Geoscience Reference
In-Depth Information
Do you see why this latter theorem explains that the same number of colors
suffice for coloring maps on these two different surfaces? Consider any map
on the surface of a sphere. Poke a hole in the interior of any region on the
map on the spherical surface. Use that hole as the “North projection Pole”
from which to project stereographically the map on the spherical surface into
a plane tangent at the antipodal point to the North projection Pole. The map
in the plane can be colored using x colors. Use inverse stereographic projec-
tion to pull the map in the plane back to the spherical surface. The entire map
on the spherical surface, except the North projection Pole, is now colored.
Color the missing point on the spherical surface with the same color as the
region that surrounds it (hence the need for selecting an interior point as the
North projection Pole). Thus, however many colors were needed in the plane
is the same number as would be needed on the surface of a sphere. Does this
result surprise you? Do you see that the surface of a sphere is, as is the plane,
a two-dimensional surface? This result was known well in advance of proving
the actual number needed; yet another instance of precision before accuracy!
10.6 Putting it all together: Theory
The issues that we face as a society in the twenty-first century, such as popu-
lation change, sustainable agriculture, natural hazards, energy, water quality
and availability, climate change, crime, and more, vary widely in scale and
discipline. Yet they share several key characteristics: They are all complex
issues that are of global importance and yet increasingly affect our everyday
lives. They are all inherently geographic issues—they all have to do with
“where,” and more important, the “whys of where.” They all depend heavily
on mathematics—to represent, process, and manipulate data. They all rely
on the kinds of critical thinking, inquiry, and problem-solving skills that we
have emphasized in this topic. As such, gaining skills in spatial mathematics
will enable anyone to be a more effective decision maker. Given the issues
outlined above, the integration of mathematics and geography will become
even more important in the future.
We hope the reader has enjoyed seeing a small sample of what “spatial math-
ematics” has to offer. This topic has presented classical materials, such as the
material associated with Eratosthenes, it has presented contemporary materi-
als such as the population potential maps of Grayson, and it suggests direc-
tion into the future in the use of non-Euclidean geometry in perspective map
projection. The reader interested in learning about graph theory, network
analysis, topology, and more, is referred to a number of references or may
await future works in this series that make even more direct linkages between
spatial mathematics and these fields.
By studying disparate spatial examples within a broad mathematical context,
we enlarge our own capabilities to think logically and clearly. Our goal is that
others feel that their intellectual horizons have been broadened, as well!
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