Message #2362

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Thu, 26 Jul 2012 20:23:53 -0500

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
>
>