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7.3 Thinking outside the pixel
In an earlier section, we employed a classical model, from pen and ink and
paper cartography, to think about the parallels on the computer screen. Now,
we stretch that style of thinking to begin with a different classical model to
suggest possible futures: A model of classical central place theory reflecting
an abstract characterization of how cities might share space. The interpreta-
tion here is that square pixels form a square lattice on the computer screen.
One might imagine, instead, that a triangular lattice with hexagonal (or tri-
angular) pixels could take advantage of this tightest method of packing poly-
gons into two-dimensional space (Fejes-Toth, 1968). To consider this idea and
imagine it cast in a contemporary setting, we look at some of the elements of
the geometry of layers of hexagons—as tessellations that partition the plane.
To see how layers of hexagons of varying size might create different pattern
depending on orientation and relative cell size, consider the two different
images in Figure 7.3. Each is based on three different layers of hexagonal
cells: Large, medium, and small. The nesting patterns, while regular, are quite
different from each other.
To gain an appreciation for working in a hexagonal grid, try an interactive
online game based on such a grid. One on a hexagonal board with extra
complexity, involving adjacency as well as jumping, may be found at: http://
neave.com/hexxagon/ . To view some of the intricacies of circle packing,
consider the interactive graphic at: http://jsxgraph.uni-bayreuth.de/wiki/
index.php/Apollonian_circle_packing . These are Apollonian circle packings.
Figure 7.3 Two different overlay patterns of three hexagonal grids, with large (green),
medium (blue), and small (red) cell size. Source: Arlinghaus, S. L. and W. C.
Arlinghaus. 2005. Chapter 2, Spatial Synthesis: Volume I, Centrality and Hierarchy.
Book 1. Ann Arbor: Institute of Mathematical Geography. http://www-personal.
umich.edu/~copyrght/image/books/Spatial%20Synthesis2/chapter2.html.
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