  A city is a whole, as are all cities in a country. Within a whole, individual
cities possess different degrees of wholeness, defined by Christopher Alexander
as a life-giving order or simply a living structure. To characterize the
wholeness and in particular to advocate for wholeness as an effective design
principle, this paper develops a geographic representation that views cities as
a whole. This geographic representation is topology-oriented, so fundamentally
differs from existing geometry-based geographic representations. With the
topological representation, all cities are abstracted as individual points and
put into different hierarchical levels, according to their sizes and based on
head/tail breaks - a classification scheme and visualization tool for data with
a heavy tailed distribution. These points of different hierarchical levels are
respectively used to create Thiessen polygons. Based on polygon-polygon
relationships, we set up a complex network. In this network, small polygons
point to adjacent large polygons at the same hierarchical level and contained
polygons point to containing polygons across two consecutive hierarchical
levels. We computed the degrees of wholeness for individual cities, and
subsequently found that the degrees of wholeness possess both properties of
differentiation and adaptation. To demonstrate, we developed four case studies
of all China and UK natural cities, as well as Beijing and London natural
cities, using massive amounts of street nodes and Tweet locations. The
topological representation and the kind of topological analysis in general can
be applied to any design or pattern, such as carpets, Baroque architecture and
artifacts, and fractals in order to assess their beauty, echoing the
introductory quote from Christopher Alexander.
  Keywords: Wholeness, natural cities, head/tail breaks, complex networks,
scaling hierarchy, urban design
