# Message #2363

From: Eduard Baumann <baumann@mcnet.ch>

Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}

Date: Fri, 27 Jul 2012 07:53:37 +0200

Very, very nice !

:-)

Ed

—– Original Message —–

From: Roice Nelson

To: 4D_Cubing@yahoogroups.com

Sent: Friday, July 27, 2012 3:23 AM

Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}

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:<br>
color each tri in order of increasing distance (of tri center,<br>
in hyperbolic space) from some fixed<br>
starting point, breaking ties arbitrarily.<br>
When choosing a color for a tri,<br>
at most n-1 of its 3*(n-2) "neighbor" tris have already been colored<br>
(I haven't proved this, but it seems to hold,<br>
from looking at a {3,7} and {3,8}).<br>
So color the new tri with any color other than<br>
the at-most-(n-1) colors used by its already-colored neighbors.
Don
```