Message #2352

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Mon, 23 Jul 2012 15:40:22 -0400

On Thu, Jul 19, 2012 at 01:39:35AM -0400, Don Hatch wrote:
…
>
> I wonder if there are more surprises
> if you do the stereographic projection
> from different point, that’s not in any of the {3,n}’s?
> I think I can imagine what it would look like
> if you chose a point on the boundary of one of the {3,n}’s
> (I think it would follow the structure of a {n,3} in a
> poincare-half-plane).
>
> But what if you choose a point that’s not
> even on the boundary of any of them?
> I’m not even sure how to find the coords of such a point…
> however I suspect the complement of the union of the {3,n}’s
> has positive fractal dimension, which would imply
> if you just pick a point at random, there’s a nonzero probability
> that it’s not on or in any of the {3,n}’s.

Melinda pointed out that I misspoke here…
I meant to say I think the set has nonzero area in the plane,
i.e. fractal dimension between 2 and 3.

To find such a point:
Start with a picture of the {3,3,7}
as we’ve been looking at,
that has lots of the apparent {3,7}’s
along the x axis.
Let p0 and p1 be the centers any two of these {3,7}s.
Let p2 be the center of the largest {3,7} between p0 and p1.
Let p3 be the center of the largest {3,7} between p1 and p2, etc.
The limit of this sequence will be a point
strictly outside every {3,7}.

I guess an infinite zoom in/out of the resulting picture
would repeat in time,
showing alternating {3,7}s appearing prominently on the left and right.

A different choice of projection point
would result in the sequence of prominent {3,7}s appearing in any desired sequence
of directions from the origin.

Don