ArticlePDF Available

Abstract and Figures

According to Christopher Alexander’s theory of centers, a whole comprises numerous, recursively defined centers for things or spaces surrounding us. Wholeness is a type of global structure or life-giving order emerging from the whole as a field of the centers. The wholeness is an essential part of any complex system and exists, to some degree or other, in spaces. This paper defines wholeness as a hierarchical graph, in which individual centers are represented as the nodes and their relationships as the directed links. The hierarchical graph gets its name from the inherent scaling hierarchy revealed by the head/tail breaks, which is a classification scheme and visualization tool for data with a heavy-tailed distribution. We suggest that (1) the degrees of wholeness for individual centers should be measured by PageRank (PR) scores based on the notion that high-degree-of-life centers are those to which many high-degree-of-life centers point, and (2) that the hierarchical levels, or the ht-index of the PR scores induced by the head/tail breaks can characterize the degree of wholeness for the whole: the higher the ht-index, the more life or wholeness in the whole. Three case studies applied to the Alhambra building complex and the street networks of Manhattan and Sweden illustrate that the defined wholeness captures fairly well human intuitions on the degree of life for the geographic spaces. We further suggest that the mathematical model of wholeness be an important model of geographic representation, because it is topological oriented that enables us to see the underlying scaling structure. The model can guide geodesign, which should be considered as the wholeness-extending transformations that are essentially like the unfolding processes of seeds or embryos, for creating beautiful built and natural environments or with a high degree of wholeness. KEYWORDS: Centers, ht-index, head/tail breaks, big data, complexity, scaling
Content may be subject to copyright.
1
Wholeness as a Hierarchical Graph to Capture the Nature of Space
Bin Jiang
Faculty of Engineering and Sustainable Development, Division of Geomatics
University of Gävle, SE-801 76 Gävle, Sweden
Email: bin.jiang@hig.se
(Draft: July 2012, Revision: December 2014, February, March 2015)
Abstract
According to Christopher Alexander’s theory of centers, a whole comprises numerous, recursively
defined centers for things or spaces surrounding us. Wholeness is a type of global structure or
life-giving order emerging from the whole as a field of the centers. The wholeness is an essential part
of any complex system and exists, to some degree or other, in spaces. This paper defines wholeness as
a hierarchical graph, in which individual centers are represented as the nodes and their relationships as
the directed links. The hierarchical graph gets its name from the inherent scaling hierarchy revealed
by the head/tail breaks, which is a classification scheme and visualization tool for data with a
heavy-tailed distribution. We suggest that (1) the degrees of wholeness for individual centers should
be measured by PageRank (PR) scores based on the notion that high-degree-of-life centers are those
to which many high-degree-of-life centers point, and (2) that the hierarchical levels, or the ht-index of
the PR scores induced by the head/tail breaks can characterize the degree of wholeness for the whole:
the higher the ht-index, the more life or wholeness in the whole. Three case studies applied to the
Alhambra building complex and the street networks of Manhattan and Sweden illustrate that the
defined wholeness captures fairly well human intuitions on the degree of life for the geographic
spaces. We further suggest that the mathematical model of wholeness be an important model of
geographic representation, because it is topological oriented that enables us to see the underlying
scaling structure. The model can guide geodesign, which should be considered as the
wholeness-extending transformations that are essentially like the unfolding processes of seeds or
embryos, for creating beautiful built and natural environments or with a high degree of wholeness.
Keywords: Centers, ht-index, head/tail breaks, big data, complexity, scaling
1. Introduction
It is commonly understood that science is mainly concerned with discovery, but only to a lesser extent,
with creation. For example, physics, biology, ecology, and cosmology essentially deal with existing
things in the physical and biological world and the universe, whereas architecture, music, and design
are about creating new things. This polarization between science and the humanities, or between
scientists and literary intellectuals, often referred to as the two cultures (Snow 1959), still persists,
despite some synthesis and convergence (Brockman 1996). However, significant changes have
happened. First, the emergence of fractal geometry (Mandelbrot 1989) created a new category of art
for the sake of science (Mandelbrot 1989, Pertgen and Richter 1987). All those traditionally beautiful
arts, such as Islamic arts and carpet weaving, medieval arts and crafts, and many other folk arts and
architecture, found a home in science. Fractal geometry and chaos theory for nonlinear phenomena
constitute part of a new kind of science called complexity science. The second change is large
amounts of data, so called big data (Mayer-Schonberger and Cukier 2013), harvested from the Internet
and, more recently, from social media such as Facebook and Twitter. This data has created all kinds of
complex patterns, collectively known as visual complexity (Lima 2011). These two changes are
closely interrelated. On the one hand, fractal geometry is often referred to as the geometry of nature,
being able to create generative fractals that mimic nature, such as mountains, clouds, and trees. On the
other hand, big data are able to capture the true picture of society and nature. In essence, nature and
society are fractal, demonstrating the scaling pattern of far more small things than large ones. Both the
2
generative fractals and visual complexity can consciously or unconsciously evoke a sense of beauty in
the human psyche.
This kind of beauty evoked by fractals and visual complexity is objective, exists in the deep structure
of things or spaces, and links to human feelings and emotions (Alexander 1993, 2002–2005,
Salingaros 1995). The feeling is not idiosyncratic, but as a connection to human beings. It sounds odd
that beauty is objective, because beauty is traditionally considered to be in the eye of the beholder.
The beauty is an objective phenomenon, i.e., objectively structural, but we human beings do have
subjective experience of it which may vary. While attempting to lay out the scientific foundation for
the field of architecture, Alexander (2002–2005) realized that science, as presently conceived, based
essentially on a positivist’s mechanical world view, can hardly inform architecture because of a lack
of shared notion of value. This is why most 20th-century architecture created all kinds of slick
buildings, which continued into the 21st century in most parts of the world. Under the mechanical
world view, feeling or value is not part of science. The theory of centers (Alexander 2002–2005)
adopts some radical thinking, in which shared values and human feelings are part of science,
particularly that of complexity science. In this theory of centers, wholeness is defined as a global
structure or life-giving order that exists in things and that human beings can feel. What can be felt
from the structure or order is a matter of fact rather than that of cognition, i.e., the deep structure that
influences, but is structurally independent of, our own cognition. To characterize the structure or
wholeness, Alexander (2002-2005) in his theory of centers distilled 15 structural properties to glue
pieces together to create a whole (see Section 2 for details), and described the wholeness as a
mathematical problem yet admitted in the meantime no mathematical model powerful enough to
quantify the degrees of wholeness or beauty.
This paper develops a mathematical model of wholeness by defining it as a hierarchical graph, in
which the nodes and links respectively represent individual centers and their relationships. The graph
provides a powerful means for computing the degree of wholeness or life. First, the graph can be
easily perceived as a whole of interconnected centers, enabling a recursive definition of wholeness or
centers. Second, spaces with a living structure demonstrate a scaling hierarchy of far more
low-degree-of-life centers than high-degree-of-life ones. The life or beauty of individual centers can
be measured by PageRank (PR) scores (Page and Brin 1998), which are based on a recursive
definition that high-degree-of-life centers are those to which many high-degree-of-life centers point.
For the graph as a whole, its degree of life can be characterized by the ht-index derived from the PR
scores; the higher the ht-index, the higher degree of life in the whole. The ht-index (Jiang and Yin
2014) was initially developed to measure the complexity of fractals or geographic features in
particular, and it was actually induced by head/tail breaks as a classification scheme (Jiang 2013a),
and a visualization tool (Jiang 2015a). Things of different sizes can be ranked in decreasing order and
broken down around the average or mean into two unbalanced parts. Those above the mean,
essentially a minority, constitute the head, and those below the mean, a majority, are the tail. This
breaking process continues recursively for the head (or the large things) until the notion of far more
small things than large ones is violated.
The contribution of this paper can be seen from several aspects. We illustrate the 15 structural
properties using a generative fractal and an urban layout based on the head/tail breaks. We define
wholeness as a hierarchical graph to capture the nature of space, with two suggested indices for
measuring the degrees of life: PR scores for individual centers, and ht-index for a whole. The
mathematical model of wholeness captures fairly well human intuitions on a living structure, as well
as Alexander’s initial definition of wholeness. Through the head/tail breaks, this paper helps bridge
fractal geometry and the theory of centers towards a better understanding of geographic space in terms
of both the underlying structure and dynamics. The mathematical model of wholeness can be an
important model for geographic representation in support of geospatial analysis, since it goes beyond
the current geometric and Gaussian paradigm towards topological and scaling thinking.
The remainder of this paper is structured as follows. Section 2 illustrates the 15 structural properties
using the Koch snowflake and a French town layout. Section 3 defines the wholeness as a hierarchical
graph a
n
Section
country
mathem
Finally,
2. The
1
Followi
n
2005) d
i
patterns
structur
e
a high
d
putting
a
the mir
r
Wu 201
b
ecause
of cent
e
and litt
l
admitte
d
are fou
n
(Alexan
Koch s
n
Figur
e
Note:
(1982)
Levels
of
As the
b
snowfla
k
based o
ht-inde
x
than lar
g
n
d sugges
t
s
h
4 presents t
h
for measuri
n
atical mode
l
Section 6 d
r
1
5 propertie
n
g his classi
c
i
stilled 15 p
r
(e.g., Alex
a
e
comprises
d
egree of li
fe
a
pair of pat
t
r
o
r
-of-the-se
l
5). In this c
such a feeli
n
e
r has receiv
e
l
e experime
n
d
t
hat the 1
5
n
d of great
der 1993,
M
n
owflake an
d
e
1: Koch sn
o
Pattern (a)
o
, while patte
of
scale
b
uilding blo
c
k
e has four
n head/tail
b
x
(Jiang and
Y
g
e ones recu
r
h
ow to quan
t
h
ree case st
u
n
g degrees
o
l
of wholene
s
r
aws conclus
i
s
c
work of th
e
r
ofound stru
a
nder et al.
