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Abstract

The wholeness, conceived and developed by Christopher Alexander, is what exists to some degree or other in space and matter, and can be described by precise mathematical language. However, it remains somehow mysterious and elusive, and therefore hard to grasp. This paper develops a complex network perspective on the wholeness to better understand the nature of order or beauty for sustainable design. I bring together a set of complexity-science subjects such as complex networks, fractal geometry, and in particular underlying scaling hierarchy derived by head/tail breaks – a classification scheme and a visualization tool for data with a heavy-tailed distribution, in order to make Alexander’s profound thoughts more accessible to design practitioners and complexity-science researchers. Through several case studies (some of which Alexander studied), I demonstrate that the complex-network perspective helps reduce the mystery of wholeness and brings new insights to Alexander’s thoughts on the concept of wholeness or objective beauty that exists in fine and deep structure. The complex-network perspective enables us to see things in their wholeness, and to better understand how the kind of structural beauty emerges from local actions guided by the 15 fundamental properties, and by differentiation and adaptation processes. The wholeness goes beyond current complex network theory towards design or creation of living structures. KEYWORDS: Theory of centers, living geometry, Christopher Alexander, head/tail breaks, and structural beauty
1
A Complex-Network Perspective on Alexander’s Wholeness
Bin Jiang
Faculty of Engineering and Sustainable Development, Division of GIScience
University of Gävle, SE-801 76 Gävle, Sweden
Email: bin.jiang@hig.se
(Draft: February 2016, Revision: April, July 2016)
“Nature, of course, has its own geometry. But this is not Euclid’s or Descartes’ geometry.
Rather, this geometry follows the rules, constraints, and contingent conditions that are,
inevitably, encountered in the real world.”
Christopher Alexander et al. (2012)
Abstract
The wholeness, conceived and developed by Christopher Alexander, is what exists to some degree or
other in space and matter, and can be described by precise mathematical language. However, it
remains somehow mysterious and elusive, and therefore hard to grasp. This paper develops a complex
network perspective on the wholeness to better understand the nature of order or beauty for
sustainable design. I bring together a set of complexity-science subjects such as complex networks,
fractal geometry, and in particular underlying scaling hierarchy derived by head/tail breaks – a
classification scheme and a visualization tool for data with a heavy-tailed distribution, in order to
make Alexander’s profound thoughts more accessible to design practitioners and complexity-science
researchers. Through several case studies (some of which Alexander studied), I demonstrate that the
complex-network perspective helps reduce the mystery of wholeness and brings new insights to
Alexander’s thoughts on the concept of wholeness or objective beauty that exists in fine and deep
structure. The complex-network perspective enables us to see things in their wholeness, and to better
understand how the kind of structural beauty emerges from local actions guided by the 15
fundamental properties, and by differentiation and adaptation processes. The wholeness goes beyond
current complex network theory towards design or creation of living structures.
Keywords: Theory of centers, living geometry, Christopher Alexander, head/tail breaks, and beauty
1. Introduction
Nature, or the real world, is governed by immense orderliness. The order in nature is essentially the
same as that in what we build or make, and underlying order-creating processes of building or making
of architecture and design are no less important than those of physics and biology. This is probably the
single major statement made by Alexander (2002–2005) in his theory of centers, in which he
addressed the fundamental phenomenon of order, the processes of creating order, and even a new
cosmology – a new conception of how the physical universe is put together. In the theory of centers or
living geometry (Alexander et al. 2012), the wholeness captures the meaning of order and is defined
as a life-giving or living structure that appears to some degree in every part of space and matter; see
Section 3 for an introduction to wholeness and wholeness-related terms. As the building blocks of
wholeness, centers are identifiable coherent entities or sets that overlap and nest each other within a
larger whole. Unlike the previous conception of wholeness focusing on the gestalt of things (Köhler
1947), the wholeness of Alexander (2002-2005) is not just about cognition and psychology, but
something that exists in space and matter. Different from the wholeness in quantum physics mainly
for understanding (Bohm 1980), Alexander’s wholeness aims not only to understand the phenomenon
2
of order, but also to create order in the built world or art. The wholeness is defined as a recursive
structure. Based on this definition, Jiang (2015) developed a mathematical model of wholeness as a
hierarchical graph with indices for measuring degrees of life or beauty for both individual centers and
the whole. This model helps address not only why a design is beautiful, but also how much beauty it
has. However, this previous study had some fundamental issues on the notions of centers and
wholeness unaddressed. Specifically, what are the centers, how are they created, and how do they
work together to contribute to the life of wholeness? In addition, the wholeness remains somehow
mysterious, particularly within our current mechanistic worldview (Alexander 2002–2005). To
address these fundamental issues, this paper develops a complex-network perspective on the
wholeness.
A complex network is a graph consisting of numerous nodes and links, with unique structures that
differentiate it from its simple counterparts such as regular and random networks (Newman 2010).
Simple networks have a simple structure. In a regular network, all nodes have a uniform degree of
connectivity. In a random network, the degrees of connectivity only vary slightly from one node to
another. As a consequence, a random network hardly contains any clusters, not to say overlapping or
nested clusters. On the contrary, complex networks, such as small-world and scale-free networks
(Watts and Strogatz 1998, Barabási and Albert 1999), tend to contain many overlapping and nested
clusters that constitute a scaling hierarchy (Jiang and Ma 2015; see a working example in Section 2).
The scaling hierarchy is a distinguishing feature of complex networks or complex systems in general.
For example, a city is a complex system, and a set of cities is a complex system (Jacobs 1961,
Alexander 1965, Salingaros 1998, Jiang 2015c), both having scaling hierarchy seen in many other
biological, social, informational, and technological systems. This paper demonstrates that the
wholeness bears the same scaling hierarchy as complex networks or complex systems in general.
Relying on the complex network perspective, this paper aims to demonstrate that wholeness is not just
in cognition and psychology, but something that exists in space and matter. It also aims to show that
the concept of wholeness is important not just for understanding the phenomenon of order, but also
for creating order with a high degree of wholeness through two major structure-preserving
transformations: differentiation and adaptation. I argue that there are major differences between the
whole as a vague term and wholeness as a recursive structure. The wholeness comprises recursively
defined centers induced by itself, whereas the whole, as we commonly perceive, comprises
pre-existing parts. The mantra that the whole is more than the sum of its parts should be more truly
rephrased as the wholeness is more than the sum of its centers. This paper examines the notions of
wholeness and centers from the perspective of complexity science. It also discusses two types of
coherence respectively created by differentiation and adaptation processes, which are consistent with
the spatial properties of heterogeneity and dependence for understanding the nature of geographic
space.
The remainder of this paper is structured as follows. Section 2 briefly introduces complex networks
and scaling hierarchy using head/tail breaks – a classification scheme and a visualization tool for data
with a heavy-tailed distribution (Jiang 2013a, Jiang 2015a). Section 3 compares related concepts, such
as whole and parts versus wholeness and centers, and discusses the theory of centers using two
examples of a cow and an IKEA desk. Section 4 presents three case studies to show how wholeness
emerges from space and how it can be generated through the two major structure-preserving
transformations of differentiation and adaptation. Section 5 further discusses implications of the
complex-network perspective and wholeness. Finally, Section 6 draws a conclusion and points to
future work.
