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

From:John Vertical <johnvertical@...>
Date:Tuesday, January 31, 2006, 19:20
tomhchappell wrote:
>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.)
(from another message)
>The problem is that a _regular_, _planar_ graph can't have the >degrees of all its nodes greater than 6. > >A graph is "regular" if every node has just as many connections as >every other node.
>Tom H.C. in MI
That isn't enuff... A graf of arbitrarily high regular connectivity (equalling an arbitrarily large negative curvature) can be drawn on a Euclidean plane, if the nodes are allowed have arbitrary distances from each another. An easy way to reach a connectivity of m is a series of concentric circles, with m^n nodes on the nth circle. These conditions also allow fitting an infinite graf within finite space - M. C. Escher's Circle Limit series is a good example of this. Similar examples with hyperbolic grafs probably also exist. Only if all nodes must be equidistant to all of their neibors (or, in case of space-filling nodes, of same shape) is 6 the maximum connectivity. John Vertical