From: | David G. Durand <dgd@...> |
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Date: | Monday, January 30, 2006, 16:14 |

toplogically,staggered triangles are squares.You can see this by drawing a dot in each figure and connecting it to its neighbors -- mathematically this is called the dual network to the one describing the edges you started with. If you do the same with staggered squares you will see an underlying triangular grid of connections, with 6 lines impinging on each point. The hexagonal grid, and non-staggered triangular grids are dual to each other. The square grid is self-dual. If you allow star shapes, asymmetrical shapes, etc. you can get much more complex topologies, and even more complex if different cells can have different shapes. Jefferson signed off the list, finding it unfriendly and uninteresting, which makes me regret my sharpness, as I think with communication in place, the discussion could have continued in an interesting direction. Like Sai, I'd never considered making a language out of a uniform grid and connectivity pattern that could be filled with symbols as needed. I think he means only some neighbors to be actually connected, but the decision to have an underlying regular convex grid does limit the maximum connections as he said. -- David On 1/30/06, Jim Henry <jimhenry1973@...> wrote:> On 1/30/06, David G. Durand <dgd@...> wrote: > > > Jefferson is assuming that the only proper way such a system can work > > is as a space-filling network of regular adjacent convex cells, (a > > convex tesselation of the plane) in which case 6 is the maximum > > connectivity, when connection is defined by sharing an edge. This is > > one way to create such a system of signs, in which case you can have a > > per-cell branching factor of three (triangles), four (squares), or six > > (hexagons). You could increase the options to include 1,2,5 if you > > agree to allow some sides of cells not to be used. > > Actually, if you have rows of squares that are offset > from one another, then you could have squares that > share (at least part of) an edge with six other squares. > > --------------------- > | | | | | | > --------------------- > | | | | | > --------------------- > | | | | | | > --------------------- > > If you have triangles in rows > that are offset from one another so the vertices in one > row don't touch the vertices in the next, > > \/\/\/\/\/\/\/\/\ > ----------------- > \/\/\/\/\/\/\/\/\ > ----------------- > \/\/\/\/\/\/\/\/\ > > Each triangle shares a full edge with two other > triangles in the same row, and half of an > edge with two triangles in the row above. > > -- > Jim Henry > http://www.pobox.com/~jimhenry >-- -- David

Tim May <butsuri@...> |