Message #2327

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Thu, 12 Jul 2012 01:52:44 -0400

On Thu, Jul 12, 2012 at 01:18:27AM -0400, Don Hatch wrote:
>
> On Wed, Jul 04, 2012 at 02:48:31PM -0500, Roice Nelson wrote:
…
> > Here is an image close to the the picture you describe. The
> > difference is that each circle in the gasket is filled with a {3,inf}
> > rather than a {3,7}.
>
> I’m having trouble getting my email client to see or detach the image
> you’re
> referring to, so I can only imagine it at the moment…
…
> And then, it seems to me that the exact same construction
> goes through for any {3,3,n}/{n,3,3},
> leading to exactly the same Appolonian gasket for each of them…
> and so presumably {3,3,inf}/{inf,3,3} would yield the same Appolonian
> gasket as well
> (though I’m having a hard time visualizing
> the {3,3,inf} directly).

Ah, I see your image…
I thought it was an attachment, but it was a link
which didn’t come out in my dumb e-mail client:
http://www.gravitation3d.com/roice/math/%7Binf,3,3%7D_sphere_at_inf.png
That gives me a *much* better feeling for the {3,3,inf} and {inf,3,3}.

Beautiful!
This picture is precisely the intersection of the {3,3,inf}
with the plane-at-infinity, in the poincare half-space model of H3, right?
Totally frickin awesome.

If we focus attention on any particular triangle
in he picture, and consider it and its 3 reflected images
in adjacent circles of the gasket, than that’s what we see of one particular {3,3} cell.

(And I believe the exact same statement can be made
about the analogous picture for {3,3,7}, which I’m imagining…
same gasket, with each circle filled with a {3,7} instead of {3,inf}… I think)

Don