2. Complex networks and the underlying scaling hierarchy
Small-world and scale-free networks are two typical examples of complex networks, which
fundamentally differ from their regular and random counterparts. A small-world network is a middle
status between the regular and random networks, so it has some nice properties of its regular and
random
clusteri
n
length (
W
their de
g
nodes t
h
have th
e
highest
that inv
compon
Comple
x
2004).
A
sub-stru
scaling
h
we ado
p
comple
x
commu
n
far mor
e
commu
n
of sizes
organs
t
More g
e
human
b
as cohe
r
notion
o
harmon
y
3 for m
o
(N
o
relati
comm
u
The un
d
in gene
r
b
iologi
c
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x
through
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e
comple
x
counterpart
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n
g coefficie
n
W
atts and
S
g
ree of conn
e
h
an well-con
e
highest de
g
and the low
e
olves a lar
g
ents are tho
s
x
networks
A
cluster has
cture that a
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h
ierarchy o
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nerally, a c
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ody growi
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ent parts ar
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o
f adaptatio
n
y
or coheren
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re details) a
Figure 1: (C
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erlying scal
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9
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ctivity de
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). Scale-fre
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n
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tain many
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r
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undant an
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d
t
tice to refe
r
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n then. In
3
o
cal efficien
of the rand
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n
y nodes ha
v
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its.
c
omponents
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n
ities than l
a
r
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i
g
of 34 nod
e
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, 5, 3, 2, 2,
2
T
here are al
s
y
of size 15,
w
h
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t parts, but
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sembled fro
m
e
ed or an e
m
o
ntex
t
s as a
w
e
tween adja
c
a
les. Cluster
s
wholeness.
u
b network
b
o
nsisting of 3
4
2
8, 15, 10, 6,
5
i
es were dete
c
2
013a, Jiang
2
o
rtant prope
r
e
ty of natur
a
ones. This
s
d
overlappin
g
d
er (1965) n
i
r
to the scal
i
this respect
,
cy of regula
o
m network
s
a
re a specia
l
i
stribution, i
n
99). In othe
r
v
e the lowest
precisely
be
t
s. The nod
e
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m
e
links, so c
o
a
complex n
e
a
rge ones (J
i
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dely studie
d
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s and 78 li
n
2
, 1, 1, 1 an
d
s
o many nes
t
w
hich is furt
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m
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x
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t
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u
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7
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,
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ted by an alg
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015a).)
r
ty of compl
a
l and societ
a
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caling hier
a
g
links. Thi
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cely articul
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,
Alexande
r
r
networks
c
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l
type of co
m
n
dicating far
r
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y
degree, and
e
cause of th
e
e
s are the b
a
m
munities
o
nstitutes a c
o
e
twork nest
e
i
ang and M
a
d
in the lit
e
n
ks, can be
b
1 (Figure 1)
t
ed relations
h
h
er
b
roken d
o
sters, which
the contex
t
a
l parts, but
t
he design p
x
ander 1964
)
t
s and syste
m
h
e concept o
f
u
tually neste
d
8 links. Pan
e
1, 1, 1 and 1
o
rithm inspir
e
e
x networks
a
l complex s
y
a
rchy is not
s
insight wa
s
a
ted in the c
l
y
simply be
c
was far ah
e
c
haracterize
d
b
y short ave
r
m
plex netw
o
r
more less-c
y
few nodes
,
some in be
t
e
ir complex
asic units,
w
o
r clusters (
N
o
herent sub
-
e
ach othe
r
,
f
a
2015). To
i
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rature of s
o
b
roken dow
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)
. Apparently
h
ips. For exa
m
o
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are similar
t
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of the hu
m
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oint of vie
w
)
. This intro
d
m
s, and hel
p
f
centers (se
e
d
communitie
s
e
l b depicts n
e
with far mor
e
e
d by head/tai
l
s
or comple
x
y
stems such
tree-like bu
t
s
originally
l
assic A Cit
y
cause the c
o
e
ad of the ti
m
d
by high
r
age path
o
rks, and
onnected
,
or hubs,
ween the
structure
w
hile the
N
ewman
-
whole or
f
orming a
i
llustrate,
o
cial and
n
into 14
there are
m
ple, the
m
munities
t
o human
m
an body.
a tree or
w
, clusters
d
uces the
p
s create
e
Section
s
e
sted
e
small
l
breaks
x
systems
as social,
t
rather a
observed
y
is Not a
o
ncept of
m
e when
4
complex networks were understood. More importantly, his thoughts are not just limited to
understanding surrounding things, just as what current complex network theory does, but aim for
creating things with living structure (i.e., buildings, communities, cities or artifacts). The scaling
hierarchy can be further seen from the relevance or importance of individual nodes within a complex
network. Through Google’s PageRank (Page and Brin 1998), nodes’ relevance or importance can be
computed, as there are far more less-important nodes than more-important ones. PageRank is
recursively defined and resembles the wholeness as a recursive structure. This will be further
discussed in the following section.
3. The wholeness and the theory of centers
The general idea of wholeness, or seeing things holistically, can be traced back to the Chinese
philosopher Chuang Tzu (360 BC), who saw the structure of a cow as a complex whole, in which
some parts were more connected (or coherent) than others. The butcher who understands the cow’s
structure always cuts the meat from the soft spots and the crevices of the meat, so makes the meat fall
apart according to its own structure. The butcher therefore can keep his knife sharp for a hundred
years. In the 20th century, wholeness was extensively discussed by many writers prominent in Gestalt
psychology (Köhler 1947), quantum physics (Bohm 1980), and many other sciences such as biology,
neurophysiology, medicine, cosmology and ecology. However, none of these writers prior to
Alexander (2002–2005) showed how to represent or formulate wholeness in precise mathematical
language. According to the Merriam-Webster dictionary, whole is something complete, without any
missing parts. On the other hand, wholeness is the quality of something considered as a whole.
However, both whole and wholeness have deeper meanings and implications in the theory of centers
(Alexander 2002–2005). In particular, the notion of wholeness is so subtle and profound that
Alexander (1979) previously referred to it as ‘the quality without a name’. He struggled with different
names, such as alive, comfortable, exact, egoless and eternal, but none of these captured the true
meaning of the quality. In this paper, as in Alexander (2002-2005), the three terms wholeness, life, and
beauty are interchangeably used when appropriate to indicate order or coherence. Things with a high
degree of wholeness are called living structure.
