Message #2330

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Fri, 13 Jul 2012 15:06:59 -0500

On Thu, Jul 12, 2012 at 12:52 AM, Don Hatch wrote:

>
> 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.
>
>
yeah :D I had thought of it as the intersection of {3,3,inf} with the
sphere-at-infinity in the ball model, then stereographically projected…
equivalent, but I like your image better.

I misnamed the pic by labeling it {inf,3,3}, since that would result in the
gasket only. My understanding was less clear at the time.

On Thu, Jul 12, 2012 at 7:12 PM, Don Hatch wrote:

> Hey Roice,
>
> I take it all back.
> Now I think you’re totally right…
> it’s not an Apollonian gasket for n < infinity–
> the circles don’t kiss.
> I think my reasoning was good up through where I said
> the sphere is covered by a disjoint union of {3,7}’s…
> that much is true, but it doesn’t imply the circles kiss…
> and in fact I now think you’re right, they don’t.
> (It’s more obvious if we consider n=6 too…
> maybe you were thinking of that that all along, but I missed it.)
> I think the circles are dense on the surface of the sphere, but no two
> of them kiss each other.
>
> Now I really want to see a picture of this thing!
>
> Don
>
>
Thanks for all your thoughts here and in previous posts. I’m understanding
this all better and better as I mull over these emails. I’ve been
imagining one cell of the {n,3,3} in the Poincare ball model as a sort of
umbrella: closed when n=6, and opening more as n increases (towards a
maximum by ever decreasing amounts). It did feel there must be *some *kind
of difference at the ball boundary for the different {n,3,3} honeycombs,
though I was still unsure about whether the circles kissed or not. At this
point, what you say sounds correct to me (dense, but not kissing unless n
is infinite).

I’d love to see a picture of this thing too. Consider the {7,3,3} such
that a vertex is at the origin, so 4 cells meet there. If we could
calculate the size of the circle associated with one of these cells (I
don’t know how to do this), we could start with that one. We’d generate a
{3,7} tiling inside that circle. I suspect the triangles in it are
precisely the same as those in the Poincare disk (?). Then we use Mobius
transformations to copy this template {3,7} tiling all over the plane.

I think we could leverage the Apollonian gasket to generate the list of
needed Mobius transforms, because even though the {3,7} boundary circles
aren’t kissing, the (non-Euclidean) centers of all the circles are still
the same as that of the gasket. So the list of transforms will be the same
list used to generate an Apollonian from a starting circle.

seeya,
Roice