Message #2346

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Thu, 19 Jul 2012 01:39:35 -0400

These pictures totally rock.
And yeah, the fact that the overall structure
on the infinity-plane of the poincare half-space
ends up following a {n,3} is a total surprise.
I have no intuition at all about why that would happen.

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.

Don

On Wed, Jul 18, 2012 at 06:13:30PM -0500, Roice Nelson wrote:
>
>
> Just one more set. Here are the {3,3,7}, {3,3,8}, and {3,3,11} boundaries
> from a wider viewpoint, more like that of the first {3,3,inf} picture.
> http://www.gravitation3d.com/roice/math/337_wide_view.png
> http://www.gravitation3d.com/roice/math/338_wide_view.png
> http://www.gravitation3d.com/roice/math/3311_wide_view.png
> It’s pretty neat how the the locations of the individual {3,n}
> tessellations fall into the face pattern of an {n,3} tessellation. I
> hadn’t noticed that before with the {3,3,inf}. I’m thinking this pattern
> wouldn’t be noticeable if you rendered these on the surface of the
> Poincare ball.
> Roice
>

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