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&#58;<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&#42;(n-2) &quot;neighbor&quot; tris have already been colored<br>
(I haven't proved this, but it seems to hold,<br>
from looking at a &#123;3,7&#125; and &#123;3,8&#125;).<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