Table 1: Comparison of whole and wholeness using two examples of a cow and an IKEA desk
Whole(noun+adjective)Wholeness(structure+measure)
onthesurfacedeeperbelow thesurface
consistsofparts(seebelowsubsection
)
consistsofcenters(seebelow subsection
)
easytosee,thewholeofacowisthecow
itself
hardtoseeasarecursivestructure,the
wholenessofacow
hardtosense,onethingismorewholethan
another
hardtosenseasthedegreeofcoherence
somehowliketemperature
easytosense,acowismorewholethana
desk
easytosense,acowhasahigherwholeness
thanadesk
awholeassembledfrompartslikeadesk adeskhas lowwholenessorlowcoherence
awholeunfoldedfromanembryolike acow acowhas highwholenessorhighcoherence
PartsCenters
preexistinginthewholeinducedbythewholeness
onthesurfacedeeperbelowthesurface
easytoseehardtosee
mechanicalorganic
nonrecursiverecursive
simplecomplex
Wholeness is defined as a recursive life-giving structure that exists in space and matter, and it can be
described by precise mathematical language (Alexander 2002-2005). Although whole and wholeness
seem different, they are closely related, and sometimes refer to the same thing. The whole is on the
surface and is referred to informally, while wholeness is below the surface and is referred to formally
(Table 1). The term whole can be both a noun and an adjective. For example, a cow is a whole, and a
cow is
m
structur
e
spatial
s
cow its
e
atoms,
m
that on
e
assembl
higher
d
of whic
h
to mech
a
simple.
b
ecause
the part
s
and the
o
(No
t
To furt
h
whole,
o
the left
wholen
e
objects
o
picture
o
prefer,
w
pick on
e
our fee
l
exists
b
phenom
centers,
than th
e
experi
m
cannot
p
wholen
e
wholen
e
m
ore whole
t
e
, and as a
m
s
e
t
and easy
e
lf. The wh
o
m
olecules, c
e
e
thing has
a
ed from par
t
d
egree of wh
h
the whole
anical piece
s
Centers are
they exist
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s
and center
s
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rganic, wh
i
Fig
u
t
e: The left is
Levelso
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o
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e
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m
ent is som
e
p
rovide a p
r
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t
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m
easure for
to see,
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eca
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lls, tissues,
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g
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e
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and wholen
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eeper
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show two
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u
re 2: Two ca
r
more whole t
h
Table 2
:
f
scale
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nters
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ndaries
ngrepetition
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pace
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on the wh
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er degree o
f
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igher degre
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aptured thr
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deepest or
l
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s
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n
r
ed among
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lly in the
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o
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m
and local s
y
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ore of th
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e
how like r
e
r
ecise meas
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mplex thin
g
2
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O
n the othe
r
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u
se it appea
r
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t
and organs
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g
ree of who
l
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ss whole th
n
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ss respecti
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s
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t
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y the
w
w
the surface,
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ifferent wo
Alexander’
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r
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h
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p
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o
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e
, or is mor
e
o
ugh the mi
r
i
de-by-side i
truest self a
s
s
yncratic, ac
c
n
ne
r
self. T
h
p
eople and
w
orld and
p
m
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e
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r
-of-
t
e
lying on h
u
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t
of
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r
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w
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r
s on the su
r
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f
orming a sc
l
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s
h
an things gr
o
can also co
m
v
ely consist.
a
use they ap
w
holeness a
n
,
and are rec
u
o
rld views: t
h
s
radical tho
u
ff
erent degree
or the left ha
s
d
amental pr
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e
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etries
r
lockandam
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u
s examine
w
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e
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t
, and
s
a whole.
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counting fo
r
h
is part of h
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a
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t
he-self qua
u
man sense
t
emperature
.
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ow and a t
r
h
ierarchical
eness has t
w
coherence.
r
face. For e
x
ard to see,
s
aling hierar
c
s
more who
l
o
wn from e
m
m
pare the t
w
Parts usuall
y
pear on the
s
n
d refer to o
r
u
rsive and c
o
h
e mechanis
t
u
gh
t
.
of wholeness
s
a higher deg
o
perties of
w
Ro
u
Ech
o
b
iguity The
Sim
Not
w
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e
r
pets posses
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a
n the one o
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lf experim
e
you are ask
e
T
he experim
e
r
10 percent
o
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e
u
res (Alexan
a
lly inside
t
t
al properti
e
The left car
p
li
t
ies than
t
to compare
.
Equally, it
r
ee. In this r
e
g
raph for
r
w
o differen
t
m
A whole is
ample, the
w
s
ince it lies
c
h
y
. It is usu
a
e than anot
h
m
bryos. For
e
w
o related ter
m
y
pre-exist i
n
s
urface and
a
r
ganic piece
s
o
mplex. Th
e
t
ic, to whic
h
(Alexander 1
r
ee of whole
n
holeness
ghness
o
es
void
p
licityandin
n
separatenes
s
e
two carpe
t
s
a high deg
r
n
the right.
T
e
n
t
(Alexan
d
e
d to choose
e
n
t
is not to
o
f our feelin
g
e
nce, accoun
t
d
er 2002–2
0
t
he human
e
s such as l
e
p
e
t
has mor
e
t
he right. T
h
two tempe
r
is usually
h
e
gard, the m
a
e
presenting
m
eanings: a
r
a relatively
w
hole of a c
o
deeper con
s
u
ally difficul
t
h
e
r
. Howev
e
e
xample, a
c
r
ms parts an
d
n
the whole
a
re non-recu
r
s
that are h
a
e
differences
h
we are acc
u
993)
n
ess than the
r
n
ercalm
s
t
s (Figure 2
)
r
ee of whol
e
This pheno
m
d
er 2002-20
0
one that re
p
choose one
gs. Instead,
y
t
ing for 90
p
0
05). The
w
self. Physi
c
e
vels of sca
l
e
of the 15
p
he mirro
r
-o
f
r
atures. Ou
r
h
ard to co
m
athematical
the whole
n
r
ecursive
coherent
o
w is the
s
isting of
t
to sense
e
r, things
c
ow has a
d
centers,
and refer
r
sive and
a
rd to see
between
u
stomed,
ight.)
)
is more
e
ness, but
m
enon of
0
5). Two
p
resents a
tha
t
you
y
ou must
p
ercent of
w
holeness
c
ally, the
e, strong
p
roperties
f
-the-self
feelings
m
pare the
m
odel of
n
ess, can
accurat
e
perspec
t
4. The
w
This se
c
of its c
e
paper w
i
case stu
whole
o
structur
e
4.1 A p
a
The wh
o
present
e
the sam
e
placed
o
is not
h
center o
remains
least 20
corners,
only th
e
wholen
e
also su
p
(Figure
as indic
a
(Note:
T
p
aper
rectang
l
the b
o
going
r
diagon
a
The pa
p
paper c
o
e
ly measure
t
ive on whol
e
w
holeness f
r
c
tion present
e
nters, and h
o
ith a tiny do
t
dies, we de
m
o
r wholenes
s
e
-preserving
a
per with a
t
o
leness is v
e
e
d a simple
e
e
example t
o
o
n the pape
r
.
h
ard to imag
f the pape
r
.
without m
u
latent cent
e
and differe
n
e
paper and
e
ss. Seen in
p
por
t
ing rel
a
4a). The w
h
a
ted by the
d
Figure 3: (
C
T
he dot in the
s
itself, (2) the
l
e, (7) the top
o
ttom right co
r
r
ight, (16) th
e
a
l ray to the b
o
p
er with the
d
o
nstitutes a
the degree
e
ness.
r
om a comp
l
s three case
o
w centers
a
t
, followed
b
m
onstrate th
a
s
can be cr
e
transformat
i
t
iny dot
e
ry subtle, b
u
e
xample inv
o
o
examine h
o
A blank pa
p
ine because
However, a
f
u
ch change
o
e
rs, such as
n
t rays fro
m
dot pre-exi
s
Figure 3, t
h
a
tionships a
m
h
ole or the o
v
d
ot sizes.