many, if not
fe
, beauty, a
n
t
erns or thin
g
l
f test that u
s
onnection,
h
n
g is shared
e
d some ha
r
n
tal evidenc
e
5
properties
a
use to vi
s
M
ehaffy and
S
d
a French to
w
o
wflake wit
h
o
r Euclidean
rn (b) posse
s
prese
n
c
ks of a wh
o
scales: 1, 1/
3
b
reaks (Fig
u
Y
in 2014),
o
r
s.
t
itatively me
u
dies applie
d
o
f life or be
a
s
s related to
i
ons and poi
n
e
pattern lan
g
ctural prope
r
1977, Tho
m
all, of thes
e
n
d wholenes
g
s side by si
d
s
es human
b
h
uman judg
m
by a majori
t
r
sh criticism
s
e
to prove
t
a
re somewh
a
s
ual aestheti
S
alingaros 2
0
w
n layou
t
(
F
h
different d
e
compl
e
shapes in g
e
s
ses a living
n
ce of many
Table 1: Th
e
o
le, centers
a
3
, 1/9, and
1
u
re 2f). In
g
or
the numbe
r
3
e
asure degre
e
d
to an archi
t
a
uty in geog
r
b
eauty, crea
t
n
ts to future
guage, and
o
rties (Table
m
pson 1917
)
e
15 fundam
e
s. We can j
u
d
e, such as t
h
b
eings as a
m
m
ent or feeli
t
y of people.
s
, e.g., the c
o
t
he theory (
A
at elusive a
n
i
c research
0
15). This p
a
F
igure 2) as
w
e
grees of lif
e
e
xity) than p
a
e
neral appea
r
structure ac
c
of the 15 st
r
e
15 structu
r
are defined
1
/27 (Figure
s
g
eneral, the
r
of times th
e
s of life for
t
ectural plan
r
aphic space
t
ion/design,
work.
o
ver the cou
r
1) that can
h
)
with so c
a
e
ntal proper
t
u
dge which
h
e two sno
w
m
easuring in
ng is not id
i
Despite of
t
o
ncept of li
f
A
lexander
2
n
d hard to g
r
and design
a
per first illu
w
orking exa
m
e
: Pattern b s
h
a
ttern a.
r
s to be cold
c
ording to A
l
r
uctural pro
p
r
al propertie
s
at different
l
s
2a, 2b), w
h
levels of s
c
at the scalin
g
individual c
e
and street n
e
s. Section 5
b
ig data, an
d
r
se of 30 yea
r
h
elp generat
e
a
lled living
s
ies, and ther
e
one has a h
i
w
flakes sho
w
s
trumen
t
(A
l
i
osyncratic,
b
h
e empirica
l
f
e is accuse
d
2
003). Alex
a
r
asp. Howe
v
for a bett
e
s
trates the 1
5
m
ples.
h
ows a high
e
and dry in t
e
l
exander (2
0
p
erties.
s
l
evels of sc
a
h
ile the axia
l
c
ale can be
g
pattern of
f
enters and t
h
etworks of
a
further disc
u
d
complexit
y
a
rs, Alexand
e
e
all kinds
o
structure. T
h
r
efore it poss
e
i
gh degree
o
w
n in Figure
lexander 20
0
b
u
t
reliable
l
evidence, t
h
d
of being s
u
a
nder (2005
)
v
er, the 15
p
e
r built en
v
5
properties
u
e
r degree of
e
rms of Man
0
02-2005) d
u
a
le. For exa
m
l
map has fi
v
characterize
f
ar more sm
a
h
e whole.
a
city and
u
sses the
y
science.
e
r (2002–
o
f “good”
h
e living
e
sses has
o
f life by
1
. This is
0
2–2005,
evidence
h
e theory
u
bjective,
)
himself
p
roperties
ironment
u
sing the
life (or
d
elbrot
u
e to the
m
ple, the
v
e scales
d by the
a
ll things
Strong
c
A stron
g
not sep
a
the patt
e
stronge
s
and blu
e
Thick b
o
Centers
snowfla
k
The fiv
e
colors (
F
the hea
d
Figure
four
b
oun
d
space b
e
with re
d
and the
related
p
The axi
a
red as
and t
h
spaces
,
patt
e
conve
x
Alterna
t
Centers
exists i
n
as in t
h
strength
or cent
e
indicati
n
Positive
The co
n
town la
y
c
enters
g
center is s
u
a
rable in for
m
e
rn (Figure
2
s
t center of t
h
e
lines (Figu
r
o
undaries
are often d
i
k
e (Figure
2
e
hierarchica
F
igure 2f),
w
d
/tail breaks
p
2: (Color o
n
levels of sc
a
d
aries
b
ut wi
t
e
tween buil
d
d
borders; (
e
ir adjacency
p
ositive spa
c
a
l lines are i
n
the longest
l
h
e shortes
t
.
I
,
because co
n
e
rn of far m
o
x
spaces hav
e
t
ing repetiti
o
are strengt
h
n
the snowfl
a
h
e axial ma
p
ened to for
m
e
rs (Figure
2
n
g an altern
a
space
n
cept of posi
t
y
ou
t
(Figur
e
u
pported by
m
ing a cohe
r
2
b), and it is
h
e axial ma
p
r
e 2f).
i
fferentiated
2
a) and the
c
l levels of t
h
w
hich appare
n
p
rocess mig
h
n
line) Koch
s
a
le: 1, 1/3, 1
/
t
h identified
d
ing blocks i
s
e
) the positiv
(blue lines)
c
es are appro
n
five hierar
c
l
ines, blue a
s
I
t should be
n
n
nectivity o
f
o
re small thi
n
e
to be furth
e
o
n
h
ened if the
y
a
ke with the
p
. The shor
t
m
the lowest
2
f). The no
t
a
ting repetiti
o
t
ive space a
p
e
2c), in w
h
other surro
u
r
ent whole.
T
supported b
y
p
is the red li
n
by thick bo
u
c
onvex spac
e
h
e axial map
n
tly lack thi
c
h
t be consid
e
s
nowflake a
n
/
9, and 1/27
w
centers of d
i
s
partitioned
e spaces are
forming a
be
ximated int
o
c
hical levels
s
the shortes
t
n
oted that th
e
f
axial lines
r
n
gs than lar
g
e
r aggregate
d
y
are repeate
d
surroundin
g
t
est lines (
b
hierarchical
t
ion of far
m
o
n, or repeat
i
p
plies to bot
h
h
ich all the
4
u
nding cent
e
T
he stronge
s
y
six, 18, a
n
ne, and it is
u
ndaries. F
o
e
s of the ur
b
can be perc
e
c
k boundari
e
e
red thin bo
u
n
d the Frenc
h
w
ith thick b
o
d
ifferent size
s
into many
p
perceived o
r
e
ady ring st
r
o
axial lines,
, perceived
a
t
lines, and
o
e representa
t
r
ather than t
h
g
e ones. This
d
into the ax
town plan.
d by the pr
o
g
alternation
s
b
lue)
b
earin
g
level as a c
e
m
ore short
l
i
ng the head
/
h
the figure
convex spa
c
e
rs in a conf
i
s
t cente
r
of
t
n
d 54 other c
e
recursively
s
o
r example,
t
b
an layout h
a
e
ived as cen
t
e
s. In this re
g
u
ndaries.
h
town of G
a
o
undaries; (
b
s
; (c) the pla
n
p
ositive or c
o
r
represente
d
r
ucture (Hill
i
which can
b
a
s five cente
r
o
ther colors
f
t
ion axial li
n
h
at of conve
x
can be clea
r
ial lines in o
o
perty of alt
e
s
of indents
a
g
this prop
e
e
nter, which
l
ines than l
o
/
tail contras
t
and ground
c
es are pos
i
i
guration as
t
he snowfla
k
e
nters in a r
e
s
upported b
y
t
he differen
t
a
ve thick bo
u
t
ers, represe
n
g
ard, the dif
f
a
ssin: (a) the
b
) the same s
n
n
of the tow
n
o
nvex spaces
d
as individu
a
er and Hans
o
e perceived
a
r
s based on
h
fo
r the lines
b
n
es is better t
h
x
spaces de
m
r
ly seen in P
a
r
der
t
o capt
u
e
rnating rep
e
a
nd outcrop
s
e
rty of alter
n
suppor
t
s ot
h
o
ng ones re
c
statistically
o
f a space.
T
i
tive (Figur
e
a whole. C
e
k
e is in the
m
e
cursive
m
a
n
y
yellow, gr
e
t
triangle si
z
u
ndaries (Fi
g
n
ted by five
f
erent mean
s
snowflake
c
s
nowflake w
i
n
of Gassin;
s
- the
b
lack
p
al centers (r
e
o
n 1984); a
n
as individua
h
ead/tail bre
a
b
etween the
h
an that of
c
m
onstrate the
a
nels e and
f
u
re wholene
s
e
tition. This
s
of the edg
e
n
ating repe
t
h
er hierarchi
c
c
urs four ti
m
rather than
s
T
his is obvi
o
e
2d), repre
s
e
nters are
m
iddle of
n
ner. The
en, cyan,
z
es in the
g
ure 2d).
different
s
used for
c
ontains
i
thout
(d) the
p
olygons
e
d dots),
n
d (f) the
l centers.
a
ks, with
l
ongest
c
onvex
scaling
f
. The
s
s of the
property
e
, as well
t
ition are
c
al levels
m
es, also
s
trictly.
o
us in the
s
ented as
5
individual centers forming a beady ring structure (Figure 2e). An axial line is an approximation of a
set of adjacent positive spaces along a same direction; refer to Hillier and Hanson (1984) for more
details.
Good shape
The concept of good shape is one of the most difficult properties to grasp. Alexander (2002–2005)
suggested a recursive rule, in which parts of any good shape are always good shapes themselves. This
sounds very much like self-similarity or alternating repetition. The snowflake is a good shape because
it consists of many good triangular shapes (Figure 2a). The axial map is a good shape because it
consists of many good shapes of axial lines (Figure 2f).
Local symmetries
Local symmetries refer to symmetries at individual levels of scale, rather than only at the global level.
The snowflake shows both local and global symmetry (Figure 2a). The Alhambra plan (c.f., Section 3)
is a very good example of local symmetries, so has a higher degree of life than that of the snowflake
(Figure 2a) and axial map (Figure 2f).
