Biology Reference
In-Depth Information
CHAPTER
4
Theory of Shape
This chapter covers the basic theory of shape, beginning with the definition of shape
and proceeding through the characterization of several theoretical spaces. Some of the
mathematics may look a bit difficult, but it is important to grasp the basic ideas, which
we present verbally as well as mathematically. These ideas will reappear in the next
chapter, because they form the core of geometric morphometrics. Interestingly, many of
the techniques used in geometric morphometrics were developed independently of this
theory even though they are justified by it. As the theory matured, it became possible
to synthesize a large body of techniques that had been developed independently of
each other and to explicate their interrelationships. Perhaps most importantly, this the-
ory also allows us to judge whether or not particular methods are valid. The theory
provides the underlying justifications for all our techniques, thereby allowing us to
make inferences about shape without worrying that those inferences are somehow based
on arbitrary or mathematically faulty choices that we happened to make in the course
of our analyses. Freed of such concerns, we can concentrate on the biological meaning
of the results.
It would be possible to learn techniques without understanding any of this theory
but don't. Without the theory it is impossible to say why some methods are right and
others are not. In effect, you would have to memorize a list of “dos” and “don'ts” by
rote without understanding why the “dos” are “dos” and the “don'ts” are “don'ts.”
Learned in that way, it might seem that there are lots of picky rules and dogma, but
these rules are not picky and they are not a matter of dogma. Rather, they all logically
(and mathematically) follow from the mathematical theory of shape. In fact, they follow
from the definition of shape. Because this definition is central to geometric morphomet-
ric theory, we begin there, developing it further than in previous chapters.
