Sorry, the word "territory" was merely an (unsuccessful) attempt at a pun on the notion of "atlas" used in the theory of manifolds. (In your example, the straig

Sorry, the word "territory" was merely an (unsuccessful) attempt at a pun on the notion of "atlas" used in the theory of manifolds. (In your example, the straight line segments are the "pages" in such atlas, and the circle - the manifold - is the "territory.") It would seem to me that the properties of the circle are not emergent properties of a bunch of line segments; rather, the circle is something else entirely, whose shape is forced upon it by its own nature. Another example that comes to mind is the so-called "fiber bundles," whose own topology in general can be much more complex than that of a "trivial bundle" (simply the product of two spaces - the "base" and the "fiber"), although locally they all do look "trivial." None of this, I am afraid, has much to do with what you said you are working on. In the realm of software development, the only emerging properties (or non-trivial topologies) are bugs.