Deep interlock and ambiguity
The figure and ground can be hardly differentiated while looking at the snowflake along its boundary
(Figure 1b) because they interpenetrate each other, forming a deep interlock. This property of deep
interlock and ambiguity is closely related to figure-ground reversal in Gestalt psychology (Rubin
1921). The same phenomenon appears in the town layout (Figure 2c and 2d), in which the building
blocks and the pieces between them interpenetrate each other, creating ambiguity in visual perception.
Contrast
Contrast recurs between adjacent centers, thus strengthening the related centers. This kind of contrast
appears between a big dot and its surrounding small dots (Figure 2b), and between the building blocks
and the positive spaces (Figure 2d). There are many other different pairs of contrast, such as between
the head and tail, red and blue, warm and cold colors, and a minority in the head and a majority in the
tail.
Gradients
The centers gradually strengthen from the smallest to the largest scale, from the shortest to the longest
line, from blue to red (Figure 2f), from the smallest to biggest dots (Figure 2b), and from the
least-connected to the most-connected lines. This property of gradients can also be referred to as the
scaling hierarchy ranging from the smallest to the largest.
Roughness
Roughness is what differentiates fractal geometry from Euclidean geometry and is synonymous to
messiness or chaos. The border of the snowflake (Figure 2a) is rough, and an axial line is a rough
representation of individual set of convex spaces (Figure 2f). Things with roughness may look messy
or chaotic, but they possess a degree of order. It should be noted that both mathematical snowflake
(Figure 2a) and real snowflake are rough, yet the former being strictly rough, while the latter
statistically rough.
Echoes
The property of echoes can be compared to that of self-similarity in fractal geometry. In the snowflake,
the triangle shape echoes again in different parts and in different sizes (Figure 2a). The scaling pattern
of far more short lines than long ones (or the head-tail contrast) recurs or echoes four times (Figure
2f).
Void
Void is defined as an empty center at the largest scale, surrounded by many other smaller centers.
Under this definition, the largest center in the snowflake is a void (Figure 2b), as is the longest axial
line (Figure 2f). In general, the highest class (which may involve multiple elements) induced by the
head/tai
l
Simplic
i
The de
g
number
similar
c
(Figure
centers
(
Not-
s
ep
a
A cente
r
scales,
f
A whol
e
univers
e
why bo
t
The 15
p
more th
e
through
life, we
3. Who
l
The id
e
neurop
h
(2002-2
0
Alexan
d
quantif
y
degrees
nodes a
n
the grap
for our
m
This se
c
good pr
o
that a l
high-de
g
ht-inde
x
Figure
3
Note:
T
l
breaks con
s
i
ty and inne
r
g
ree of life o
f
of differen
t
c
enters to o
n
2b). This is
(
Figure 2f).
W
a
rateness
r
is not sepa
r
f
rom the sm
a
e
connec
t
s
t
e
. A whole c
t
h the snowf
l
p
roperties
bi
e
properties
,
our elabora
t
define the
w
l
eness as a
h
e
a of whol
e
h
ysiology, m
e
0
05) has e
v
d
er’s definiti
o
y
the degree
of life, but
i
n
d links rep
r
p
h, we can c
o
m
odel is tha
t
c
tion presen
t
o
xy of degr
e
i
ving struct
u
g
ree-of-life
x
: the higher
3
: (Color on
l
T
he dot size
s
s
titutes a vo
i
r
calm
f
a center d
e
centers. Th
i
n
e class. The
the same
fo
W
ithin each
r
able from i
t
a
llest to the
l
t
o the nearb
y
onnects to
h
l
ake (Figure
i
nd all cent
e
,
the higher
t
ion that the
s
w
holeness as
a
h
ierarchical
e
ness has b
e
e
dicine, cos
m
v
er formul
a
o
n of whole
n
of life of a
r
i
t lacks of t
h
r
esent identi
f
o
mpute the
d
t
it captures
f
t
s the two
m
e
es of life o
r
u
re demonst
r
centers; an
d
the ht-index
l
ine) Whole
n
s
indicate d
e
i
d.
pends on its
i
s process c
a
six, 18, an
d
o
r the axial
class, there
i
t
s surroundi
n
l
arges
t
, are
e
y
wholes to
h
uman being
s
2a) and axia
e
rs together
i
t
he degree
o
e properties
a
hierarchic
a
graph
e
en discuss
e
m
ology, and
ted and de
f
n
ess, some
p
r
chitecture.
T
h
e recursive
f
ied centers
egrees of li
fe
f
airly well t
h
m
easures, th
e
r
beauty. In t
h
r
ates a scali
n
d
the degre
e
,
the higher
d
n
ess as a hie
r
e
gree central
i
6
simplicity
a
a
n be achie
v
d
54 red dot
s
map, whic
h
i
s a sense of
n
g centers.
T
e
ssential and
recursively
s
in their de
e
a
l map (Figu
r
i
nto a whol
e
o
f life or th
e
are real and
a
l graph to
m
e
d in a va
r
ecology (e.
g
fined i
t
in
p
revious eff
o
T
he propose
d
property.
W
and their re
l
fe
for the ind
i
h
e recursive
e
PR scores
t
he next sect
i
n
g hierarch
y
e
of the sc
a
d
egree of lif
e
r
archical gra
p
i
ty in Figure
a
nd inner ca
l
v
ed through
can be clus
t
h
is classifie
d
simplicity a
n
T
his propert
y
not separab
l
form even
e
p psyche, e
v
r
e 2f) look b
e
e
to develop
wholeness.
identifiable.
m
easure the
d
r
iety of sci
e
g
., Bohm 19
8
precise m
a
rts have bee
n
d
measure
L
W
e represent
a
l
ationships
w
i
vidual cent
e
n
ature of w
h
and h
t
-inde
x
i
on, we furt
h
y
of far mor
e
a
ling hierarc
e
or wholen
e
p
h of (a) the
3a and
b
et
w
m
, or the pr
o
the head/tai
l
t
ered into th
r
d
into five
h
n
d inner cal
m
y
has severa
l
l
e in formin
g
larger whol
e
v
oking a se
n
e
autiful.
a high degr
e
We have s
h
To further
q
d
egree of life
e
nces such
8
0), but no o
n
t
hematical
l
n
made (e.g.
L
does indic
a
a
whole as
a
w
ithin the w
h
e
rs and the
w
h
oleness as
d
x
, and argue
h
er illustrate
e
low-degre
e
h
y can be
c
e
ss.
snowflake,
a
w
eenness cen
t
o
cess of red
u
l
breaks by
r
ee classes
o
h
ierarchical
m
.
l
other mea
n
g
a scaling
h
e
s, toward t
h
n
se of beaut
y
e
e of whole
n
h
own in Fig
u
q
uantify the
d
e
or wholene
s
as physics,
n
e prior to
A
l
anguage.
F
.
, Salingaros
a
te approxi
m
a
graph, in
w
h
ole (Figure
w
hole. What
d
efined by A
e
s why they
through ca
s
e
-of-life ce
n
characterize
d
a
nd (b) the
a
trality in Fi
g
u
cing the
grouping
o
r centers
levels or
n
ings. All
h
ierarchy.
h
e entire
y
. That is
n
ess. The
u
re 2 and
d
egree of
s
s.
biology,
A
lexander
F
ollowing
1997) to
m
ately the
w
hich the
3). With
i
s unique
lexander.
can be a
s
e studies
n
ters than
d
by the
xial map
g
ure 3b.
3.1 Me
a
Accordi
n
strength
cumulat
a whol
e
social s
t
status o
f
centrali
t
or prest
i
The def
i
definiti
o
our frie
n
comput
a
iterativ
e
operate
s
Essenti
a
and the
r
more v
o
in whic
h
r(i) and
the dam
p
Note:
deg
r
The PR
which s
u
the sno
w
lines di
f
centers
a
can deri
v
3.2 Me
a
Ht-inde
x
features
The sno
world, t
h
derive t
heavy-t
a
a
suring the
d
ng to the
t
ened by its
ive properti
e
e
. Given the
t
atus of a p
e
f
a person,
w
t
y), but also
w
i
ge that dete
r
i
nition of P
R
o
n is recursi
v
n
ds of frien
d
a
tionally int
e
e
until conve
r
s
on a web
a
lly, voting
f
r
efore the P
R
o
tes. Formall
y


∈
h
n is the tot
a
r(j) are PR
s
p
ing factor,
w
Figure 4:
(
The degrees
r
ee, blue as
t
scores capt
u
u
rrounding
c
w
flake’s cen
t
f
fer from th
e
a
re assigne
d
v
e the ht-in
d
a
suring the
d
x
is a head/
t
in particula
r
wflake has
f
h
e scales ar
e
t
he inherent
a
iled distrib
u
d
egrees of l
i
t
heory of c
e
surroundin
g
e
s, implying
recursion i
n
e
rson within
w
e should n
o
w
ho are the
r
mines
t
he p
e
R
can be exp
r
v
e. To make
a
d
s, and so o
n
e
nsive beca
u
r
gence is re
a
graph, in
w
f
or incomin
g
R
scores. Ho
w
y
, PR is defi
n


a
l number o
f
s
cores of no
d
w
hich is us
u
(
Color onlin
e
of life are v
t
he lowes
t
d
e
u
re the spiri
t
c
enters poin
t
t
ers look th
e
e
ir length s
h
d
degrees of
d
ex as an in
d
d
egree of li
f
t
ail breaks i
n
r
(Jiang and
f
our scales 1
e
not as clea
r
scales. In
t
u
tion. We di
v
i
fe using the
e
nters (Ale
x
g
centers.
M
that the deg
r
n
herent in
w
a social ne
t
o
t just ask h
o
p
eople. In o
t
e
rson’s statu
s
r
essed as i
m
a
n analogue
,
n
until virtu
a
u
se it involv
e
a
ched (see L
a
w
hich dire
c
g
links deter
m
w
ever, the
v
n
ed as follo
w
f
nodes; ON
(i
d
es i and j; n
j
ally set to 0.
e
) Degrees
o
isualized
b
y
e
gree, and o
t
t
of wholen
e
t
to a centra
l
e
same as th
e
own previo
u
life measur
e
icator for th
e
f
e using ht-i
n
n
duced inde
x
Yin 2014).