C
olor online)
A
s
heet of pape
r
dot, (3) the h
a
rectangle, (8
)
r
ner, (12) the
r
e
white cross
b
o
ttom left cor
n
d
ot is more
w
simple net
w
of wholene
s
l
ex-networ
k
studies to s
h
a
re created
b
b
y studies of
a
t wholenes
s
e
ated step
b
i
ons.
u
t also very
o
lving a bla
n
o
w the who
l
p
e
r
is a whol
it is pretty
f
ter a tiny d
o
o
n the surfa
c
a halo aro
u
m
the dot to
t
st
. All othe
r
h
ese centers
a
m
ong the c
e
v
erall config
u
A
simple who
r
created at le
a
a
lo, (4) the b
o
)
the top left
c
r
ay going up,
b
y these four
r
n
er, (19) the
d
th
e
w
hole than t
h
w
ork contain
6
s
s or life.
T
k
perspectiv
e
h
ow how w
h
b
y wholenes
s
the Alhamb
r
s
exists in s
p
b
y step
b
y
f
concrete. T
o
n
k paper wit
h
l
eness emer
g
l
e, which is
e
simple, co
n
ot
(no more
t
c
e,
b
ut its
w
u
nd the dot,
t
he corners
(
r
s are induc
a
re real, no
t
e
nters, whic
h
g
uration of t
h
o
leness emerg
e
a
st 20 entities
o
ttom rectang
l
c
orner, (9) the
(13) the ray
g
r
ays, (17) the
d
iagonal ray t
o
e
top left cor
n
h
e blank pa
p
n
ing only fi
v
T
he model i
s
e
h
oleness is r
e
s
. We begin
r
a plan and
t
p
ace. More i
m
f
ollowing t
h
o
discuss w
h
h
a tiny dot
(
g
es and cha
n
e
asy to see.
T
n
sisting of f
o
t
han 0.0001
o
w
holeness c
h
the four re
c
(
Panels c-h
o
ed by the
o
t
just in cog
n
h
constitute
h
e space is t
h
e
d from a bla
n
, listed accor
d
l
e, (5) the lef
t
-
top right cor
n
g
oing down, (
diagonal ray
t
o
the top right
n
er.)
p
er itself. Th
v
e nodes (Fi
s
essentiall
y
e
presented a
w
ith the si
m
h
e Sierpins
k
m
portantly,
w
h
e 15 funda
m
h
oleness, Al
e
Panels a an
d
n
ges before
a
T
he wholen
e
o
ur corners
a
o
f the sheet
)
h
anges dram
a
c
tangles aro
u
o
f Figure 3)
.
o
verall confi
g
n
ition and p
s
a complex
h
e source of
n
k paper with
d
ing to their r
e
-
hand rectang
l
n
er, (10) the b
o
1
4) the ray g
o
t
o the bottom
corner, and (
2
e
wholeness
g
ure 4b). H
o
y
a complex
a
s a comple
x
m
ples
t
case s
t
k
i carpe
t
. Th
r
we elaborat
e
mental pro
p
e
xander (20
0
d
b of Figur
e
a
nd after a t
i
e
ss of the bl
a
and the gra
v
)
is placed, t
h
a
tically. Th
e
u
nd the dot,
.
Among th
e
guration of
sycholog
y
.
T
network as
the centers’
a dot place
d
e
lative streng
t
l
e, (6) the rig
h
ottom left co
r
o
ing left, (15)
right corner,
(
2
0) the diago
n
of the origi
n
H
owever, the
-network
network
t
udy of a
r
ough the
e
on how
p
erties or
0
2–2005)
e
3). I use
ny dot is
a
nk paper
v
itational
h
e whole
e
re are at
the four
e
centers,
space or
T
here are
a whole
strength,
t
h: (1) the
ht
-hand
r
ner, (11)
the ray
(
18) the
n
al ray to
n
al blank
network
represe
n
b
lank p
a
whole
o
hierarc
h
differen
t
4a, the
s
such as
diagona
l
Figure
4
(Note: T
h
with d
o
Compar
As a ru
relation
s
are furt
h
it is we
a
18, 19
a
is very
s
4.2 The
The Al
h
centers.
salient
p
Figure
5
space.
F
many d
i
sub-wh
o
asymm
e
The left
sub-wh
o
level ca
n
The wh
o
relation
s
eventua
l
The pla
n
n
ting the wh
a
pe
r
(Figur
e
o
f the paper
h
y, i.e., far m
t
communiti
e
s
trongest ce
n
centers 8,
9
l
rays. Thes
e
4: (Color onl
i
h
e first netwo
o
t sizes indic
a
r
ed to the co
m
u
le, weak ce
n
s
hip is cum
u
h
er enhance
d
a
ker than ce
n
a
nd 20 are th
s
teep, while
t
Alhambra
p
h
ambra plan
Thick boun
d
p
roperties. T
h
5
). There ar
e
F
or example
,
i
fferent leve
l
o
le, the mi
d
e
trical, cont
a
sub-whole
c
o
les compris
n
be further
o
le and its
n
s
hips, usuall
y
l
ly formed,
a
n
lacks glob
a
oleness of t
h
e
4a). First,
t
with the do
t
ore less-con
n
e
s, such as t
h
n
te
r
number
1
9
, 10, and 1
1
e
sub-whole
s
i
ne) The netw
o
rk consists o
f
a
ting their str
e
highe
r
m
plex netw
o
n
ters tend t
o
u
lative or ite
r
d
by some in
-
n
ter 1
b
ecaus
e weakes
t
,
b
t
he one in Fi
p
lan
possesses
m
d
aries, local
h
ere are at l
e
e
more othe
r
,
t
he whole
o
l
s. It consist
s
d
dle sub-w
h
a
ining furthe
r
c
omprises t
h
es three sub
b
roken do
w
n
umerous su
b
y
with small
a
nd i
t
is the
s
a
l symmetr
y
h
e paper wi
t
t
he whole o
f
t
con
t
ain th
e
n
ected node
s
h
e four corn
e
1
is followe
d
1
as corners
constitute s
o
o
rks of the w
h
and
(
f
the 20 nodes
,
ngth. It is ob
v
r
degree of
w
o
rk presente
d
o
support st
r
r
ative. This
i
-
links in an
i
e of the iter
a
b
ecause they
gure 4b is v
e
m
any of the
symmetries
,
e
ast 720 pos
i
centers cre
a
o
f the plan
c
s
of three su
b
h
ole, and t
h
r
three sub-
w
h
ree sub-wh
o
-
wholes: to
p
w
n into sub-
w
b
-wholes (o
r
surroundin
g
s
ource of lif
e
y
, but it is fu
l
7
t
h the dot is
f the blank
e
20 nodes.
