# Message #2318

From: Don Hatch <hatch@plunk.org>

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

Date: Fri, 06 Jul 2012 05:05:42 -0400

The formula for cell in-radius is cosh(inradius({p,q})) = cos(pi/q)/sin(pi/p).

For the {n,ultrainf} based on {6,4},

I think we want n such that cosh(inradius({3,n})) == cosh(inradius(6,4)),

that is:

cos(pi/n)/sin(pi/3) = cos(pi/4)/sin(pi/6)

=> cos(pi/n)/(sqrt(3)/2) = (sqrt(2)/2)/(1/2)

=> n = +-pi/acos(sqrt(3/2))

= 4.770984191560898 i

So the {3,ultrainf} based on {6,4}

is {3, 4.770984191560898 i}.

Not a nice number like I had hoped.

Someone check my math?

Don

On Fri, Jul 06, 2012 at 04:16:04AM -0400, Don Hatch wrote:

>

>

> 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/

>

>

–

Don Hatch

hatch@plunk.org

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