Message #2332

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Fri, 13 Jul 2012 18:25:47 -0500

>
>
> 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.
>
>
I don’t think the construction I suggested works. I think it was incorrect
of me to assume the centers of the {7,3,3} circles would coincide with the
centers of the gasket (this is perhaps only true for the first 4 circles).
Using the Mobius transforms of the Apollonian gasket as I suggested would
leave empty space.

So I’m not sure how one would go about constructing the {3,3,7} picture.
This stuff can be hard to think about!

Roice