I
, 1/3, 1/9, a
n
r
as in the sn
h
e axial m
a
v
ided all 39
l
7
e
PageRank
x
ander 200
2
M
ore import
a
r
ee of life o
f
w
holes or ce
n
t
work (Katz
ow many p
e
t
her words,
i
s
. This thou
g
m
portant pag
e
,
our import
a
a
lly all peo
p
e
s all the no
a
ngville and
c
ted links i
n
m
ines the i
m
v
oting is not
w
s:
(
i) is the outl
i
j
denotes th
e
85.
o
f life of (a)
t
dot sizes an
d
t
her colors a
s
e
ss, or the d
e
l
one for co
m
e
ir sizes (Fi
g
u
sly
b
ecaus
e
e
d by PR sc
o
e
degree of l
i
n
dex for th
e
x
for measu
r
It
reveals th
e
n
d 1/27, so t
h
n
owflake. W
e
a
p for exam
p
l
ines around
scores for t
h
2
–2005), th
e
a
ntly,
t
he li
fe
f
a center rel
i
n
ters, the d
e
1953, Bon
a
e
ople this p
e
i
t is not onl
y
g
ht underlies
e
s to which
m
a
nce depend
s
p
le on the pl
a
des of a gr
a
Meyer 200
6
n
dicate hotl
i
m
portance a
n
one page o
n
[1]
i
nk nodes (t
h
e
number of
o
t
he snowfla
k
d
the spectr
a
s
degrees be
t
e
gree of life
.
m
puting the
P
g
ure 4). Ho
w
e
PR is recu
r
o
res. Based
i
fe of the w
h
e
wholeness
r
ing the co
m
e
number of
i
h
e ht-index
i
e
must cond
u
p
le, the len
g
the average
h
e centers
e
degree of
fe
(or intens
i
i
es on those
o
e
gree of life
a
cich 1987).
r
son knows
popularity,
b
Google’s P
R
m
any import
s
on the imp
o
a
net are inc
l
p
h. Howeve
6
for more d
e
nks from o
n
d relevance
n
e vote, but
i
h
ose nodes t
h
o
utlink node
s
k
e and (b) th
e
a
l colors, wit
h
t
ween the hi
g
.
We sugges
t
P
R scores.
T
w
ever the d
e
r
sively defi
n
on degrees
o
h
oleness.
m
plexity of f
r
i
nherent sca
l
i
s 4. For mo
s
u
ct the head/
t
g
th of the
a
length (first
f
life of a
c
i
ty) has rec
u
of all other
c
can be co
m
To assess t
h
(which is t
h
b
ut also soc
i
R
algorithm.
t
ant pages p
o
o
rtance of o
u
l
uded. It so
u
e
r, the comp
u
e
tails). The
P
o
ne page to
e
of individ
u
i
mportan
t
p
a
h
at point to
n
s
of node j;
a
e
axial map
h red as the
h
g
hest and lo
w
t
a directed
T
he degrees
e
grees of lif
e
n
ed. To this
p
o
f life of ce
n
fr
actals or g
e
l
es in a set o
r
st patterns i
n
/
tail breaks
p
a
xial lines e
mean) into
t
c
enter is
u
rsive or
c
enters in
m
pared to
h
e social
h
e degree
i
al powe
r
o
int. This
u
r friends,
u
nds very
u
tation is
P
R model
another.
al pages,
a
ges have
n
ode i);
a
nd d is
h
ighes
t
w
es
t
.
g
raph, in
of life in
e
of axial
p
oint, all
n
ters, we
e
ographic
r
pattern.
n
the real
p
rocess to
x
hibi
t
s a
t
wo par
t
s.
Those l
i
tail. Th
i
(for exa
m
that the
if the re
the ht-i
n

in whic
h
In the s
a
scores t
o
higher
d
of life (
h
Why ca
n
scaling
p
comple
x
is what
higher
d
more sc
a
Not
e
4. Case
To furt
h
b
uildin
g
New Y
o
map of
M
used s
m
b
etwee
n
amount
s
The tw
o
mappin
g
4.1 The
The Al
h
degree
o
degree
o
i
nes longer t
h
i
s head/tail
b
m
ple, < 40
p
head is a m
i
sulting hea
d
n
dex (h) is d
e
1
h
m(r) is the
a
me way as
o
derive h a
s
d
egree of lif
e
h
=5) than th
a
n
the ht-ind
e
p
attern, or
r
x
, or involve
the ht-inde
x
d
egree of lif
e
a
les involve
d
Figure 5:
e
: A growing
A
m
studies: Co
m
h
er demonstr
a
g
, cit
y
, and
c
o
rk City), an
d
M
anhattan,
a
m
all data, wit
h
n
the center
s
s
of auto-ge
n
o
data sets
w
g
and map g
e
plan of Alh
h
ambra is o
n
o
f life accor
d
o
f life of the
h
an the mea
n
b
reaks proce
p
ercent). Ea
c
i
nority o
r
, al
t
d
is not a mi
n
e
fined as fol
l
number of
v
the axial-li
n
s
an indicat
o
e
or the whol
a
t of the sno
w
e
x indicate t
h
r
ecurring ti
m
s more scal
e
x
refers to.
O
e
or the who
l
d
during the
Intuitive ex
a
mouse foot
m
sterdam fro
m
m
puting th
e
a
te the math
e
c
ountry scal
e
d
the countr
y
a
nd the stre
e
h
which we
m
s
for comp
u
n
erated axia
w
ere previo
u
e
neralizatio
n
ambra
n
e of the m
d
ing to Ale
x
architectur
a
n
constitute
ss continue
s
c
h time a w
h
t
ernativel
y
,
t
n
ority. The
h
l
ows:
alid means
d
n
e length,
w
o
r of the deg
e
ness. Acco
r
w
flake (h=4
)
h
e degree o
f
m
es of far
m
e
s, than Figu
O
ur intuition
l
eness. For
e
formation a
n
a
mples of a
h
i
n four days
m
the 16th to
e
degrees of
e
matical mo
d
e
s: the Alha
m
y
of Sweden
e
t network o
f
m
anually id
e
u
tation and
l
lines and
s
u
sly studied
n
.
ost beautifu
l
x
ande
r
(200
2
a
l plan and i
t
8
the head, a
n
s
recursively
h
ole or head
t
he mean m
u
ht
-index is t
h
d
uring the h
e
w
e run the h
e
g
ree of life o
r
r
ding to our
)
.
f
life or the
w
m
ore small t
h
u
re 1a. This
c
also suppor
t
e
xample, the
n
d develop
m
h
igher ht-in
d
from day 1
2
the 18th ce
n
life
d
el of the w
h
m
bra
b
uildi
n
. We rely o
n
f
Sweden to
e
ntified abo
u
analysis. T
h
s
treets extra
c
by Jiang (
2
l building c
2
–2005) an
d
t
s numerous
n
d those that
for the hea
d
is broken i
n
u
st be valid.
h
e number o
f
[2]
e
ad/tail brea
k
e
ad/tail brea
k
r
the whole
n
calculation,
t
w
holeness?
h
ings than l
a
c
omplexity,
o
t
s the notio
n
embryo an
d
m
en
t
(Figure
5
d
ex meaning
2
to day 15,
a
n
tury (Alexa
n
h
oleness, w
e
n
g complex,
n
t
he archite
c
compute th
e
u
t 700 center
s
h
e second
a
c
ted from
O
2
013b) and
J
omplexes i
n
d
Salingaros
centers to s
e
are shorter
t
d
until it is
n
n
to two parts
I
n other wo
r
f
valid mean
s
k
s process fo
r
k
s process
u
n
ess. The hi
g
t
he axial ma
p
The ht-inde
x
a
rge ones. F
o
r the numb
e
n
that the hi
g
d
city
b
ecom
e
5
).
a higher de
g
a
nd the evol
u
n
der 2002–2
0
applied it t
o
the island
o
c
tural plan o
f
e
ir degrees o
s
and more t
h
a
nd third st
u
O
penStreetM
a
J
iang et al.
n
the world
(1997). We
e
e if our mo
d
t
han the me
a
no longer a
s
, we must
m
r
ds, a mean
i
n
s plus one.
F
r the PR sco
u
sing the ce
n
g
her the h
t
-i
n
p
has a high
e
x
is a meas
u
F
igure 1b lo
o
e
r of scales
i
g
her the h
t
-i
n
e
more com
p
g
ree of life
u
tion of the
c
005)
o
three case
s
o
f Manhatta
n
f
Alhambra,
o
f life. The f
i
h
an 800 rel
a
u
dies invol
v
a
p (OSM)
d
(2013) for
c
and has a
v
tried to co
m
del of whol
e
a
n are the
minority
m
ake sure
i
s invalid
F
ormally,
r
es.
n
ters’ PR
n
dex, the
e
r degree
u
re of the
o
ks more
i
nvolved,
n
dex, the
p
lex with
c
ity of
s
tudies in
n
(part of
the axial
i
rst study
t
ionships
v
ed large
d
atabases.
c
ognitive
v
ery high
m
pute the
e
ness can
capture
the faca
d
and mi
d
convex
their rel
a
connect
i
above.
T
graph t
o
The pro
The set
not be
u
all spac
e
visual i
n
support
not use
a
data. In
some a
u
indicate
centers
properti
e
space a
m
all the c
e
F
i
Note: T
h
are div
i
life for
4.2 The
The ne
x
city wa
s
generat
e
line-to-l
to, or s
u
set, incl
u
(Jiang a
n
the intuitio
n
des and inte
r
d
dle), or nin
e
spaces fro
m
a
tionships.