S
s
than well-
c
ers, the four
d
by centers
2
; 12, 13, 1
4
o
me implici
t
h
oleness resp
e
(b) the blank
p
,
while the se
c
vious that t
h
w
holeness, th
a
d
in Section
2
t
rong ones,
a
i
s why cent
e
i
terative ma
n
a
tive nature
o
only contai
n
e
ry flat.
15 fundame
n
,
levels of s
c
i
tive spaces
o
a
ted
b
y the
w
c
onsists of
m
b
-wholes at
t
h
e right su
b
w
holes, so t
h
o
les: left, m
i
p
, middle an
d
w
holes or c
e
r
centers) c
o
g
centers poi
e
or beauty o
l
l of local s
y
far more c
o
p
aper conta
i
S
econd, the
c
onnected o
n
rays, and t
h
2
, 3, and so
o
4
, and 15 as
t
relationshi
p
e
ctively emer
g
p
ape
r
c
ond network
h
e paper wit
h
a
n the blank
2
, the netwo
r
a
s shown i
n
e
r 1 is the st
r
n
ner. Althou
g
o
f support r
e
n
ou
t
-links.
T
n
tal properti
c
ale, and str
o
o
r centers, a
n
w
holeness o
r
m
any nested
t
he first lev
e
b
-whole. E
a
h
ere are nin
e
i
ddle and ri
g
d
bottom. T
h
e
nters at the
o
nstitute mu
t
nting to big
f
the buildin
g
y
mmetries a
t
o
mplex or
w
i
ns only fiv
e
20 nodes fo
r
n
es. Third, th
e top three
n
o
n. There ar
e
rays; and 1
p
s within the
g
ed from (a) t
h
consists of o
n
h
the dot is
m
paper.)
r
ks for the
w
n
Figure 4.
T
r
ongest, wit
h
g
h center 2
h
lationships.
T
T
he hierarch
y
es that help
o
ng centers
a
n
d 880 relati
o
r
the overall
sub-wholes
l, which ca
n
a
ch of the
e
sub-whole
s
g
h
t
. Each of
h
e nine sub-
w
third or fou
r
t
ually reinfo
r
central ones
g
complex.
different le
v
w
hole than t
h
e
nodes, wh
e
r
m
a strikin
g
h
e 20 nodes
c
n
odes. Seen
i
e
several su
b
7, 18, 19,
a
complex ne
h
e paper wit
h
n
ly the four n
o
m
ore whole,
o
w
holeness ar
e
T
his kind o
f
h
nine in-lin
k
h
as also nin
e
The centers
y
shown in
F
identify m
a
a
re probably
onships (Jia
n
l
configurati
o
or centers
d
n
be named
a
three sub-
w
s
at the sec
o
the middle
a
w
holes at t
h
r
th level, re
c
rcing and s
u
. Scaling hi
e
v
els of scal
e
h
at of the
e
reas the
g
scaling
c
onstitute
i
n Figure
b
-wholes,
n
d 20 as
t
work.
h
the dot,
o
des, both
o
r has a
e
directed.
f
support
k
s, which
e
in-links,
of 16, 17,
F
igure 4a
a
ny latent
the most
n
g 2015b,
o
n of the
d
efined at
a
s the left
w
holes is
o
nd level.
a
nd right
h
e second
c
ursively.
u
pporting
e
rarchy is
e
, such as
the thre
symmet
r
Each o
f
Overall,
in an ite
the spa
c
recurrin
g
The pla
n
presenc
e
iterativ
e
differen
t
wide ra
n
and eac
h
underli
e
Figure
(Note:
aliv
e
adapte
d
contin
u
4.3 The
The Si
e
possess
e
assesse
d
meets a
to far m
o
At each
Section
1/9, an
d
dimensi
o
constitu
t
two kin
d
consecu
t
centers
o
also ne
s
relation
s
eight m
i
e sub-whol
e
r
ies are mai
n
f
the local
s
the plan ha
s
e
rative or rec
c
e. Eventual
g
in the pla
n
n
was likel
y
e
of the 15 f
u
e
ly applyin
g
t
iation and a
n
ge of scale
s
h
wholeness
e
s the structu
5: (Color onl
i
It is essenti
a
e
and beauti
f
d
to its surro
u
u
ous or itera
t
Sierpinski
c
e
rpinski car
p
e
s a high d
e
d
from two a
power-law
r
o
re small th
i
scale, all pi
e
5 for furthe
r
d
1/27. Eac
h
o
n of log(8)
t
e a comple
x
ds of relati
o
tive scales t
h
o
nes are en
h
s
ted relatio
n
s
hips, the ce
n
i
ddle square
s
e
s at the fir
s
n
ly created
s
ymmetries
a
s
a good sha
p
ursive man
n
ly the large
n
contribute
a
y
to be desi
g
u
ndamental
p
g
the 15
p
daptation in
s
from the s
m
develops, o
re-preservin
g
i
ne) The com
p
a
lly the dens
f
ul, forming
u
nding, and
t
t
ive differen
t
c
arpet
p
e
t
, althoug
h
e
gree of wh
o
spects: acro
s
r
elationship
w
i
ngs than lar
g
e
ces are the
s
r
discussion
o
h
of these sc
/log(1/3) =
1
x
network a
s
o
nships: tho
s
h
at are dire
c
h
anced by n
e
n
ships amon
g
n
tral square
s
and the 64
s
t level and
b
y numero
u
a
dapts its i
n
p
e, because
i
n
e
r
. Some of
number of
a
great deal t
o
g
ned unself
c
p
roperties. I
n
p
roperties,
n
a step-by-st
e
m
allest to t
h
r more trul
y
g
o
r
wholen
e
p
lex network
o
ity of the ce
n
so called liv
t
here are far
m
t
iation proc
e
than the rigi
d
h
being on
e
o
leness. Th
e
s
s all scales,
w
ith the lev
e
g
e ones (see
s
ame size,
w
o
f spatial de
p
ales respect
i
1
.89 (Mand
e
s
a whole (P
a
s
e among a
ted. The su
r
e
ighboring
c
g
the 73 in
d
has the hig
h
smallest sq
u
8
the nine s
u
u
s walls or t
h
n
dividual lo
c
i
t comprises
the shapes
e
local sym
m
o the life of
c
onsciously
n
this regard
namely tra
n
e
p fashion.
T
h
e largest. E
a
y
unfolds, fr
o
e
ss-extendin
g
of the 720 ce
n
n
ters, and th
e
v
ing structur
e
more small
c
e
sses. The co
d tree (Alex
a
e
of many
s
e
wholeness
and at one s
e
ls of scale
(
Section 5 f
o
w
hich should
p
endence).
