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Re: Non-linear full-2d writing (again)

From:tomhchappell <tomhchappell@...>
Date:Tuesday, January 31, 2006, 2:05
--- In, Jefferson Wilson <jeffwilson63@F...>
>wrote: > Sai Emrys wrote: >>On 1/28/06, Jefferson Wilson <jeffwilson63@f...> wrote: >>>[snip] >>>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. >> >>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.
Well, this could stand some elaboration and clarification. It certainly doesn't matter whether or not they are square. It probably does matter whether or not they're allowed to overlap. If all the symbols are the same size, then, on a plane of zero curvature, each symbol can be contiguous simultaneously with at most six other symbols; if the curvature is positive (like a sphere), the limit is five. To get more than six, you'd need negative curvature (a saddle-shaped surface, for instance.) However Jeff Wilson's Glyphica Arcana, if I have attributed it to the right author, demonstrates that each symbol could have more than six places for other symbols to connect to it, even if they are all the same size and shape and not allowed to overlap; it's only that, at most six of those places can be occupied at a time. For instance, suppose they are all squares, oriented the same way (that is, with parallel sides; for instance, all sides either horizontal or vertical). A given square can be contiguous to another along any one of its four edges. On each edge, they can be contiguous along the whole edge, or along either half of the edge. That gives 4*3=12 places to attach another square.* But only six at a time can actually be occupied; otherwise, two of the attaching squares will have to overlap each other. *[If we allow the corners, as well (and we shouldn't, to be fair to the "six at most" question), that would make 16 places.]
>>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.
Hexagonal close-packing is precisely the realization of the maximum connectivity of a regular planar graph.
>You've failed to define what you mean by "arbitrary degree of >branching." Mathematically, space-filling two-dimensional >arrangements are limited to six connections.
Only if negative curvature is prohibited -- am I right? A negative- enough curvature would allow a tiling by heptagons, for example -- wouldn't it? But if you let it get really big -- allowed enough heptagons -- it would get hard to keep your paper in just one room, unless you started shrinking the heptagons.
>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.
Do you mean by that, that you can throw in a finite, suitably sparse set of exceptional "tiles" with more than 6 neighbors? In fact, some bathroom floors are tiled with octagons and "diamonds" (tilted squares), the octagons having 4 octagonal neighbors horizontally and vertically, and 4 diamond neighbors diagonally; every diamond having 4 octagonal neighbors. This graph is not regular, however. A regular planar graph -- every vertex the same as every other vertex -- can't have degree greater than 6.
> Jefferson >
Thanks for writing. Tom H.C. in MI


Patrick Littell <puchitao@...>
John Vertical <johnvertical@...>