Message #2345

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Thu, 19 Jul 2012 01:21:25 -0400

On Wed, Jul 18, 2012 at 12:47:21PM -0500, Roice Nelson wrote:
>
>
> On Wed, Jul 18, 2012 at 4:43 AM, Don Hatch wrote:
>
> > I have a different question though…
> > When I was describing how I’d draw this thing,
> > I thought the little triangles would be
> > spherical triangles, that is, bounded by geodesics
> > (i.e. arcs of great circles).
> > But that’s not true… they are actually
> > bounded by non-geodesic circular arcs on the sphere.
> > But, I thought, it would be reasonable to
> > draw the first picture of it with them approximated
> > by geodesics… or even by straight line segments.
> > I see you didn’t draw straight line segments,
> > but I can’t tell– are you drawing geodesics?
> > Or are you drawing the real things?
> >
>
> I think the answer is that I’m drawing the real things, but must admit I’m
> taking a leap of faith in Math God by saying that. I didn’t assume the
> little triangles were spherical (bounded by geodesics), though I did
> reason they had circular arcs, and would therefore have circular arcs in
> the plane too.
> What I did was use the inradius of the {7,3,3} to calculate the midpoint
> of an arc segment of one of these triangles on the sphere. I didn’t even
> go through the effort to calc an endpoint, as you laid out. I already had
> a function to generate a {3,7} starting triangle in the plane, so I used
> my calculated midpoint to scale that template triangle to the right size.
> It did feel like a jump to assume the geometry would lead to a standard
> {3,7} triangle at the origin. But since all the geometrical relations
> (and stereographic projection) would preserve circles, it seemed it had to
> be. This was the leap of faith.

It looks like you’re totally right
(more obvious in the {3,3,8})–
if I locate three segments from the same 2d face,
their curvatures in the picture are such that they are all part of a common circle,
as required.
I wasn’t expecting that at all (I thought they were going to be curved
in strange unfamiliar ways).
Excellent.

Don