Message #2365

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Sat, 28 Jul 2012 11:34:27 -0400

Ah, I think I see your point…
I’m now looking at the two pictures side-by-side as you suggested
(interesting!)
and the checkerboard regions do help me get my bearings
as I correlate the two pictures. They are helpful
in locating a particular *edge* of the {3,3,8}.

They confuse me as I try to locate a particular cell, though.

Don

On Fri, Jul 27, 2012 at 01:26:39PM -0400, Don Hatch wrote:
>
>
> 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/
>
>

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