TITLE: Thom's Principle: rich structures are more numerous in low dimension
QUESTION [15 upvotes]: Marcel Berger states Thom's Principle as:

"rich structures are more numerous in low dimension,
  and poor structures are more numerous in high dimension."

This is in
Geometry II
(Springer-Verlag, Berlin, 1987.

Google books), pp.39-40, after a discussion of regular polytopes
shows that there are only three convex regular polytopes in dimensions
larger than 4, but six in dimension 4, five in dimension 3, and
an infinite number in dimension 2.
He then lists further examples of the principle:
e.g., simple Lie groups illustrate rich structure in low dimensions,
and topological vector spaces illustrate poor structure in
high dimension (all homeomorphic in finite dimensions).
In so far as Thom's principle is true—or at least holds widely—my question is: 

Why should low dimensions exhibit richer structure
  than high dimensions? Is there any generic reason to expect this?

It might also be interesting to track down Thom's own formulation
of his principle, to understand the context in which he proposed it.
Edit. Here is a snapshot of Berger's examples (p.40):

REPLY [4 votes]: I am reminded of the fact that for geometries of dimension strictly higher than 2, the Desarguesian and Pappian configurations are equivalent, so that (in my view) there is a greater degree of uniformity in higher dimensional geometric spaces.  I think Thom's principle (as noted in another post) is a selection principle and tells the reader what Thom thinks of by rich structure.  If structures are essentially (in mathematical logic, some notion of conservative) extensions of incidence relations, then I am not too surprised that something like this principle should hold, since a certain amount of uniqueness is lost in a structure that should be invariant under more geometric transformations.  I am not a geometer, though, so go ask someone like Hrushovksii what they think (from a mathematical logic perspective) of Thom's principle.
Gerhard "Ask Me About System Design" Paseman, 2011.09.13