# 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