T
i
vely contai
n
e
lbrot 1982)
.
anel b of Fi
g
same scale
r
rounding c
e
c
enters. In a
d
d
uced cent
e
h
est degree
o
u
ares. Even
t
u
b-wholes a
t
h
ick bound
a
c
al contex
t
o
many good
e
cho each ot
h
m
etries and
m
the individu
a
(Alexander
,
the plan co
u
n
sformation
T
he space co
n
a
ch step cre
a
o
m the previ
g
transform
a
n
ters and 880
e
ir complex
e
or structur
a
c
enters than
mplex netw
o
a
nder 1965).
s
trictly defi
n
of the car
p
cale. Across
(
Salingaros
o
r further dis
c
be extende
d
T
o be more s
p
n
s one, eig
h
.
The 73 sq
u
g
ure 6). Wit
h
that are un
d
e
nters usuall
d
dition to th
rs (Panel c
o
f wholenes
s
t
ually, the u
n
t
the secon
d
a
ries as one
o
r need to
m
shapes at di
f
h
er in the ov
e
m
any of th
e
a
l centers an
1964), but
n
u
ld be thou
g
properties,
n
tinues to b
e
a
tes the con
t
ous one. Th
i
a
tions (Alex
a
relationships
r
elationship
s
a
l beauty. Ev
e
l
arge ones,
w
o
rk as a who
)
n
ed fractals
e
t
or the c
o
all scales, t
h
1
999), whic
h
c
ussion on s
p
to more or
l
p
ecific, ther
e
ht
, and 64 s
q
u
ares of the
t
h
the compl
e
d
irected, an
d
y
support t
h
e
support re
l
of Figure
6
s
, which is
a
n
derlying sc
a
d
levels. Th
e
of the 15 p
r
m
ake a
b
ett
e
f
ferent level
s
e
r
all config
u
e
other 15
p
n
d the plan a
s
n
o one can
g
ht of as gen
e
to the
w
e
differentiat
e
t
ext for the
n
is generatio
n
a
nder 2002–
2
of the Alham
b
s
that make
t
ery center is
w
hich arises
o
le is no less
(Mandelbr
o
o
herence ca
n
h
e number o
h
should be
patial heter
o
l
ess similar
s
e
are three s
c
q
uares, with
three differ
e
e
x network,
d
those bet
w
h
e center on
e
lationships,
6
). Because
a
ccumulated
a
ling hierarc
h
e
se local
r
operties.
e
r space.
s
of scale
u
ration of
p
roperties
s
a whole.
deny the
e
rated by
w
hole by
e
d with a
n
ext one,
n
process
2
005).
b
ra plan
t
he plan
well
from the
ordered
o
t 1982),
n
also be
f squares
extended
o
geneity).
s
izes (see
c
ales: 1/3,
a fractal
e
nt scales
there are
w
een two
e
s, or the
there are
of these
from the
h
y of the
Sierpin
s
not a tr
e
Despite
for in p
r
major d
i
is
b
y s
t
Alhamb
r
Sierpin
s
and all
additio
n
increme
n
the two
local a
n
creation
divided
to fit to
15 pro
p
structur
e
(Note: (
a
represen
t
of the
Throug
h
concret
e
recursiv
simplici
t
further
d
living g
e
the Eart
h
5. Impl
i
The co
m
fragme
n
the kin
d
like tho
s
of whol
Jiang 2
0
living s
t
from o
n
most in
t
s
ki carpet is
s
e
e, but a co
m
its high de
g
r
actical desi
g
i
fferences b
e
t
ep-by-step
ra lacks gl
o
s
ki carpet ar
e
shapes a
r
e
n
, the Sierpi
n
n
t of the nu
m
patterns ar
e
n
d global sc
a
or unfoldin
g
and differe
n
their local
c
p
erties (Ale
x
e
-preserving
Figur
e
a
) the carpet
w
t
degrees of i
n
complex net
w
h
the case s
t
e
and visibl
e
ely defined
c
ty, we adop
t
d
iscusses th
e
e
ometry is p
o
h
’s surface
m
i
cations of t
h
m
plex-netwo
r
n
ts. Comple
x
d
of proble
m
s
e in biolog
y
eness, is th
e
0
13b). Seen
t
ructure that
n
e to anothe
r
t
eresting par
t
s
hown in a t
r
m
plex networ
k
g
ree of whol
e
g
ns. Instead,
e
tween the t
w
generation;
o
bal symme
t
e
the same s
i
squares in
t
n
ski carpet
m
ber of squ
a
e
the same
a
a
les. The w
h
g
: differenti
a
n
tiated to me
e
c
ontexts. De
s
x
ander 200
2
transformat
i
e
6: (Color on
l
w
ith three sca
l
n
-links, (c) ne
s
w
ork, in whic
h
t
udies, the
s
e
, helping
r
c
enters, and
t
ed these si
m
e
implicatio
n
o
werful and
m
ore whole
o
h
e complex
-
r
k perspecti
v
x
networks
p
m
a city is (J
y
. The notion
e
same as o
n
from the w
h
determines
t
r
, but there i
t
for urban
d
r
ee structure
k
.
e
ness,
t
he Si
we seek th
e
w
o patterns.
the Sierpin
s
t
ry, but wit
h
i
ze in one s
c
t
he Sierpins
k
is too rigid
a
res across t
w
a
t the funda
m
h
oleness of
t
a
tion and ad
a
e
t the scalin
g
s
ign or maki
n
2
–2005, Sal
i
ons toward
a
l
ine) A compl
e
l
es, (b) its co
m
s
ted relations
h
h
the dot size
d
eemingly a
b
r
educe the
m
the centers
m
ple cases to
n
s of the co
m
unique in te
r
o
r more
b
ea
u
-
network p
e
v
e enables
u
p
rovide a po
w
a
cobs 1961)
that a city i
s
n
e city
b
ein
g
h
oleness or t
h
t
he image o
f
s a shared i
m
esign theor
y
9
(Panel d of
F
i
erpinski car
p
e
kind of pa
t
The Sierpin
s
ki carpet
h
h
only local
c
ale, while t
h
ki carpet,
w
in terms o
f
w
o consecut
i
m
ental level
t
he Alhamb
r
a
ptation. In
t
g
hierarchy,
a
ng is much
m
l
ingaros 20
1
a whole wit
h
ex network p
e
m
plex networ
k
h
ips among t
h
represen
t
s th
e
d
egrees of lif
e
b
stract conc
e
m
ystery of
w
are induced
illustrate t
h
m
plex netwo
r
r
ms of plan
n
u
tiful.
e
rspective a
n
u
s to see thi
n
w
erful mea
n
)
. A city is
e
s
more whol
e
g
more ima
g
h
e theory o
f
f
the city. T
h
m
age of the
y
, which has
b
F
igure 6). It
p
et is not e
x
t
tern like th
e
ski carpet is
h
as local an
symmetries
h
ey are mor
e
w
hile the Al
f
the scalin
g
i
ve scales is
in terms o
f
r
a can only
b
t
he course o
a
nd newly a
d
m
ore challe
n
1
3, Salingar
o
h
a higher d
e
e
rspective on
k
of 73 pre-ex
h
e 73 created
c
e
strength of
t
e
.)
e
pts of who
l
w
holeness.