T
i
ng door), a
n
T
hese conv
e
o
be used for
cess of ide
n
of convex s
p
u
nique. The
p
e
s are cover
e
n
spection, i.
central ones
a
ny automat
i
this regard,
t
u
tomatic pro
c
the degree
with the h
i
e
s such as
l
m
ong the 15
e
nters, usin
g
i
gure 6: (Co
l
here are 725
i
ded into thr
e
the individu
streets of
M
x
t two case s
t
s
chosen fo
r
e
d 1,800 axi
ine intersect
u
ppor
t
, long
o
u
ding 166,4
7
n
d Claramu
n
n
or percepti
o
r
nal structu
r
e
subparts se
p
m
functional
s
T
he relation
s
n
d are amo
n
e
x spaces an
computing
t
n
tifying the
c
p
aces must
b
p
rocess start
s
e
d (Hillier a
n
e., which c
e
. There is us
u
i
c process i
n
t
his case stu
c
esses were
of life for t
h
i
ghest degr
e
l
ocal symm
e
properties.
T
g
equation [2
l
or online) T
h
convex spa
c
e
e parts (left
,
al centers ar
e
M
anhattan
a
t
udies move
r
the case s
al lines for
M
ion. This int
o
nes. The s
a
7
9 streets fo
r
n
t 2004, Jia
n
o
n. Our stud
y
r
e in the ver
t
p
arated by r
e
s
paces such
hips are bet
w
n
g the cent
e
d
their relat
i
t
he degrees
o
c
onvex spac
e
b
e the least
n
s
with the fir
n
d Hanson 1
9
e
nters tend
t
u
ally little a
m
n
order to m
a
d
y comple
m
adopted. Fi
g
h
e centers.
T
e
es of life
a
e
tries,levels
T
he h
t
-index
].
h
e degrees
o
c
es, or cente
r
,
righ
t
, and
m
e
indicated
b
nd Sweden
from the ar
c
t
udy becaus
e
M
anhattan
a
e
rsection rel
a
me rule app
l
r
the entire c
n
g, Zhao and
9
y
concentra
t
t
ical directio
e
d lines in F
i
as rooms, c
w
een two sp
e
rs that belo
ionships co
n
o
f life.
e
s and their
n
umber of f
a
r
st fattest co
n
9
84). As to t
h
to suppo
r
t
o
m
biguity in
t
a
ke the relati
o
m
ents the foll
g
ure 6 show
s
T
he results
a
a
re rather
o
of scale, s
t
of the whol
e
o
f life comp
u
r
s, and 880
r
m
iddle) or s
u
b
y dot sizes,
w
6.
c
hitectural s
c
e
i
t
s street
n
a
nd convert
e
l
ationship is
l
ies to the st
r
c
ountr
y
. The
Yin 2008).
I
t
ed on its tw
o
n. The plan
i
gure 6. We
f
ourts, garde
n
aces that pe
n
n
g to the s
a
n
stitute
t
he
n
relationship
s
a
ttest conve
x
n
vex space,
t
h
eir relation
s
o
ther ones?
t
erms of the
r
o
nships as a
c
owing two
c
s
the compu
t
a
re highly i
n
o
bvious bec
a
t
rong center
s
e
ness is 6, d
e
u
ted for the
p
r
elationships
u
bparts, indi
c
w
hile the pl
a
c
ale to city a
n
n
etwork is
e
e
d them into
b
ased on th
e
r
eets of Sw
e
streets are a
I
n other wo
r
o
-dimension
a
consists of
t
f
irst manual
l
n
s, and hall
s
n
etrate each
a
me parts o
r
n
odes and li
n
s
is time-co
n
spaces; oth
e
h
e second f
a
s
hips, we m
u
For exampl
r
elationship
s
c
curate as p
o
c
ases involvi
n
t
ed degrees
o
structive. F
o
a
use of the
s
, thick bou
n
e
rived from
t
lan of the A
l
between th
e
c
ated by red
l
a
n as a whol
e
n
d country s
c
e
asily perce
i
a hierarchi
c
e
simple rule
den. The str
e
mix of nam
e
d
s, individu
a
n
al layout by
t
hree parts (
l
l
y drew all i
n
s
, and then
i
other (such
r
subparts
m
n
ks of a hi
e
n
suming an
d
e
rwise, the
s
a
ttest, and so
u
s
t
determin
e
l
e, peripher
a
s
. We delibe
r
o
ssible for s
u
ng big data,
o
f life, in w
h
o
r example,
recurring
s
n
daries, an
d
t
he degrees
o
l
hambra (ins
e
centers. Th
e
l
ines. The d
e
e
has a degr
e
c
ales. Manh
a
i
ved as a w
h
c
al graph in
that short li
n
r
eets are the
c
ed and natu
r
a
l street seg
m
ignoring
l
eft, right
n
dividual
i
dentified
as with a
m
entioned
e
rarchical
d
tedious.
s
et would
on, until
e
through
a
l centers
r
ately did
u
ch small
in which
h
ich dots
the three
s
tructural
d
positive
o
f life for
et)
e
centers
e
grees of
e
e of life
a
ttan as a
h
ole. We
terms of
n
es point
c
omplete
r
al streets
m
ents are
merged
street n
e
and the
the dire
c
were vi
s
degree
o
surprisi
n
two an
d
power l
a
scaling
h
(Note:
A
Based
o
respecti
v
intuitio
n
b
ecause
rather s
i
structur
e
than th
a
1,800 a
x
comple
x
that mo
d
variatio
n
Fi
g
Note:
T
than hi
and
o
striking
according t
o
e
tworks are t
h
directed lin
k
c
ted graphs,
s
ualized usi
n
o
f life, and
n
g that the
d
d
three deca
d
a
w exponen
t
h
ierarchy.
Table 2: P
o
A
lpha is the
p
o
n the degr
e
v
ely for Ma
n
n
that the S
w
the former
i
mple grid-li
k
e
. Our calc
u
a
t of Manha
t
x
ial lines. T
h
x
geographi
c
d
ern buildin
g
n
would hav
e
g
ure 7: (Col
o
T
he two enl
a
i
gh-degree-o
o
ther colors
f
power law
w
o
the same
n
h
en convert
e
k
s indicate r
e
we compute
n
g the spect
r
the other
c
d
egrees of li
f
d
es of pow
e
t
around 2.0
+
o
wer law sta
t
p
ower law e
x
scor
e
Man
h
Sw
e
e
es of life
o
n
hattan and
S
w
edish stree
is far more
k
e layout,
w
u
lation does
t
tan, althou
g
h
is is in cons
i
c
features or
g
s such as s
k
e
lower deg
r
o
r online) T
h
a
rged insets
(
f-life center
s
f
or degrees
o
w
ith two or
t
n
ames, and
e
d into dual
g
e
lationships
d the degree
r
al colors,
w
c
olors for d
f
e demonstr
a
e
r law fi
t
, a
n
+
is consiste
n
t
istics on P
R
x
ponent, P a
n
e
, above whi
c
A
h
attan
2
e
den
2
o
f individu
a
S
weden eac
h
t
network i
s
h
eterogeneo
u
w
hile the Alh
support this
g
h the Alha
m
i
stent with t
h
space (Jian
g
k
yscrapers a
n
ees of life o
r
e degrees o
f
Manhatt
a
(
b and c) pre
s
s
: Blue for t
h
o
f life betwe
e
t
hree decade
s
10
further mer
g
g
raphs in w
h
from short
s
e
s of life for
t
w
ith blue as
d
egrees of l
i
a
ted very st
r
n
d a very hi
n
t with the t
h
R
scores for t
h
n
index for t
h
c
h the powe
r
A
lpha
2
.39
0
2
.19
0
a
l centers,
w
h
as a whole
s
more aliv
e
us and far l
a
h
ambra as a
b
intuition, i.
m
bra has o
n
h
e definition
g and Yin
2
n
d urban str
u
r
wholeness
t
f
life shown
t
a
n (a) and S
w
sent a better
h
e lowest de
g
e
n the lowes
t
s
of power l
a
0.8) (e)
g
ed together
h
ich the nod
e
s
treets (or a
x
t
he individu
a
the lowest
d
i
fe between
r
iking powe
r
gh degree
o
h
eoretic rule
s
h
e Manhatt
a
h
e goodness
r
law is obse
r
P
X
0
.763.
7
0
.801.
2
w
e further c
o
. The result
e
than the
M
a
rger than t
h
b
uilding co
m
e
., the Alha
m
n
ly 725 con
v
of h
t
-index:
2
014). From
u
ctures such
t
han traditio
n
t
hrough the
u
w
eden (d)
sense of far
g
ree of life,
r
t
and highes
t
a
w fi
t
and v
e
to form th
e
e
s represent
t
x
ial lines) to
a
l axial lines
d
egree of li
f
the lowest
r
laws (Clau
s
f goodness
s
Salingaros
n
and Swed
e
of fit, and
xm
r
ved).
X
min
7
0E04
2
7E05
o
mputed h
t
-
appea
r
s to b
M
anhattan st
r
h
e latter. M
a
m
plex shows
m
bra has a
h
v
ex spaces,
w
the higher t
h
the case st
u
as shoppin
g
n
al architect
u
u
nderlying s
t
more low-d
e
r
ed for the hi
t
. The PR sc
o
ry high goo
d
e
natural str
e
t
he individu
a
long ones.
