Message #2364

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Fri, 27 Jul 2012 13:26:39 -0400

Hmm, I don’t know about the "help ground oneself" part…
I feel like the checkerboard areas are confusing me, more than helping,
in my effort to visually locate cells.
I really think no-two-of-same-color-at-a-vertex would be good.

One other suggestion I think I forgot to mention before…
it would be nice to see one animation
with the "stationary" {3,n} and its neighbors colored,
and another with the initially inverted {3,n} and its neighbors colored.

Don

On Thu, Jul 26, 2012 at 08:23:53PM -0500, Roice Nelson wrote:
>
>
> I found a nice periodic (though irregular) 10-color painting of the {3,8}
> using MagicTile. (aside: I think I can turn this into a vertex-turning
> puzzle, so I’ll plan on that :D)
> http://gravitation3d.com/roice/math/ultrainf/338/38_10C.png
> Here is the {3,3,8} where the cells attached to the outer circle use this
> coloring. It’s cool to look at it side-by-side with the one above.
> http://gravitation3d.com/roice/math/ultrainf/338/338_neighbors_10C.png
> The 7C vertices make it easy to distinguish individual cells, and the
> checkerboard vertices give salient areas to help ground oneself, so I
> think this coloring would work quite well for the next animation.
> Roice
>
> On Thu, Jul 26, 2012 at 1:41 AM, Don Hatch wrote:
>
> As for coloring…
> yeah it won’t be periodic,
> but I think it would be really helpful
> to get a coloring of the outer {3,n}
> in which the n tris around any vertex are n different colors.
> That would accomplish the goal of getting sufficient separation
> between any two cells of the same color in the {3,3,n},
> so that it’s easier to tell which tris are from a common cell.
> (a 2-coloring of the {3,8} wouldn’t accomplish this)
>
> I think the following coloring algorithm works:
> color each tri in order of increasing distance (of tri center,
> in hyperbolic space) from some fixed
> starting point, breaking ties arbitrarily.
> When choosing a color for a tri,
> at most n-1 of its 3*(n-2) "neighbor" tris have already been colored
> (I haven’t proved this, but it seems to hold,
> from looking at a {3,7} and {3,8}).
> So color the new tri with any color other than
> the at-most-(n-1) colors used by its already-colored neighbors.
> Don
>
>

–
Don Hatch
hatch@plunk.org
http://www.plunk.org/~hatch/