From: | Jefferson Wilson <jeffwilson63@...> |
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Date: | Sunday, January 29, 2006, 7:07 |

Sai Emrys wrote:> On 1/28/06, Jefferson Wilson <jeffwilson63@...> wrote: > >>>>What do you mean by "arbitrary degree?" If all symbols are the >>>>same size you're more-or-less restricted to six branches from a >>>>single symbol.You have yet to even try to explain what you mean by "arbitrary degree of branching." If you don't bother to try to explain what you mean I can only assume that you're babbling to hear yourself babble. I won't be responding if you can't be bothered to even begin an attempt to explain yourself.>>>Only if they're also all square AND not allowed to overlap (or 'fill' >>>a square space, like all 'ideographic' languages I know do - e.g. >>>Japanese / Chinese kanji/hanzi always "take up" one square of space, >>>no matter what they do within it). >> >>Uh, no. It doesn't matter whether they're square or allowed to >>overlap or change in size. Two-dimensional space-filling permits >>only six connections, and if you aren't talking about same-size >>space-filling then your connections aren't arbitrary in the first >>place. > > I don't believe you. Prove it?If your shapes aren't same-size their dimensions must be defined, and hence the relations involved are not arbitrary. If a pattern isn't space-filling then connections between elements must be defined by something outside the elements and the space they occupy, such definitions cannot be arbitrary. These are some of the basic _definitions_ of map theory. (That's mathematical map theory, not cartography.)> I can think of several simple counterexamples - hexagonal grids like > wargames,Limited to six connections.> my drawing a circle with a bunch of lines coming out of it > to circles all around itWhich represent defined branches and not arbitrary ones.> (distance required increases with N, if > they're all equidistant; otherwise, it becomes like atomic shells); > etc.So what's an "arbitrary degree of branching"?>>>If you have different shape of their 'personal space' - e.g. hexagonal >>>(viz. maps used for wargames) - or if they have allowance for some >>>sort of fusional morphology, then I see no reason why it cannot in >>>fact be literally to any arbitrary degree of branching / recursion. >> >>You've failed to define what you mean by "arbitrary degree of >>branching." Mathematically, space-filling two-dimensional >>arrangements are limited to six connections. Even if there's a >>higher order of symmetry (7-fold or eight-fold) there can still >>be only six or fewer local connections. Greater connectivity can >>be defined, but if it's defined it can't (by definition) be >>arbitrary. > > I don't see how you arrive at that <=6 number.Draw a circle. What is the maximum number of circles that can be drawn touching that circle, or allowed to overlap to the same degree while being distinguishable. The answer is six. The same holds true for all regular shapes, no matter how intricate. Irregular shapes don't matter, because degree to which their connectivity is measured must be defined, and thus cannot be arbitrary. (This only applies to two-dimensional connectivity. The situation is different in three dimensions.) -- Jefferson http://www.picotech.net/~jeff_wilson63/rpg/

Paul Bennett <paul-bennett@...> |