Message #2351

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Mon, 23 Jul 2012 14:39:17 -0400

On Sat, Jul 21, 2012 at 05:58:45PM -0500, Roice Nelson wrote:
>
>
> Interesting thoughts/questions! Here are some pictures of the {3,3,8}
> with rotations applied to the sphere-at-infinity. Also, I improved the
> images to have true stereographic projection of edge widths, which looks much cooler.

This is a subtle statement!

My first reaction to this statement was that it must be nonsense…
The arcs you’re drawing aren’t edges at all,
they are cross sections of faces,
and if the faces of an actual physical {3,3,8} have some finite thickness
then, in a conformal projection,
their thicknesses should look like zero everywhere at infinity,
i.e. all the arcs you’re drawing must have zero width.

But, it does make sense in a way…
if your arcs have infinitesimal width
in the middle of the apparent {3,8}’s,
then I guess they approach *infinitesimal-squared* width
when approaching the edges of that {3,8}.
So it does make sense to blow up the infinitesimal
to a finite width, resulting in your picture.

But then I find it hard to reconcile this with the internal structure
of the {3,3,8}…
If you’ve multiplied all the face thicknesses by infinity
(at least at the horizon),
then can you still draw the internal structure of the {3,3,8}
to fit with your picture in some way?
I no longer have a coherent picture of it in my mind.

> * www.gravitation3d.com/roice/math/ultrainf/338/338.png A base image
> for comparison.
> * www.gravitation3d.com/roice/math/ultrainf/338/338_half_plane.png
> This one has the structure of a poincare-half-plane. One observation
> is that the dividing line doesn’t go through the origin. This is
> because the {3,8} tilings are not great circles, so when the boundary
> of one of them gets projected to a straight line (goes through the
> north pole), it will necessarily have an offset. The amount of offset
> depends on the size of the particular {3,8} tiling (smaller tiling =
> more offset).
> * www.gravitation3d.com/roice/math/ultrainf/338/338_random_view.png I’m
> also unsure how to approach finding a point not inside and not on the
> boundary of any {3,8} tiling, but here’s a randomly picked one which
> might be along these lines. For a point like you describe, there
> would be no inverted {3,8} tiling in the picture. My guess is that if
> you were to zoom out, you’d see an ever increasing cascade of larger
> and larger {3,8} tilings (with everything still always filled in and
> dense everywhere, of course). Sort-of the opposite of what would
> happen if you zoomed into one of these "irrational" points. I’m
> curious if we can say anything about the point antipodal to one of
> these. Is it also "irrational" or not?
> * www.gravitation3d.com/roice/math/ultrainf/338/338_tet_vertex.png This
> one puts one of the green tetrahedron vertices at the origin, so the
> inverted circle is now one of the 4 largest {3,8}s. Hence, it takes
> up more of the image and the visual result is that the whole image
> looks shrunk.

Right! The last one is exactly the same as the original 338.png
(but with the blue and green lines in different places).
And if you just change the viewpoint a bit,
it will be *exactly* the same picture as the original (same size).

This is the mind-blowing thing about these pictures, to me…
*every* one of the apparent {3,8}’s that you see
are the same!
Some of them look very much like they have 8 natural "neighboring"
{3,8}’s, but they don’t– each one has an infinite
number of neighboring {3,8}’s, which become more or less
obvious with a changing viewpoint.

I’ve been staring at these pictures for a long time
visualizing how this {3,8} is the same as that one,
and which neighbors map to which neighbors.
Or, if I keep one {3,8} with its boundary fixed,
how the rest of the picture can move around.

Two {3,8}’s are neighbors
iff the corresponding cells of the superimposed {8,3,3} are neighbors;
i.e. iff there is an edge of the {3,3,8} connecting
a vertex of one {3,8} with a vertex of the other {3,8}.
So, if we focus on a particular {3,8},
its neighbor {3,8}s are in 1-to-1 correspondence
with the vertices of this {3,8}–
with exactly one edge-of-the-{3,3,8}
leading from the vertex to a vertex of the neighbor {3,8}.

