# Message #2317

From: Don Hatch <hatch@plunk.org>

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

Date: Fri, 06 Jul 2012 04:16:04 -0400

Thinking more about the parametrization…

as we increase n to infinity, the cell in-radius of {3,n}

increases, approaching a finite limit (the in-radius of {3,inf}), right?

Then you can keep increasing the in-radius towards infinity,

resulting in various kinds of what we’ve been calling {3,ultrainf}.

I wonder if we can invert the formula for in-radius in terms of n,

giving n in terms of in-radius? I guess the n’s

for various kinds of {3,ultrainf} would be imaginary or complex?

So maybe n is still a natural parameter for all of these,

as an alternative to in-radius or distance-between-pairs-of-edges.

In particular, I wonder what the parameter n is

for the picture derived from the {6,4}?

Let me find that formula again…

Don

On Wed, Jul 04, 2012 at 02:33:55PM -0500, Roice Nelson wrote:

>

>

> Sure, I’m interested in what you guys came up with

> along the lines of a {3,ultrainfinity}…

> I guess it would look like the picture Nan included in his previous

> e-mail

> (obtained by erasing some edges of the {6,4})

> however you’re free to choose any triangle in-radius

> in the range (in-radius of {3,infinity}, infinity], right?

>

> yep, our discussion finished on that picture, so Nan already shared most

> of what we talked about. I like your thought to use the inradius as the

> parameter for {3,ultrainf}, and that range sounds right to me.

>

>

> Is there a nicer parametrization of that one degree of freedom?

> Or is there some special value which could be regarded as the canonical

> one?

>

> Nan and I had discussed the parametrization Andrey mentions, the (closest)

> perpendicular distance between pairs of the the 3 ultraparallel lines.

> Since "trilaterals" have no vertices, these distances can somewhat play

> the role of angle - if they are all the same you have a

> regular trilateral. The trilateral derived from the {6,4} tiling that

> Nan shared is even more regular in a sense. Even though trilaterals have

> infinite edge length, we can consider the edge lengths between the

> perpendicular lines above. Only for the trilateral based on the {6,4} are

> those lengths equal to the "angles". So perhaps it is the best canonical

> example for {3,ultrainf}.

> seeya,

> Roice

>

>

>

–

Don Hatch

hatch@plunk.org

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