B
s
and streets,
f
e, red as th
e
and highes
t
set et al. 20
0
of fi
t
(Tabl
e
(1998) sug
g
e
n networks
m
in the min
i
-
indices of
5
b
e consistent
r
eet networ
k
a
nhattan as
a
much more
higher degr
e
w
hile Manh
a
h
e h
t
-index,
u
dies, we ca
n
g
malls with
o
u
re.
t
reet structu
r
e
gree-of-life
i
ghest degre
e
o
res exhibit
a
d
ness of fit (
p
e
ets. The
a
l streets,
B
ased on
and they
e
highest
t
. It was
0
9), with
e
2). The
g
ested fo
r
i
mum PR
5
and 7,
with our
k
. This is
a
city has
complex
e
e of life
a
ttan has
the more
n
foresee
o
ut much
e of
centers
e
of life,
a
very
p
around
11
The three case studies demonstrate that the defined wholeness, or hierarchical graph, captures fairly
well human intuitions on the degree of life. Various maps or patterns based on the head/tail breaks
classification were found to have a higher degree of life than those using natural breaks through the
mirror-of-the-self test (Wu 2015). It was also found that the kind of skills of appreciating the living
structure can be improved through training. The reader might have noticed that the hierarchical graph
goes beyond the exiting geographic representations (Peuquet 2002), since it is topological rather than
geometric (Jiang and Claramunt 2004). The topological representation enables us to see the
underlying scaling pattern, which constitutes a different way of thinking, the Paretian thinking, for
geospatial analysis (Jiang 2015b). The model of wholeness would contribute fundamentally to
geodesign by orienting or re-orienting it towards beautiful built and natural environments or with a
high degree of wholeness. Geodesign, as currently conceived (e.g., Lee, Dias and Scholten 2014),
mainly refers to a set of geospatial techniques and technologies for planning built and natural
environments through encouraging wide-range human participation and engagement. However, there
is a lack of standards in terms of what a good environment is. The model of wholeness provides a
useful tool and indices for measuring the goodness. We therefore believe that geodesign should be
considered as the wholeness-extending transformations, or something like the unfolding processes of
seeds or embryos (Alexander 2002–2005) towards a high degree of wholeness (see a further
discussion in Section 5). This idea of unfolding applies to map design as well, since maps are
essentially fractal and possess the same kind of beauty (Jiang 2015c). This is in line with what we
have discussed at the beginning of this paper that design should be part of complexity science. The
mathematical model of wholeness also points to the fact that the wholeness or degree of life is
mathematical and computational (Alexander 2002–2005), and it captures the nature of space, or
geographic space in particular. The next section adds some further discussions on the mathematical
model of wholeness and its implications in the era of big data.
5. Further discussions on the mathematical model of wholeness
Wholeness emerges from recursively defined centers, so it can be considered as an emergence of
complex structures. To sense or appreciate the wholeness, we must develop both figural and analytical
perception, or see things holistically and sequentially. However, a majority of people tend to see
things analytically rather than figuratively (Alexander 2005). These two kinds of perception help us
see a whole and its building-block centers, and perceive the degree of life or wholeness through the
interacting and reinforcing centers. These perception processes are manifested in the mathematical
model of wholeness. In other words, this model enables us to see things in their wholeness from their
fragmented, yet interconnected, parts. The wholeness, or life, or beauty is something real, rather than
a matter of opinion (Alexander 2002–2005). This kind of beauty exists in geographic space, arising
from the underlying scaling hierarchy, or the notion of far more small geographic features than large
ones (Jiang and Sui 2014). Large amounts of geographic information harvested from social media and
the Internet enable us to illustrate striking scaling patterns (Jiang and Miao 2014, Jiang 2015a) and
assess the goodness of geographic space.
The 15 properties are mainly considered as structural properties or the glue that holds space together,
through which wholeness can be constructed. Recognition of the underlying structure is just one part
of science - discovery. The other part is how to generate the kind of structure, or creation of the living
structure, which is the central theme of the second book (Alexander 2002-2005). The 15 properties
also can act as the glue for the creation, or the wholeness-extending transformations. For example, the
process of generating the snowflake (Figure 1) is not additive but transformative. At each step, we do
not just add smaller triangles, but transform the previous version as a whole, to give it more
centeredness or wholeness by inducing more triangles (or centers in general) to intensify those that
exist already. This generative process of the snowflake is the same as that of creating the life of the
column in the step-by-step fashion elaborated by Alexander (2002-2005). In this regard, the
mathematical model of wholeness is of use to guide the unfolding process because both the PR scores
and ht-index provide good indicators for the degrees of life.
The mathematical model of wholeness is not limited to measuring the degree of life in geographic
12
space. It can be applied to artifacts such as Baroque and Beaux arts, kaleidoscopes, and visual
complexity generated from big data (Lima 2011). In spite of the popularity of the generative fractals
and visual complexity, the question as to why they are beautiful has never been well-addressed.
Through our model, we are able to not only explain why visual complexity and generative fractals are
beautiful, but also measure and compare the degree of beauty. This kind of beauty exists in the deep
structure, rather than in the surface coloring or appearance. This sense of beauty belongs to 90 percent
of our self, or our feelings are all the same (Alexander 2002–2005). Importantly, the beauty has
positive effects on human well-being. Taylor (2006) found that generative fractals, much like the
natural scenes (Ulrich 1984), can help reduce physiological stress. Salingaros (2012) further argued
that well-designed architecture and urban environments should have healing effects.
With the model of wholeness, the kind of beauty becomes computable and quantifiable. We
mentioned earlier how the computed degree of life is consistent with human intuitions on living
structures. One way to verify this is through eye-tracking experiments of human attention while
watching a building plan such as the Alhambra (Yarbus 1967, Duchowski 2007). The captured
fixation points from a group of people can be analyzed and compared to the degree of life computed
using the model. This state-of-the-art methodology complements the mirror-of-the-self test, which
only captures human intuitions on degree of life for a pair of patterns or things. The digital
eye-tracking data can verify or compare our computed results on the degree of life. The beauty
constitutes part of complexity science (Casti and Karlqvist 2003, Taylor 2003) and helps bridge
science and arts in the big data era.
6. Conclusion
According to the theory of centers, all things and spaces surrounding us possess a certain degree of
order or life, and those with a high degree of order are called living structures. This order
fundamentally differs from what we are used to: regularity in terms of Euclidean geometry or
normality in terms of Gaussian statistics. To put it more broadly, we are used to the 20th-century
scientific worldview (mechanistic in essence), in which beauty is considered a matter of opinion,
rather than that of fact. The living structure that exists in nature (e.g., Thompson 1917), as well as in
what we build and make (e.g., Alexander et al. 1977) has many, if not all, of the 15 properties. This
paper illustrated the 15 properties using two examples of space: the generative fractal snowflake and
the French settlement layout. The illustration is well-supported by the head/tail breaks, a new
classification scheme and visualization tool for data with a heavy-tailed distribution. We have shown
the recurrences of the 15 properties in the living structures, making the 15 properties less elusive.
To quantify the living structure, this paper developed a model of wholeness based on Alexander’s
mathematical view of space. This model is a hierarchical graph in which numerous centers are
represented by the nodes and their interactions are the directed links. Based on the initial definition of
wholeness, particularly its recursive nature of centers, we suggested PR scores and ht-index as good
proxies for the degrees of life because of their recursive nature. The three case studies presented some
strong results. For example, the centers with the highest degrees of life in the Alhambra plan capture
fairly well human intuitions on a living structure. More importantly, the degrees of life for both
Manhattan’s and Sweden’s street networks demonstrate very striking power laws. These results are
encouraging in terms of recognizing and appreciating the living structure. However, we are still far
away from creating the kind of living structure known as the field of harmony-seeking computations
(Alexander 2005). In this regard, we believe that the mathematical model of wholeness and related
measures shed light on the wholeness-extending transformations. Our future work points in this
direction.
Acknowledgment
The initial idea of this paper emerged while I was visiting the Tokyo Institute of Technology,
supported by the JSPS Invitation Fellowship. I would like to thank Toshi Osaragi for hosting my visit,
which I enjoyed very much. I also would like to thank Jou-Hsuan Wu and Ding Ma for helping with
13
the case studies, Michael Mehaffy and the editor Brian Lees, and the three anonymous reviewers for
their constructive comments.
References:
Alexander C. (1993), A Foreshadowing of 21st Century Art: The color and geometry of very early
Turkish carpets, Oxford University Press: New York.
Alexander C. (2002-2005), The Nature of Order: An essay on the art of building and the nature of the
universe, Center for Environmental Structure: Berkeley, CA.
Alexander C. (2003), New Concepts in Complexity Theory: Arising from studies in the field of
architecture, http://www.natureoforder.com/library/scientific-introduction.pdf
Alexander C. (2005), Harmony-seeking computations: a science of non-classical dynamics based on
the progressive evolution of the larger whole, Presented at The International Workshop on Grant
Challenge in Non-Classical Computation, 18-19 April, 2005.
Alexander C., Ishikawa S., Silverstein M., Jacobson M., Fiksdahl-King I., and Angel S. (1977), A
Pattern Language: Towns, buildings and construction, Oxford University Press: New York.
Bohm D. (1980), Wholeness and the Implicate Order, Routledge: London and New York.
Bonacich P. (1987), Power and centrality: a family of measures, American Journal of Sociology, 92,
1170-1182.
Brockman J. (editor, 1996), The Third Culture, Simon and Schuster: New York.
Casti J. and Karlqvist A. (editors, 2003), Art and Complexity, Elsevier: Amsterdam
Clauset A., Shalizi C. R., and Newman M. E. J. (2009), Power-law distributions in empirical data,
SIAM Review, 51, 661-703.
Duchowski A. (2007), Eye Tracking Methodology, Springer: Berlin.
Hillier B. and Hanson J. (1984), The Social Logic of Space, Cambridge University Press: Cambridge.
Jiang B. (2013a), Head/tail breaks: A new classification scheme for data with a heavy-tailed
distribution, The Professional Geographer, 65(3), 482 – 494.
Jiang B. (2013b), The image of the city out of the underlying scaling of city artifacts or locations,
Annals of the Association of American Geographers, 103(6), 1552-1566.
Jiang B. (2015a), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77.
Jiang B. (2015b), Geospatial analysis requires a different way of thinking: The problem of spatial
heterogeneity, GeoJournal, 80(1), 1-13.
Jiang B. (2015c), The fractal nature of maps and mapping, International Journal of Geographical
Information Science, 29(1), 159-174.
Jiang B. and Claramunt C. (2004), Topological analysis of urban street networks, Environment and
Planning B: Planning and Design, 31(1), 151-162.
Jiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based
social media, The Professional Geographer, xx(xx), xx-xx, DOI:
10.1080/00330124.2014.968886
Jiang B. and Sui D. (2014), A new kind of beauty out of the underlying scaling of geographic space,
The Professional Geographer, 66(4), 676–686.
Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic
features, Annals of the Association of American Geographers, 104(3), 530–541.