T
or created
b
e subtlety o
f
r
k perspecti
v
n
ing and rep
a
n
d wholene
s
n
gs in their
w
n
s to further
e
ssentially p
r
e
than anoth
e
g
inable or l
e
f
centers, it i
h
e image of
city for all
b
een persist
e
should be n
o
x
actly the ty
p
e
Alhambra
p
b
y creation
,
d
global sy
; all center
s
e
or less sim
i
h
ambra has
g
ratio at p
r
by exactly
e
f
wholeness
o
b
e obtained
f
design, a
w
d
ded centers
n
ging than u
n
o
s 2015) p
r
gree of who
l
t
he Sierpinsk
i
i
sting centers
,
c
enters, and (
d
h
e 73 pre-exi
s
l
eness and
c
T
he wholen
e
b
y the whole
n
f
the wholen
v
e to argue
w
a
iring our en
v
s
s
w
holeness, r
a
study whol
e
r
oblems of
o
e
r, or that a
c
e
gible than
a
s
the city it
s
t
he city for
i
p
eople. Thi
s
e
ntly criticiz
e
o
ted that the
p
e of patter
n
p
lan. There
,
while the
A
y
mmetries,
w
s
(or square
s
ilar in the
A
different s
h
r
ecisely 1/3,
e
ight times.
H
or coherenc
e
through ste
p
w
hole is con
t
s
or scales a
r
n
derstandin
g
r
ovide gui
d
l
eness.
i
carpe
t
, in which th
e
d
) the scaling
s
ting centers
o
c
enters beco
m
n
ess compri
s
e
ness. For th
e
n
ess. The ne
x
w
hy the wh
o
n
vironment o
a
ther than a
s
e
ness and u
n
o
rganized c
o
c
ity has a hi
g
a
nother (Ly
n
s
elf or the u
n
i
ndividuals
m
s
shared im
a
ed for lacki
n
whole is
n
we look
a
re some
A
lhambra
w
hile the
s
) of the
A
lhambra;
h
apes. In
and the
H
owever,
e
at both
p
-by-step
t
inuously
r
e created
g
, but the
d
ance for
dot sizes
hierarchy
o
r their
m
e more
s
es many
e
sake of
x
t section
leness or
r making
s
parts or
n
derstand
o
mplexity
g
h degree
n
ch 1960,
n
derlying
m
ay vary
a
ge is the
n
g a solid
10
and robust scientific underpinning (Jacobs 1961, Marshall 2012). The complex network perspective or
more truly the wholeness itself will inject scientific elements into urban planning and design.
The Sierpinski carpet is an image of the Earth’s surface metaphorically. This is because there are far
more small things than large ones across all scales – the spatial property of heterogeneity, and related
things are more or less similar in terms of magnitude at each scale – the spatial property of
dependence. Both heterogeneity and dependence are commonly referred to as spatial properties about
geographic space or the Earth’s surface (Anselin 1989, Goodchild 2004). The carpet is also an ideal
metaphoric image toward which our built environment should be made. It implies that any space
ought to be continuously differentiated to retain the scaling hierarchy across all scales ranging from
the smallest to the largest, and any building or city ought to be adapted to its natural and built
surroundings at each scale. The two processes of differentiation and adaptation enable us to create a
whole with a high degree of wholeness. In this regard, the wholeness or the theory of centers would
have enormous effects on geography, not only for better understanding geographic forms and
processes, but also for planning and repairing geographic space or the Earth’s surface (Mehaffy and
Salingaros 2015, Mehaffy 2007). Built environments must adapt to nature, and new buildings must
adapt to their surroundings.
The two spatial properties of heterogeneity and dependence constitute a true image of the Earth’s
surface at both global and local scales. Globally, spatial phenomena vary dramatically with far more
small things than large ones across all scales, while locally they tend to be dependent or
auto-correlated with more or less similar things nearby. These two properties are the source of the two
kinds of harmony or coherence respectively across all scales and at every scale. For example, there
are far more small cities than large ones across all scales globally (Zipf 1949), whereas nearby cities
tend to be more or less similar in terms of the central place theory (Christaller 1933, 1966). However,
the geography literature focuses too much on dependence, formulated as the first law of geography
(Tobler 1970), but very little on heterogeneity. More critically, spatial heterogeneity is mainly defined
for spatial regression with limited variation, governed by Gaussian thinking (Jiang 2015d). This
understanding of spatial heterogeneity is flawed, given the fractal nature of geographic space or
features. Spatial heterogeneity should be formulated as a scaling law because it is universal and
global.
The wholeness or the theory of centers in general brings a new perspective to the science of complex
networks. For example, the detection of communities of complex networks can benefit from the
wholeness as a recursive structure. Instead of a flat hierarchy of community structures, a complex
network contains numerous nested communities of different sizes, or far more small communities than
large ones (Tatti and Gionis 2013, Jiang and Ma 2015). This insight about nested communities can be
extended to classification and clustering. Current classification methods can be applied to mechanical
assembly, so mechanical parts can be obtained. For a complex entity such as complex networks, the
parts are often overlapping and nested inside each other. This resembles both wholeness and centers as
a recursive structure. From a dynamic point of view, a complex network is self-evolved with
differentiation and adaptation. A complex network has the power to evolve toward more coherence at
both global and local scales, and is self-organized from the bottom up, rather than imposed from the
top down. This insight about self-organization reinforces Alexander’s piecemeal design approach
through step-by-step unfolding or transformations for a living structure with a high degree of
wholeness.
6. Conclusion
This paper develops a complex network perspective on the wholeness to make it more accessible to
both designers and scientists. I discussed and compared related concepts such as whole and parts
versus wholeness and centers to make them explicitly clear. A whole is a relatively coherent spatial set,
while wholeness is a life-giving structure – not something about the way they are seen, but something
about the way they are (Alexander 2003, p. 14). The wholeness is made of centers rather than
11
arbitrarily identified parts. The centers are created or induced by the wholeness and made of other
centers, rather than just those pre-existing in the whole. Given these differences, the mantra that the
whole is more than the sum of its parts should be more correctly rephrased as the wholeness is more
than the sum of its centers. I demonstrated that the complex-network perspective enables us to see
things in their wholeness. More importantly, I elaborated on design or making living structure through
wholeness-extending transformation or unfolding in step-by-step differentiation and adaptation.
Although understanding the nature of wholeness is essential, the ultimate goal of Alexander's concept
of wholeness or living geometry is to allow us to make artifacts and built environments with the same
order and beauty of nature itself.
