Message #2341

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Wed, 18 Jul 2012 12:47:21 -0500

Hi Don,

Some inlines and a fixed image below!

On Wed, Jul 18, 2012 at 4:43 AM, Don Hatch wrote:

> After staring at your picture for a few more minutes,
> I’m now pretty sure that the empty spaces
> must be a mistake.
> Here’s my reasoning…
>
> The center cell has its 4 "feet" in each of
> 4 different {3,7}’s on the sphere surface.
> These 4 {3,7}’s are arranged so that they
> are equally spaced on the sphere surface,
> in a tetrahedral pattern
> (as we see in your picture).
>
> Now imagine the cell we get by reflecting
> the center cell about one of its faces.
> The "far" foot of this new cell
> will be directly opposite a foot of the original cell.
> But that’s right in the center of an empty space
> in your picture.

ah, thanks! This was exactly the clue needed to find the right image.
Here is a corrected version.

http://www.gravitation3d.com/roice/math/733_sphere_at_inf.png

In case you’re interested in what was wrong before, here are images of the
initial tet plus one reflection, before and after.

before:
http://www.gravitation3d.com/roice/math/733_incorrect_one_reflect.png
after: http://www.gravitation3d.com/roice/math/733_correct_one_reflect.png

As far as the amount of filling in, this is about the best possible given
the numerical accuracy of the MagicTile codebase (leaves much to be
desired). I had to tweak tolerance parameters to keep this from falling
apart.

>
> > 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.

The last steps were to use a tetrahedral tiling to make the 3 other legs of
the tet, then to reflect this tet around in the plane. The code reflects
using circle inversions across the arcs of the triangle edges.

Best,
Roice