A Fractal Perspective on Scale in Geography

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 this confusion and frustration mainly arise from Euclidean geometric thinking, with which locations, directions, and sizes are considered absolute. It is time to reverse 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 how small. However, 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 introduce topological and scaling analyses based on street-related concepts such as natural streets, street blocks, and natural cities. We further contend that spatial heterogeneity, or the fractal nature of geographic features, is the first and foremost effect of two spatial properties, because it is global and universal.

KEYWORDS: Scaling, spatial heterogeneity, conundrum of length, MAUP, topological analysis

Full text

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.


2. Frac
t
Euclide
a
with the
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two geo
m
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e
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s
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m
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a
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och curve
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och curve
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ely applyi
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s
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och curve,
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cursive
a
nd

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


(Zipf 1
9
a great
e
pattern.
thinkin
g
pattern
o
3. Falla
c
Geogra
p
Batty a
n
for the
p
Becaus
e
absolut
e
for a to
coastlin
e
lack of
a
people
o
fallacie
s
results
a
3.1 Coa
The len
g
the mea
s
coastlin
e
of lengt
h
long ti
m
measur
e
fractal
n
length a
n
(Note
slope
o
c). Th
26.9, a
n
d), t
h
9
49); the firs
t
exten
t
a fra
c
Therefore,
g
(McKelve
y
o
f far more
s
c
ies of scale
p
hic feature
s
n
d Longley 1
p
urpose of
m
e
of this ass
u
e
. This is a c
o
pographic s
u
e
. The same
a
tangent li
n
o
ften treat
t
s
commonly
a
cross scales
.
st length, is
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g
th of a lin
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urement de
p
e
length ten
d
h
, also com
m
m
e. Geograp
e
geographic
n
ature of g
e
n
d further d
e
Figure
2
: The length
o
f the surfac
e
h
e five slope
n
d 26.9–40.9
h
e slopes lo
o
t
number is
c
tal pattern
i
at the fund
a
y
and Andri
a
s
mall things
t
in geograp
h
s
,
b
oth natu
r
994, Chen
2
m
easurement
u
mption, geo
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is true for a
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t
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u
seen in geo
g
.
l
and area,
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e
ar geograph
p
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a
d
s to increas
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m
only know
n
hers attemp
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features on
e
ographic fe
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e
veloped fra
c
2
: (Color on
l
of the coast
l
e
change wit
h
classes bet
w
degrees. D
e
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1, the secon
d
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s more a st
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ni 2005, Ji
a
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han large o
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al and man
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2
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,
we have t
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raphic feat
u
cy in geogr
a
m
putable.
H
topographic
mplies that
c
u
rements as
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raphy on th
e
a
nd surface
s
i
c feature s
u
a
p scale or i
m
e
exponenti
a
n
as
p
arado
x
t
ed to solve
maps. How
e
a
tures. Man
d
c
tal geometr
y
l
ine) Illustra
t
l
ine changes
h
boxes and
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een blue an
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with dif
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5
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atistical pat
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e, fractal t
h
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ng 2015b)
b
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es.
m
ade, are f
r
2015a), alth
o
o
assume ge
o
u
res are con
s
a
phy. In add
i
H
owever, a
t
surface dev
o
curve lengt
h
something
e
sizes of ge
o
s
lope
u
ch as a coa
s
m
age resolut
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a
lly at the sa
m
x
of length (
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this proble
e
ver,
t
he pr
o
delbrot (19
8
y
.
t
ion of how
m
with yardst
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digital elev
a
d
red are res
p
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same area
b
ff
erent resol
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r
-resolution
D
the third nu
t
ern that me
e
h
inking is n
o
b
ecause bot
h
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actal in na
t
o
ugh within
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graphic fea
t
s
idered mea
s
tion, it is co
m
t
angent can
n
o
id of tange
n
h
s and surfa
c
absolute. I
n
o
graphic fea
t
s
tline is not
i
on. As map
m
e pace (Ri
Steinhaus 1
9
m (Perkal
1
o
blem is ess
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8
2) eventual
m
easuremen
t
i
cks (Panel
a
a
tion model
(
p
ectively, 0
–
b
ut with diff
e
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tions, such
a
D
EMs.)
m
ber is 1/3,
e
ts the scali
n
o
t much di
ff
h
are conce
r
u
re (Goodc
h
a limited sc
a
t
ures as Euc
l
s
urable, or t
h
m
monly con
n
ot be defin
e
n
t planes (M
c
e slopes ar
e
n
this sectio
t
ures and on
measurable.
scale expon
e
c
hardson 19
9
83) puzzle
d
966, Nystu
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e
ntially unso
l
l
y uncovere
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changes wi
a
), the area o
f
DEM) resol
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–
6.8, 6.8–12.
4
e
rent resolut
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s more area
s
and so on.
H
ng law tha
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f
ferent from
r
ned with th
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ild and M
a
a
ling range.
H
l
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61). This co
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fo
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n
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u
tions (Pan
e
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ence,
t
o
n
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a
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owever,
n
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m
ent to be
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int of a
9
82). The
r
able, yet
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nt some
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tatistical
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precise,
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ases, the
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ionship, i.e.
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. If there is
data has ch
a
e
d. One exa
m
g
ood soluti
o
i
s (Tobler 1
9
eal with sca
l
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ssues
things wo
u
y
deal with
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nships, an
d
n
ships of t
h
t
hings than
tree or a ri
v
p
ological an
d
n
g analytical
o
r example,
p
resentation
o
f street se
g
t
ation offer
s
m
s of lengt
h
e
. Therefore
,
t
ure of indiv
s
of natural
s
e
rs in Panel
e
25 city bl
o
n
ected ones,
t
reets and th
e
y
graph; th
e
t blocks (or
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mbers sho
w
r
, or the
b
or
d
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hy and eco
l
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n
to other sc
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,
the ratio o
f
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ple of this
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n to effecti
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9
89). There
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ld inevita
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scale issue
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d
from me
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ings and e
n
w
ell-connec
t
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er network
d
scaling an
a
insights int
o
a street ne
t
of stree
t
ne
t
g
ments. Des
p
little analy
t
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or similar
j
,
let us shift
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dual streets
.
s
treets and a
n
(a), are cr
e
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cks, which
and far mo
r
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ir six inters
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e
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s
field blocks
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b
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logy, there
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les, either s
c
e
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f
the change
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uation, in w
h
is how citi
e
v
ely dealing
f
ore,
t
he ne
x
b
ly resul
t
i
n
s
, we must
a
surement t
o
n
ables us t
o
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ted ones, i
n
k
. We exem
p
a
lyses that a
r
o
geospatial
twork is a
t
works is E
u
p
ite its use
fu
t
ical insight
s
j
unctions in
from the ge
o
.
natural city
e
ated from
are red in
P
r
e small stre
ections are
c
s
how the d
e
in gray) in
P
b
locks indi
c
s usually
l
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c
aling up
changes.
in length
keep the
h
ich data
e
s can be
with the
x
t section
n
errors,
shift our
o
scaling
o
see the
n
analogy
p
lify this
r
e free of
data, but
graph of
u
clidean-
u
lness in
s
into the
terms of
o
metrical
19 street
P
anel (b).
e
t blocks
c
onverted
e
grees of
P
anel (b)
c
ate 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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