Jiang B., Zhao S., and Yin J. (2008), Self-organized natural roads for predicting traffic flow: a
sensitivity study, Journal of Statistical Mechanics: Theory and Experiment, July, P07008.
Katz L. (1953), A new status index derived from sociometric analysis, Psychometrika, 18(1), 39-43.
Langville A. N. and Meyer C. D. (2006), Google's PageRank and Beyond: the science of search
engine rankings, Princeton University Press: Princeton, N.J.
Lee D. J., Dias E. and Scholten H. J. (2014, editors), Geodesign by Integrating Design and Geospatial
Sciences, Springer: Berlin.
Lima M. (2011), Visual Complexity: Mapping Patterns and Information, Princeton Architectural Press:
New York.
Mandelbrot B. (1982), The Fractal Geometry of Nature, W. H. Freeman and Co.: New York.
Mandelbrot B. B. (1989), Fractals and an art for the sake of science, Leonardo, 2, 21-24.
Mayer-Schonberger V. and Cukier K. (2013), Big Data: A Revolution That Will Transform How We
14
Live, Work, and Think, Eamon Dolan/Houghton Mifflin Harcourt: New York.
Mehaffy M. W. And Salingaros N. A. (2015), Design for a Living Planet: Settlement, science, and the
human future, Sustasis Press: Portland, Oregon.
Page L. and Brin S. (1998), The anatomy of a large-scale hypertextual Web search engine,
Proceedings of the Seventh International Conference on World Wide Web, 107–117.
Pertgen H.-O. and Richter P. H. (1987), The Beauty of Fractals: Images of complex dynamical
systems, Springer-Verlag: Berlin.
Peuquet D. J. (2002), Representations of Space and Time, The Guilford Press: New York.
Rubin E. (1921), Visuell Wahrgenommene Figuren: Studien in psychologischer Analyse [Visually
Perceived Figures: Studies in phychological analysis], Gyldendalske boghandel: Copenhagen.
Salingaros N. (1995), The laws of architecture from a physicist’s perspective, Physics Essays, 8,
638—643.
Salingaros N. (1997), Life and complexity in architecture from a thermodynamic analogy, Physics
Essays, 10(1), 165-173.
Salingaros N. (1998), A scientific basis for creating architectural forms, Journal of Architectural and
Planning Research, 15, 283-293.
Salingaros N. A. (2012), Fractal art and architecture reduce physiological stress, Journal of
Biourbanism, 2, 11-28
Snow C. P. (1961), The Two Cultures and the Scientific Revolution, Cambridge University Press: New
Yo rk .
Taylor R. P. (2003), Fractal expressionism - where art meets science, In: Casti J. and Karlqvist A.
(editors, 2003), Art and Complexity, Elsevier: Amsterdam, 117 - 144.
Taylor R. P. (2006), Reduction of physiological stress using fractal art and architecture, Leonardo, 39,
245-250.
Thompson D. W. (1917), On Growth and Form, Cambridge University Press: Cambridge.
Ulrich R. S. (1984), View through a window may influence recovery from surgery, Science,
224,420-422.
Wu J. (2015), Examining the New Kind of Beauty Using Human Beings as a Measuring Instrument,
Master thesis at the University of Gävle.
Yarbus A. L. (1967), Eye Movements and Vision, Plenum Press: New York.
... To this point, we have computed the degree of order for both the living structure and its substructures. In addition to the score 18, the degree of order for the living structure can also be characterized by the ht-index (Jiang, 2015b;Jiang & Yin, 2014). That is, to take the order scores in t4 and apply the head/tail breaks 2.0 (with the 40% on average as the threshold) to them, the resulting htindex is three. ...
... Under the organismic worldview, we put forward the new geographic representation to compute the degree of order inherent in geographic space. Compared to existing geographic representations (e. g., Goodchild et al., 2007;Jiang, 2015b;Yuan, 2001), in which spaces are externally rather than internally connected, the new representation that is built on the internal connection of recursively defined substructures shows several advantages. First, space bears an inherent hierarchy or spatial heterogeneity, so it can be hierarchically organized. ...
Article
Full-text available
As Christopher Alexander discovered, all space or matter – either organic or inorganic – has some degree of order in it according to its structure and arrangement. The order refers to a kind of structural character, called living structure, which is defined as a mathematical structure that consists of numerous substructures with an inherent hierarchy. Across the hierarchy, there are far more small substructures than large ones, while on each level of the hierarchy the substructures are more or less similar in size. In this paper we develop a new approach to representing geographic space as a hierarchy of recursively defined subspaces for computing the degree of order. A geographic space is first represented as a hierarchy of recursively defined subspaces, and all the subspaces are then topologically represented as a network for computing the degree of order of the geographic space, as well as that of its subspaces. Unlike conventional geographic representations, which are mechanical in nature, this new geographic representation is organic, conceived, and developed under the third view of space; that is, space is neither lifeless nor neutral, but a living structure capable of being more living or less living. Thus, the order can also be referred to as life, beauty, coherence, or harmony. We applied the new representation to three urban environments, 253 patterns, and 35 black-white strips to verify it and to demonstrate advantages of the new approach and the new kind of order. We further discuss the implications of the approach and the order on geographic information science and sustainable urban planning.
... In other words, the number of low values is far greater than the number of high values. In previous studies, pixel values of nighttime images, the density of street nodes, and the density of social media location data have been confirmed as datasets that follow a heavy-tailed distribution [37,38]. ...
Article
Full-text available
Understanding and quantifying urban expansion is critical to urban management and urban planning. The accurate delineation of built-up areas (BUAs) is the foundation for quantifying urban expansion. To quantify urban expansion simply and efficiently, we proposed a method for delineating BUAs using geographic data, taking Guangzhou as the study area. First, Guangzhou’s natural cities (NCs) in 2014 and 2020 were derived from the point of interest (POI) data. Second, multiple grid maps were combined with NCs to delineate BUAs. Third, the optimal grid map for delineating BUA was determined based on the real BUA data and applying accuracy evaluation indexes. Finally, by comparing the 2014 and 2020 BUAs delineated by the optimal grid maps, we quantified the urban expansion occurring in Guangzhou. The results demonstrated the following. (1) The accuracy score of the BUAs delineated by the 200 m × 200 m grid map reaches a maximum. (2) The BUAs in the central urban area of Guangzhou had a smaller area of expansion, while the northern and southern areas of Guangzhou experienced considerable urban expansion. (3) The BUA expansion was smaller in all spatial orientations in the developed district, while the BUA expansion was larger in all spatial orientations in the developing district. This study provides a new method for delineating BUAs and a new perspective for mapping the spatial distribution of urban BUAs, which helps to better understand and quantify urban expansion.
... This shared notion of goodness or the quality without a name (Alexander 1979) should become the hard and universal standard of sustainable urban planning. Fortunately, a mathematical model of living structure or wholeness (Jiang 2015a(Jiang , 2019 has been developed to address not only why a structure is beautiful, but also how beautiful the structure is. The mathematical model relies on two factors to determine the degree of livingness or goodness: (1) the number of substructures, and (2) their inherent hierarchy. ...
Presentation
Full-text available
A commentary reflects on Christopher Alexander's (1936-2022) pursuit of living structure in cities, published online in Buildings and Cities: https://www.buildingsandcities.org/insights/commentaries/christopher-alexander.html
... Many heavy-tailed distributions have been found in previous studies, such as the density of street nodes, the density of social media location data, and the area of street blocks (Jiang 2015). For example, in a previous study, we found that the density of POIs also follows the heavy-tailed distribution, which can be used to derive NCs, and the NCs derived from POIs are used to study urban expansion (Yang et al. 2019). ...
Article
Full-text available
Studying the structure of polycentric cities can promote a better understanding of urban development and contribute to urban planning. In this study, we identified polycentric cities in China and evaluated the urban centre development level of polycentric cities from new data and method. We used Luojia-1A night-time light (NTL) data, combined with the concept of natural cities (NCs), to identify urban centres and thus identify polycentric cities in China. In addition, we used the urban centre development index (UCDI) to quantify the urban centre development level (UCDL) that represents the overall urban centre development level within a polycentric city. The polycentric cities in China are characterized by the spatial distribution pattern of a larger number in the east and fewer in the west. There are a large number of polycentric cities in eastern China, and the closer to the coastal areas, the more polycentric cities there are. The distribution of UCDL in China’s polycentric cities is characterized by significant spatial heterogeneity. UCDLs are generally smaller in polycentric cities in western China. In addition, polycentric cities in northeastern China have smaller UCDL. Polycentric cities with high UCDL are concentrated in the central and coastal regions of China.
... Under the organismic worldview, we put forward the new geographic representation to compute the degree of order inherent in geographic space. Compared to existing geographic representations (e.g., Yuan 2001, Goodchild et al. 2007, Jiang 2015b, in which spaces are externally rather than internally connected, the new representation that is built on the internal connection of recursively defined substructures shows several advantages. First, space bears an inherent hierarchy or spatial heterogeneity, so it can be hierarchically organized. ...
Preprint
Full-text available
As Christopher Alexander discovered, all space or matter – either organic or inorganic – has some degree of order in it according to its structure and arrangement. The order refers to a kind of structural character, called living structure, which is defined as a mathematical structure that consists of numerous substructures with an inherent hierarchy. Across the hierarchy, there are far more small substructures than large ones, while on each level of the hierarchy the substructures are more or less similar in size. In this paper we develop a new approach to representing geographic space as a hierarchy of recursively defined subspaces for computing the degree of order. A geographic space is first represented as a hierarchy of recursively defined subspaces, and all the subspaces are then topologically represented as a network for computing the degree of order of the geographic space, as well as that of its subspaces. Unlike conventional geographic representations, which are mechanical in nature, this new geographic representation is organic, conceived, and developed under the third view of space; that is, space is neither lifeless nor neutral, but a living structure capable of being more living or less living. Thus, the order can also be referred to as life, beauty, coherence, or harmony. We applied the new representation to three urban environments, 253 patterns, and 35 black-white strips to verify it and to demonstrate advantages of the new approach and the new kind of order. We further discuss the implications of the approach and the order on geographic information science and sustainable urban planning.