The correspondence is particularly clear
in your "half-plane" view…
the vertices of the "big" {3,8} below the horizon line
clearly correspond (by reflection)
to the neighbor {3,8}’s above the horizon line,
and it’s easy to visualize the edge-of-the-{3,3,8} joining them
(it’s a semicircle orthogonal to the plane).

That would also look neat wrapped back around the sphere,
with one {3,8} covering the southern hemisphere
and the rest of the picture covering the northern hemisphere:
the correspondence between vertices-of-3,8 below the equator
and {3,8}’s above the equator would be even more obvious.

Thinking along these lines,
I believe I finally do understand why your {3,3,8}
pictures seem to roughly follow the structure of an {8,3} in the plane.
The "primary" {3,8}s in your picture
(i.e. the ones that appear to correspond to the faces of the structural {8,3})
are precisely the neighbors of the "outer" {3,8},
and they are naturally arranged corresponding
to the vertices of the outer {3,8}–
i.e. these primary {3,8}s are arranged like the vertices of a {3,8},
i.e. like the cells of an {8,3}.

For me, this is, again, easiest to think about
if I imagine the point of view in the poincare ball model
in which one of the {3,8}s exactly covers the southern hemisphere.
A vertex below the equator corresponds to a {3,8} above the equator,
with an edge-of-the-{3,3,8} joining them inside the sphere.
I’d like to see that picture.

> It would be cool to see a animation with a smoothly changing viewpoint,
> but that would take a long time to generate. I’ve been using 100k tets
> (400k triangles) for these, and each takes a minute or so to produce on my
> laptop. For an animation, each frame needs to be generated all anew (as
> the areas that need filling in change depending on the view).

Yes! I’d love to see an animation.
Especially an animation in which the boundary of one of the {3,8}s is fixed
and the rest of the picture moves.

If the fixed {3,8} is the "outer"
(or lower-half-plane, or southern hemisphere) one,
then we’ll get the usual effect of panning around in hyperbolic 2-space:
(both within the {3,8} itself, and, reflected, in the rest of the picture).

But if we fix a *different* {3,8}…
that’s what I’m really wanting to see.
I think that would help me break my mind’s insistence
on thinking the {3,8} has 8 "special" closest neighbors,
when it really doesn’t.

Hey, one mundane question about these pictures–
there seems to be some strange artifact
that makes parts of the picture
look visibly darker than other parts–
do you know what’s going on there?
At first I thought this was just due to there being
more refinement in some areas than others,
but looking closer, I don’t think that explains it…
e.g. in your 338_half_plane image,
focusing on the largest complete {3,8}
(to the upper-right of the center),
its top-most edges seem clearly bolder
than its bottom-most edges.
And towards the left side of the 338_random_view image,
it looks like it’s even filled in some
parts with black that should clearly be white.

Don

> seeya,
> Roice
>
> On Thu, Jul 19, 2012 at 12:39 AM, Don Hatch <hatch@plunk.org> wrote:
>
> These pictures totally rock.
> And yeah, the fact that the overall structure
> on the infinity-plane of the poincare half-space
> ends up following a {n,3} is a total surprise.
> I have no intuition at all about why that would happen.
>
> I wonder if there are more surprises
> if you do the stereographic projection
> from different point, that’s not in any of the {3,n}’s?
> I think I can imagine what it would look like
> if you chose a point on the boundary of one of the {3,n}’s
> (I think it would follow the structure of a {n,3} in a
> poincare-half-plane).
>
> But what if you choose a point that’s not
> even on the boundary of any of them?
> I’m not even sure how to find the coords of such a point…
> however I suspect the complement of the union of the {3,n}’s
> has positive fractal dimension, which would imply
> if you just pick a point at random, there’s a nonzero probability
> that it’s not on or in any of the {3,n}’s.
> Don
>
>

–
Don Hatch
hatch@plunk.org
http://www.plunk.org/~hatch/