# 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

>

>