10,871 research outputs found
Wholeness as a Hierarchical Graph to Capture the Nature of Space
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.Comment: 14 pages, 7 figures, 2 table
Defining and Generating Axial Lines from Street Center Lines for better Understanding of Urban Morphologies
Axial lines are defined as the longest visibility lines for representing
individual linear spaces in urban environments. The least number of axial lines
that cover the free space of an urban environment or the space between
buildings constitute what is often called an axial map. This is a fundamental
tool in space syntax, a theory developed by Bill Hillier and his colleagues for
characterizing the underlying urban morphologies. For a long time, generating
axial lines with help of some graphic software has been a tedious manual
process that is criticized for being time consuming, subjective, or even
arbitrary. In this paper, we redefine axial lines as the least number of
individual straight line segments mutually intersected along natural streets
that are generated from street center lines using the Gestalt principle of good
continuity. Based on this new definition, we develop an automatic solution to
generating the newly defined axial lines from street center lines. We apply
this solution to six typical street networks (three from North America and
three from Europe), and generate a new set of axial lines for analyzing the
urban morphologies. Through a comparison study between the new axial lines and
the conventional or old axial lines, and between the new axial lines and
natural streets, we demonstrate with empirical evidence that the newly defined
axial lines are a better alternative in capturing the underlying urban
structure.
Keywords: Space syntax, street networks, topological analysis, traffic,
head/tail division ruleComment: 10 pages, 7 figures, and 2 tables, one figure added + minor revisio
R\'enyi Mutual Information for Free Scalar in Even Dimensions
We compute the R\'enyi mutual information of two disjoint spheres in free
massless scalar theory in even dimensions higher than two. The spherical twist
operator in a conformal field theory can be expanded into the sum of local
primary operators and their descendants. We analyze the primary operators in
the replicated scalar theory and find the ones of the fewest dimensions and
spins. We study the one-point function of these operators in the conical
geometry and obtain their expansion coefficients in the OPE of spherical twist
operators. We show that the R\'enyi mutual information can be expressed in
terms of the conformal partial waves. We compute explicitly the R\'enyi mutual
information up to order , where is the cross ratio and is the
spacetime dimension.Comment: 29 pages; More discussion on the partition function of primary
operators, the contribution from spin-1 operator has been correcte
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