The complex-network perspective enables us to develop new insights into planning and repairing
geographic space. For example, differentiation and adaptation are two major processes for making
living structures or geographic space in particular. This study showed that these processes underlie the
two unique properties of geographic space: spatial heterogeneity and spatial dependence. These are
respectively formulated as the scaling law and the first law of geography. Globally across all scales,
there are far more small things than large ones. In contrast, locally at every scale, things tend to
depend on each other with more or less similar sizes. These two spatial properties are the source of the
two types of coherence respectively at global and local scales. In this connection, the theory of centers
or living geometry would significantly contribute to understanding and making geographic space. Our
future work points to this direction on how to rely on wholeness-extending transformations to create
geographic features with a high degree of wholeness.
Acknowledgement
This work is substantially inspired by the life’s work of Christopher Alexander, which aims to develop
a solid and robust scientific underpinning for architecture. Unfortunately, the mainstream architecture
profession has yet to grasp, or put to use, the more profound aspects of his thought. I would like to
thank Michael Mehaffy for his constructive comments and Maggie Moore Alexander for sharing some
of the unpublished manuscripts of Alexander. I would also like to thank the four anonymous referees
for their useful comments, and Zheng Ren for helping with some of the figures.
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... For the two squares shown in pair (c), the left one, with a tiny dot, is able to induce some 20 centers, whereas the right (empty square) one has only five centers. More importantly, there is a very steep hierarchy among the 20 centers, but a very flat hierarchy with the five centers ([10] Volume 1, p. 81-82, [32], p. 479-480). The snowflake (d) is differentiated from the triangle, so the former (or the left) has a higher degree of goodness than the latter (or the right). ...
Article
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Conceived and developed by Christopher Alexander through his life's work, The Nature of Order, wholeness is defined as a mathematical structure of physical space in our surroundings. Yet, there was no mathematics, as Alexander admitted then, that was powerful enough to capture his notion of wholeness. Recently, a mathematical model of wholeness, together with its topological representation, has been developed that is capable of addressing not only why a space is good, but also how much goodness the space has. This paper develops a structural perspective on goodness of space (both large-and small-scale) in order to bridge two basic concepts of space and place through the very concept of wholeness. The wholeness provides a de facto recursive definition of goodness of space from a holistic and organic point of view. A space is good, genuinely and objectively, if its adjacent spaces are good, the larger space to which it belongs is good, and what is contained in the space is also good. Eventually, goodness of space, or sustainability of space, is considered a matter of fact rather than of opinion under the new view of space: space is neither lifeless nor neutral, but a living structure capable of being more living or less living, or more sustainable or less sustainable. Under the new view of space, geography or architecture will become part of complexity science, not only for understanding complexity, but also for making and remaking complex or living structures. In a good system, we would expect to find the following conditions: Any identifiable subsystems, we would hope, would be well-that is to say, in good condition. And we would hope that the larger world outside the complex system is also in good order, and well. Thus, the mark of a good system would be that it helps both the systems around it and those which it contains. And the goodness and helping towards goodness is, in our ideal complex system, also reciprocal. That is, our good system, will turn out to be not only helping other systems to become good, but also, in turn, helped by the goodness of the larger systems around it and by the goodness of the smaller ones which it contains.
... All space and matter (either organic or inorganic) have some degree of life in it -"every brick, every stone, every person, every physical structure of any kind at all" (Alexander 2002-2005)according to its underlying structure and arrangement. The phenomenon that all space has some degree of life has been extensively studied in computer science, architecture and urban science (e.g., Gabriel 1998, Salingaros 2014, Jiang 2016, Mehaffy 2017, Gabriel and Quillien 2019. If the degree of life is too low, the structure is called a dead or nonliving structure; otherwise, it is called a living structure. ...
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It is a reprint by the magazine Coordinates from the previous journal article published by MDPI.
... All space and matter (either organic or inorganic) have some degree of life in it -"every brick, every stone, every person, every physical structure of any kind at all" (Alexander 2002(Alexander -2005) -according to its underlying structure and arrangement. The phenomenon that all space has some degree of life has been extensively studied in computer science, architecture and urban science (e.g., Gabriel 1998, Salingaros 2014, Jiang 2016, Mehaffy 2017, Gabriel and Quillien 2019. If the degree of life is too low, the structure is called a dead or nonliving structure; otherwise, it is called a living structure. ...
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The Earth's surface or any territory is a coherent whole or subwhole, in which the notion of "far more small things than large ones" recurs at different levels of scale ranging from the smallest of a couple of meters to the largest of the Earth's surface or that of the territory. The coherent whole has the underlying character called wholeness or living structure, which is a physical phenomenon pervasively existing in our environment and can be defined mathematically under the new third view of space conceived and advocated by Christopher Alexander: space is neither lifeless nor neutral, but a living structure capable of being more alive or less alive. This paper argues that both the map and the territory are a living structure, and that it is the inherent hierarchy of "far more smalls than larges" that constitutes the foundation of maps and mapping. It is the underlying living structure of geographic space or geographic features that makes maps or mapping possible, i.e., larges to be retained, while smalls to be omitted in a recursive manner (Note: larges and smalls should be understood broadly, in terms of not only sizes, but also topological connectivity and semantic meaning). Thus, map making is largely an objective undertaking governed by the underlying living structure, and maps portray the truth of the living structure. Based on the notion of living structure, a map can be considered to be an iterative system, which means that the map is the map of the map of the map, and so on endlessly. The word endlessly means continuous map scales between two discrete ones, just as there are endless real numbers between 1 and 2. The iterated map system implies that each of the subsequent small-scale maps is a subset of the single large-scale map, not a simple subset but with various constraints to make all geographic features topologically correct.
... This course is intended to give students a broader understanding on complexity and order in other sciences and how these concepts are applied to the study of the built environment. It includes modules on the systems view of life [9], complex networks, space and centrality [10], the new science of cities [11,12] and complexity in architectural theory [13]. ...
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The wholeness, conceived and developed by Christopher Alexander, is what exists to some degree or other in space and matter, and can be described by precise mathematical language. However, it remains somehow mysterious and elusive, and therefore hard to grasp. This paper develops a complex network perspective on the wholeness to better understand the nature of order or beauty for sustainable design. I bring together a set of complexity-science subjects such as complex networks, fractal geometry, and in particular underlying scaling hierarchy derived by head/tail breaks-a classification scheme and a visualization tool for data with a heavy-tailed distribution, in order to make Alexander's profound thoughts more accessible to design practitioners and complexity-science researchers. Through several case studies (some of which Alexander studied), I demonstrate that the complex-network perspective helps reduce the mystery of wholeness and brings new insights to Alexander's thoughts on the concept of wholeness or objective beauty that exists in fine and deep structure. The complex-network perspective enables us to see things in their wholeness, and to better understand how the kind of structural beauty emerges from local actions guided by the 15 fundamental properties, and by differentiation and adaptation processes. The wholeness goes beyond current complex network theory towards design or creation of living structures.
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It is a re-print of the original article by the magazine Coordinates
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