# 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/