Article
Christopher Alexander has been a leading pioneer of academic research on architectural and urban design since the early 1960s. He is also a practicing architect and builder with a passion for creating and restoring life and beauty to our physical environment. In this essay I review, evaluate, and reflect on some of his particularly fruitful, promising, or problematic ideas. I will put forth some ideas of my own for clarification, and to indicate avenues for future research. I argue that Alexander’s notion of patterns (a verbal medium for capturing and conveying design knowledge in a systematic, reusable form) is in need of conceptual development along lines I suggest, even though Alexander downplayed the significance of patterns as he moved on to other theoretical ideas (mainly about aesthetics) later in his career. While I go into some detail about selected parts of Alexander’s work, the intended readership of this essay is not restricted to specialists. I have made an effort to provide guidance and background information to readers not already familiar with Alexander’s comprehensive body of theory.
Article
Full-text available
要說「美在欣賞者眼中」意味著美在很大程度上是主觀的,因此美與否的答 案因人而異。雖然這種主觀性觀點是普遍持有的,但也有一種客觀性觀點試圖以 某種定量的方式衡量美。克里斯多夫.亞歷山大早就發現,美或連貫性與次對稱 性或子結構的數量高度相關,並表明,無論其信仰,文化和種族如何,不同人 甚至不同民族之間都有一個共同的關於美的概念—結構美。這種結構美的概念 直接產生於活力結構或整體性,這是一種物理和數學結構,存在於所有空間和物 質。基於活力結構的概念,本文開發了一種根據自動生成的子結構的數量(S) 及其固有的等級層次(H)來計算圖像結構美或活力(L)的方法。為了驗證這 種方法,我們進行了一系列案例研究,適用於八對圖像,包括達文西的《蒙娜麗 莎》和波洛克的《藍極》。我們發現,《藍極》在結構上比《蒙娜麗莎》更美麗, 傳統建築在結構上通常比現代主義建築更美麗。這一發現意味著事物或圖像的美 在很大程度上是一個客觀事實,而不是像傳統上設想的那樣,是某種觀點或個人 偏好。有關結構美的研究對許多學科都有深遠的影響,如圖像理解和電腦視覺、 建築和都市設計、人文和藝術,神經生理學和心理學。在這些學科中,如何理解 美是一個關注的焦點。
Chapter
The paper presents a theory for the architecture of the city. The aim of the theory, based on the imitation of cities that grow spontaneously, is to propose a method for designing a city. In doing so, we apply the concept of pattern derived from Alexander’s design method. Using the pattern method, we conceive the city as a tool to solve the problem of communication between its inhabitants. We then explain the basic center-area pattern as a practical way of solution. The hierarchical combination of these patterns results in the urban structure. The method is shown with an example of a city design made through a Netlogo computer platform. With the help of the computer, the road network is then designed and its characteristics are studied using graphical topological surveys that allow the definition of squares and urban axes, the main components of the urban space. Based on centrality measures, we establish the use of the land and the shape of urban blocks. The conclusions underline the need for a comprehensive approach to the conception of the city.KeywordsTown designComplex systemsZipf’s lawPattern language
Article
Full-text available
Sustainable urban design or planning is not a LEGO-like assembly of prefabricated elements, but an embryo-like growth with persistent differentiation and adaptation towards a coherent whole. The coherent whole has a striking character-called living structure-that consists of far more small substructures than large ones. To detect the living structure, this paper develops a new approach for uncovering the underlying living structure of urban environments. The approach takes an urban environment as a whole and recursively decomposes it into meaningful subwholes at different levels of hierarchy or scale ranging from the largest to the smallest. This approach helps us not only better understand an urban environment as a living structure, but also better plan or transform the urban environment to be living or more living, or equivalently to be beautiful or more beautiful. 关键词 城市设计/规划;结构美;空间句法;自然街道;活力;整体
Article
Map generalization is a process of hierarchically reorganizing features whereby the global shape of the original datasets can be transferred in different scales. We propose a stroke and centrality-based method to hierarchically extract the skeleton structures from buildings aiming to support generalization. Firstly, the strokes are generated from refined proximity graph network. Next, by regarding the strokes as dual graph, three centrality indices are calculated for each stroke whereby an integrated factor is created to measure the importance level of the strokes. Finally, the hierarchical skeleton structures are extracted based on the stroke importance levels through different selection ratios. By classifying the buildings into different categories, different generalization operators are selected considering their characteristics. The experimental results demonstrate that the extracted hierarchical skeleton structures can represent the global shape of the entire region. Through this support, the global and local patterns of the original buildings can be both preserved.
Article
Full-text available
The things surrounding us vary dramatically, which implies that there are far more small things than large ones, e.g., far more small cities than large ones in the world. This dramatic variation is often referred to as fractal or scaling. To better reveal the fractal or scaling structure, a new classification scheme, namely head/tail breaks, has been developed to recursively derive different classes or hierarchical levels. The head/tail breaks works as such: divide things into a few large ones in the head (those above the average) and many small ones (those below the average) in the tail, and recursively continue the dividing process for the large ones (or the head) until the notion of far more small things than large ones has been violated. This paper attempts to argue that head/tail breaks can be a powerful visualization tool for illustrating structure and dynamics of natural cities. Natural cities refer to naturally or objectively defined human settlements based on a meaningful cutoff averaged from a massive amount of units extracted from geographic information. To illustrate the effectiveness of head/tail breaks in visualization, I have developed several case studies applied to natural cities derived from the points of interest, social media location data, and time series nighttime images. I further elaborate on head/tail breaks related to fractals, beauty, and big data. Keywords: Big data, social media, nighttime images, natural cities, fractals, and ht-index
Article
Full-text available
A fractal can be simply understood as a set or pattern in which there are far more small things than large ones, e.g., far more small geographic features than large ones on the earth surface, or far more large-scale maps than small-scale maps for a geographic region. This paper attempts to argue and provide evidence for the fractal nature of maps and mapping. It is the underlying fractal structure of geographic features, either natural or human-made, that make reality mappable, large-scale maps generalizable, and cities imageable. The fractal nature is also what underlies the beauty of maps. After introducing some key fractal concepts such as recursion, self-similarity, scaling ratio, and scaling exponent, this paper demonstrates that fractal thought is rooted in long-standing map-making practices such as series maps subdivision, visual hierarchy, and Töpfer's radical law. Drawing on previous studies on head/tail breaks, mapping can be considered a ranking and head/tail breaking process; that is to recursively divide things, according to their geometric, topological and/or semantic properties, into the head and the tail for map generalization, statistical mapping, and cognitive mapping. Given the fractal nature of maps and mapping, cartography should be considered a perfect combination of science and art, and scaling must be formulated as a law of cartography, or that of geography in general. KEYWORDS: Scaling of geographic features, map generalization, statistical mapping, cognitive mapping, head/tail breaks
Article
Full-text available
This paper examines the former location-based social medium Brightkite, over its three-year life span, based on the concept of natural cities. The term 'natural cities' refers to spatially clustered geographic events, such as the agglomerated patches aggregated from individual social media users' locations. We applied the head/tail division rule to derive natural cities. More specifically, we generated a triangulated irregular network, made up of individual unique user locations, and then categorized small triangles (smaller than an average size) as natural cities for the United States (mainland) on a monthly basis. The concept of natural cities provides a powerful means to develop new insights into the evolution of real cities, because there are virtually no data available to track the history of a city across its entire life span and at very fine spatial and temporal scales. Therefore, natural cities can act as a good proxy of real cities, in the sense of understanding underlying interactions, at a global level, rather than of predicting cities, at an individual level. Apart from the data produced and the contributed methods, we established new insights into the structure and dynamics of natural cities, e.g., the idea that natural cities evolve in nonlinear manners at both spatial and temporal dimensions. Keywords: Big data, head/tail breaks, ht-index, power laws, fractal, and nonlinearity
Article
Full-text available
Geospatial analysis is very much dominated by a Gaussian way of thinking, which assumes that things in the world can be characterized by a well-defined mean, i.e., things are more or less similar in size. However, this assumption is not always valid. In fact, many things in the world lack a well-defined mean, and therefore there are far more small things than large ones. This paper attempts to argue that geospatial analysis requires a different way of thinking - a Paretian way of thinking that underlies skewed distribution such as power laws, Pareto and lognormal distributions. I review two properties of spatial dependence and spatial heterogeneity, and point out that the notion of spatial heterogeneity in current spatial statistics is only used to characterize local variance of spatial dependence. I subsequently argue for a broad perspective on spatial heterogeneity, and suggest it be formulated as a scaling law. I further discuss the implications of Paretian thinking and the scaling law for better understanding of geographic forms and processes, in particular while facing massive amounts of social media data. In the spirit of Paretian thinking, geospatial analysis should seek to simulate geographic events and phenomena from the bottom up rather than correlations as guided by Gaussian thinking. KEYWORDS: Big data, scaling of geographic space, head/tail breaks, power laws, heavy-tailed distributions
Book
In Europe, the emerging discipline of geodesign was earmarked by the first Geodesign Summit held in 2013 at the GeoFort, the Netherlands. Here researchers and practitioners from 28 different countries gathered to exchange ideas on how to merge the spatial sciences and design worlds. This book brings together experiences from this international group of spatial planners, architects, landscape designers, archaeologists, and geospatial scientists to explore the notion of ‘Geodesign thinking’, whereby spatial technologies (such as integrated 3D modelling, network analysis, visualization tools, and information dashboards) are used to answer ‘what if’ questions to design alternatives on aspects like urban visibility, flood risks, sustainability, economic development, heritage appreciation and public engagement. The book offers a single source of geodesign theory from a European perspective by first introducing the geodesign framework, then exploring various case studies on solving complex, dynamic, and multi-stakeholder design challenges. This book will appeal to practitioners and researchers alike who are eager to bring design analysis, intelligent planning, and consensus building to a whole new level.
Book
This title is the result of a one-week workshop sponsored by the Swedish research agency, FRN, on the interface between complexity and art. Among others, it includes discussions on whether "good" art is "complex" art, how artists see the term "complex", and what poets try to convey in word about complex behavior in nature.