Geoscience Reference
In-Depth Information
Apollonian circle packings arise by repeatedly filling the interstices between
mutually tangent circles with further tangent circles. The problems associated
with close packing and related topics form a broad field within mathematics;
clearly, these topics are spatial in nature.
7.3.1 Hexagonal hierarchies and close packing of the
plane: Overview
Gauss (1840) proved that the densest lattice packing of the plane is the one
based on the triangular lattice. In 1968 (and earlier), Fejes-Toth proved that
that same packing is not only the densest lattice packing of the plane but is
also the densest of all possible plane packings. If one thinks, then, of circular
buffers around lattice points as if they were bubble foam, the circles centered
on a square grid pattern expand and collide to form a grid of squares (Boys,
1902)—a raster. The circles centered on a triangular grid pattern expand and
collide to form a mesh of regular hexagons, as do the cells in a slice of a hon-
eycomb of bees (de Vries, 1906)—as proximity zones. The theoretical issues
surrounding tiling in the plane are complex; even deeper are those issues
involving packings in three-dimensional space. The reader interested in prob-
ing this topic further is referred to the References section at the end of this
chapter and at the end of the topic. The interpretation of the simple triangular
grid has range sufficient to fill this document and far more.
7.3.2 Classical urban hexagonal hierarchies
One classical interpretation of what dots on a lattice might represent is found
in the geometry of “central place theory” (Christaller, 1933, 1941; Lösch, 1954).
This idea takes the complex human process of urbanization and attempts to
look at it in an abstract theoretical form in order to uncover any principles
which might endure despite changes over time, situation, cultural tradition,
and all the various human elements that are truly the hallmarks of urbaniza-
tion. Simplicity helps to reveal form: Models are not precise representations of
reality. They do, however, offer a way to look at some structural elements of
complexity. Thus, dots on a triangular lattice become populated places (often,
villages). Circles, expanding into hexagons, are areas that are tributary to the
populated places. In the traditional formulation (described after Kolars and
Nystuen, 1974) one considers four basic postulates (not one of which is “real”
but each of which is simple):
• The backdrop of land supports a uniform population density.
• There is a maximum distance that residents can easily penetrate into
the tributary area.
• There is slow and steady population growth.
• Village residents who move, as a result of growth (or for other rea-
sons), attempt to remain in close contact with their previous location
(to maintain social or other networks).
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