Scaling Relative Asymmetry in Space Syntax Analysis
Scaling Relative Asymmetry in Space Syntax Analysis
Scaling Relative Asymmetry in Space Syntax Analysis
Mrio Krger
Professor of Architecture
University of Coimbra, Portugal
Faculty of Architecture
University of Porto, Portugal
Pages: 194-203
2012
http://www.journalofspacesyntax.org/
volume: 3
issue: 2
S
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J
O
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Mrio Krger
Professor of Architecture
University of Coimbra, Portugal
Faculty of Architecture
University of Porto, Portugal
This paper reports on a study of space syntax measures and focuses on the standard deviation of the
depth from an axial map. The first section of the paper is a partial review of the original study On node
and axial maps: Distance measures and related topics (Krger, 1989). The following sections present
new developments whereby a more robust statistical approach to work with integration is used, which not
Keywords:
Space syntax
measures, axial
map analysis, depth
standard deviation.
only considers the mean values given by Relative Asymmetry (RA), but also the corresponding standard
deviation. In other words, the proposition is to work not only with a measure of centrality (1/RA), but also
with a dispersion measure in order to obtain a more complete picture of the distribution of depth in an axial
map. The result of this study on space syntax measures takes into account the standard deviation of the
depth from an axial map, proposing a new measure of Scaled Relative Asymmetry of axial line i (SRAi),
which suggests powerful correlations with natural movement.
1. Introduction
At the social level, space affects human behaviour
characteristics.
visual barriers.
Asymmetry (RRA).
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Scaling relative
asymmetry
Krger, M. & Vieira, A.
lines that cover all urban public spaces, i.e. lines that
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GM (k,
is non isomorphic; i.e. while an axial
k) GM
(k,m)m)
map corresponds to just one graph, to the same
mmax =
k(k 1)
2
(1)
Figure 1:
An example of an (8, 6)
axial map and its correspondent (6, 8) graph.
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Krger, M. & Vieira, A.
or node i
RAi =
2(MDi 1)
(k 2)
(3)
2. dij = dji ,
dij
j =1 (k 1)
k
MDi =
(2)
tation.
Mean depth measures the extent to which a
given line i is segregated from the remaining lines of
a) RAmax
b) RAmin
Figure 2:
a) Chain having end point with maximum RA.
b) Star having centre with minimum RA.
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k 1
d = m,
j =1
ij
m =1
mmax =
k(k 1)
2
MDmax
k
= .
2
MDi MDmin
MDmax MDmin
RAi =
different sizes.
The usual approach is to compare RA values
for each point with RA values of a root of a diamond
(4)
Level
N Points
Depth
20
21
22
23
24
23
22
21
20
Figure 3:
D46 - Diamond
Shape with 46
points and 9 levels of depth.
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Krger, M. & Vieira, A.
d/2
point, called the root, is put at the base and then all
S2 =
S = ( d 1)(2(
1
2
mes
S2 =
d / 21
d /2
d / 21
q =0
q =0
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d / 21
d(2 )
q
q =0
TDk = ( d 2 )(2d / 2 ) +
But, in general, as
(5)
q(2
q=0
(6)
q=0
q=0
(d q)(2 )
d / 21
d(2. ) + q(2 ) .
TDk = S1 + S2 = q(2 q ) +
q=0
TDk = q(2 ) +
)+ 2
d +1)
2
d / 21
) + 2]
(n+1)
d/2
q=0
q=0
d / 21
d / 21
d(2 )
q
q=0
(a ax n+1 )
ax =
(1 x)
k =0
n
(7)
then,
the root and q the depth, also from the root, of the
TDk = ( 3 2 )(d2d / 2 ) d
(8)
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n, i.e. by
Dk =
k + 2
n = lg 2
(13).
(9)
2 [ k(n 1) +1]
(k 1)(k 2)
(14).
If we substitute the value of n, given by expression (14), in (13) we finally obtain the RA of a
n1
k = 2n + 2 2i
Dk =
i =0
2 [(k(lg 2 ( k+2
3 ) 1) +1)]
(k 1)(k 2)
(15).
and
the second one the number of points at all
other levels.
As, in general,
2
i =0
(1 2 )
(1 2)
n+1
then, after
k = 3 2 n 2
(10)
If we substitute (10) in (9) we obtain an expression for the root total depth as a function of the
number
of diamond points (k) as well as a function
of its diameter level (n), i.e. simply as
TDk = k n
(11)
MDk = (k n) /(k 1)
in axial maps.
(12)
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k (x i x )2
k x i 2
i =1
= i =1 + (k 2)( MD ) 2
k
k 1
k 1
(16)
of their nodes.
xi
i 1
; i.e. we need to
D
STDi = i TDk i MDk MDi (21)
Dk
Dk
ing expression,
d /2
d / 21
q =0
q =0
x i 2 = q2 (2q ) +
i=1
(d q)2 (2q )
where
(17)
n1
q=0
q=0
q2(2q ) + (d q)2(2q )
x
i=1
= (3 2 n 4n 8n + 3 2
n
n +2
12) (19).
(SMDi) as
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(18).
j =1
TDk = 2n(3.2 n 1 1) ;
MDk = (k n) /(k 1) ;
SRAi =
kn 2
3 2 n n 2 4n 2 8n + 3 2 n +2 12
(k 2)
k 1
k 1
SMDi SMDmin
SMDmax SMDmin
(22),
(20).
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SMDmin
(23),
mean depth and for the total depth from the root
of a diamond shape with k nodes.
k (i k ) 2
2
Dc = i =1 k 1
STDs
= SMDs =
k 1
D
STDs = s TDk s MDk MDs
Dk
Dk
SMDs =
1
(k 1)
SMDmax =
(24),
STDc
k 1
D
STDc = c TDk c MDk MDc
Dk
Dk
STDc
(k 1)
given
by equation (22).
SMDMax = SMDc =
(26).
k 3 2k
12(k 1)
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5. Conclusions
References
ra@arq.up.pt) is Assistant
Lecturer in CAAD at Faculty
of Architecture of University
of Porto (FAUP) and is graduated in mathematics and
in architecture at the same
university. She has a Master
in urban planning by the
University of Aveiro (2007)
and is a research student
at FAUP on how learning
spaces can improve learning activities. She is also a
junior researcher in CEAU
(www.ceau.arq.up.pt) and
participates in research
projects on E- Learning
Caf and Spatial Representation and Communication
Center - FAUP.
203
with natural urban movement will produce interesting results. These tests will be undertaken in a future
study of the axial comparative analyses of urban
maps representing Portuguese settlements, as well
as Architecture Faculty buildings in Portugal. In the
latter case, since two-dimensional plane axial maps
do not apply to multi-storey buildings, the diamond
shape should be mapped onto a sphere in order to
in an axial map.
p.347-357.