Message #2321

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Fri, 06 Jul 2012 10:55:13 -0500

On Fri, Jul 6, 2012 at 4:05 AM, Don Hatch wrote:

> 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?

Wolfram Alpha verifies this, and gives some alternate solutions to this
equation as well.

http://www.wolframalpha.com/input/?i=cos%28pi%2F4%29%2Fsin%28pi%2F6%29+%3D+cos%28pi%2Fn%29%2Fsin%28pi%2F3%29

None of the solutions have simple looking numbers though. I was also
disappointed to see that for {3,3,r}, no integer r works with the
{3,ultrainf} based on the {6,4}. Now I’m left wondering if there is a
better canonical {3,ultrainf}, which can be justified as "the best" in some
other way.

Roice