# Message #2334

From: Don Hatch <hatch@plunk.org>

Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}

Date: Sat, 14 Jul 2012 15:51:39 -0400

On Fri, Jul 13, 2012 at 06:25:47PM -0500, Roice Nelson wrote:

>

> 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

If you can just figure out the coordinates

where three incident edges of one {3,3} of the {3,3,7} meet the sphere,

that will give you one of the little spherical triangles…

Then just transform that one spherical triangle

by symmetries of the {3,3,7}

(3 generators suffice, in any of several ways);

that should give the whole picture.

Don

–

Don Hatch

hatch@plunk.org

http://www.plunk.org/~hatch/