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A circle is a 1-dimensional manifold (which, indeed, could be embedded in a space that has more dimensions). This follows exactly from your own (correct) image

A circle is a 1-dimensional manifold (which, indeed, could be embedded in a space that has more dimensions). This follows exactly from your own (correct) image of it as being created from (i.e. being locally homeomorphic to) straight lines. But the entire thing ("the territory") always turns out to be more than the set of parts it is "built from," and if you ignore this, not only you are bound to lose the forest for the trees, your understanding of the whole may end up being completely wrong.

Krei-se 2y ago
Would you like to elaborate more on the territory part? As far as i understood you talk about emergence of properties beyond the sum of the elements but might be mistaken. Im fine with links to libgen.is or lengthy pdfs if necessary. I'm currently working on trees for a nested controller approach and constructing / deconstructing differing branches. So even though i don't feel stuck right now i'm always willing to take hints for taking another path.
- Koshkin 2y ago
  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.
  - Krei-se 2y ago
    Nah, that's totally fine, i for my part tried to make a joke about tree structure in a frontend/backend router. As there are multiple branches (subtrees) derived from different (single linked list) paths it's not a problem to go back to a node and diverge, it does not invalidate the tree. Im sure a lot of the joke in each part is lost in niche knowledge and translation. As for manifolds, i am drawn to stuff like a hendecachoron and we have a new fountain in Chemnitz now called "Manifold" which is made from circles and i love it! Also the local architect of the town hall has the surname "Möbius". So i'll take the hint to learn a bit more about topology!

Krei-se 2y ago

Would you like to elaborate more on the territory part? As far as i understood you talk about emergence of properties beyond the sum of the elements but might be mistaken. Im fine with links to libgen.is or lengthy pdfs if necessary. I'm currently working on trees for a nested controller approach and constructing / deconstructing differing branches. So even though i don't feel stuck right now i'm always willing to take hints for taking another path.

Koshkin 2y ago
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.
- Krei-se 2y ago
  Nah, that's totally fine, i for my part tried to make a joke about tree structure in a frontend/backend router. As there are multiple branches (subtrees) derived from different (single linked list) paths it's not a problem to go back to a node and diverge, it does not invalidate the tree. Im sure a lot of the joke in each part is lost in niche knowledge and translation. As for manifolds, i am drawn to stuff like a hendecachoron and we have a new fountain in Chemnitz now called "Manifold" which is made from circles and i love it! Also the local architect of the town hall has the surname "Möbius". So i'll take the hint to learn a bit more about topology!