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/