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Figure 7.8 The large green hexagon contains the equivalent of four blue hexagons (1
full one plus six 1/2 blue hexagons). Similarly, the intermediate-sized blue hexa-
gon contains the equivalent of six red hexagons. QR codes lead to animations.
Source: Arlinghaus, S. L. and W. C. Arlinghaus. 2005. Spatial Synthesis: Volume I,
Centrality and Hierarchy. Book 1. Ann Arbor: Institute of Mathematical Geography.
books/Spatial%20Synthesis2/singlek4bluenewcropped600.gif.
of four blue hexagons and four-squared red hexagons. The next largest hexagon,
not shown in Figure 7.8, would contain four green hexagons, four-squared blue
hexagons, and four to the third power red hexagons. Powers of four, fractions
with four or its root, and the square root of four all come into play.
As one might imagine hexagonal pixels creating a different form of raster, so
too, one can develop in parallel various numeric observations, principles, and
theorems. As shown here, some are derivative of classical central place theory.
Others use more modern techniques involving fractal geometry to capture the
pattern and characterize it completely. One might view this latter process as
parallel to developing vector images—based on mathematical characteriza-
tion. The spatial idea is the same; the mathematical tools employed are not.
Rich theory on the topic is present in the References section—it may serve as
a guide to uncovering directions for exciting future research.
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