Message #2359

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Thu, 26 Jul 2012 02:41:23 -0400

Yeah!!

I like that one seemingly-arbitrary {3,8}
stays in one place.

This particular animation path is a bit overwhelming to me though…
how about starting with a more mundane one,
in which the contents of the "stationary" {3,n}
simply pans steadily horizontally or vertically?

(Also… not your fault, but the youtube viewer
seriously sucks for viewing this!
I want to manually scrub the time back and forth slowly, and just can’t,
and it’s really frustrating.
Is there a way to download movies from youtube, or a less sucky way
to view them?)

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

On Wed, Jul 25, 2012 at 10:32:32PM -0500, Roice Nelson wrote:
>
>
> On Mon, Jul 23, 2012 at 1:39 PM, Don Hatch wrote:
>
> Yes! I’d love to see an animation.
> Especially an animation in which the boundary of one of the {3,8}s is
> fixed
> and the rest of the picture moves.
>
> If the fixed {3,8} is the "outer"
> (or lower-half-plane, or southern hemisphere) one,
> then we’ll get the usual effect of panning around in hyperbolic 2-space:
> (both within the {3,8} itself, and, reflected, in the rest of the
> picture).
>
> But if we fix a *different* {3,8}…
> that’s what I’m really wanting to see.
> I think that would help me break my mind’s insistence
> on thinking the {3,8} has 8 "special" closest neighbors,
> when it really doesn’t.
>
> Here ya go! A short video (just 150 frames). It could be improved for
> sure, but I hope this first attempt is still useful. Feel free to make
> suggestions (I don’t like the speedup at the end, for instance). The
> fixed {3,8} boundary is highlighted in blue.
>
> http://youtu.be/cQszcpIWeas
>
> seeya,
> Roice
>

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