Message #2309

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Wed, 04 Jul 2012 14:33:55 -0500

> 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