as well as in fractal geometry, we note that the definitions of scale in 
geography are very much constrained by Euclidean geometry, for measuring geographic features 
rather than for illustrating the underlying scaling pattern. Most geographic features are inevitably 
fractal, so their sizes are unmeasurable or scale-dependent. We must be aware of scale effects in 
measuring geographic features, and that their sizes change as the measuring scale changes. The 
measurement is a relative indicator, rather than something absolute.  
 
Not only their sizes, but also statistical inferences on geographic features are scale dependent. 
Statistical reasoning cannot be translated across scales, or from an aggregate scale to individual ones. 
Given these circumstances, we must adopt fractal thinking for geospatial analysis involving all scales 
rather than a single scale or a few scales. We must examine if there are far more small things than 
large ones, rather than measuring individual sizes. We must also determine if, or how, the locations 
are related, rather than measuring absolute locations. However, unlike their sizes, statistical inferences 
on geographic features have not been found to hold a simple relationship with scales within a scaling 
range. This certainly warrants further research in the future. 
 
Acknowledgment 
XXXXXX 
 
 
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