Message #2353

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Mon, 23 Jul 2012 18:52:30 -0500

Hi Don, I have a few inlines below.

On Mon, Jul 23, 2012 at 1:39 PM, Don Hatch wrote:

> 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.
>
>
Very intriguing. I’ll think of the results here as a shrinkage of the
cells, with the black lines being empty space between cells. And I see…
if we shrink the cells by an infinite amount at the ball boundary, we
definitely can’t shrink by the same amount in the interior.

For 2D tilings, I simulate edge thickness by shrinking tiles such that the
new edge lines are equidistant from the original borders of the tiles. If
we shrink the {3,3,n} cells in a similar equidistant manner from the cell
surface, the gap between cells (in the ball model) will be infinitesimal at
the boundary. Maybe the way to look at these pictures is as a sort of
"geodesic shrink" of the cells, i.e. you cut a cell back to a neighboring
geodesic surface, one that is very close in the ball interior, but diverges
away towards the boundary.

It is prettier, so we absolutely must find a good way to justify it!! :D

> * 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 is awesome! I had noticed that reflection pattern, but had no
understanding of what it meant.

> 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.
>
>
I had not even realized we could do an H3 transform to have one of the
{3,8}s cover a complete hemisphere, but it makes sense. In the picture you
describe, it’d be cool to color the neighbors of that {3,8} hemisphere
differently, to distinguish them from all the other {3,8}s.

> 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}.
>
>
Beautiful explanation, thank you!

Here is an image I can now understand a little better in this context:

http://www.gravitation3d.com/roice/math/ultrainf/338/338_tet_colored.png

Each of the four ends of a cell is assigned a different color. The
neighbor {3,8}s opposite the outer red {3,8} is only composed of the other
3 colors (I wonder how many triangles in each of the neighbor {3,8}s are
associated with the cells connected to just the outer red tiling). If you
pick any {3,8} in this picture, none of its neighbors will have a like
color. Like colors sort of repel each other (easier for me to see by
looking at the yellow circles).

> 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.

Are you interested to see only the ball boundary, or would you like to see
some {3,3} edges in the interior of the ball as well? I’m curious, since
it would affect possible approaches towards the rendering of something like
this.

>
> > 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.
>
>
Cool, these thought definitely give me some good direction for next
efforts. Btw, I shared these pics with Vladimir Bulatov (bulatov.org), and
turns out he has done some animations of H3 tilings having tiles with
infinite volume, taking the same approach of showing the patterns on the
horizon. Not the same tilings we’re looking at and trying to understand,
but still thought I’d share a couple of his videos he pointed me to.

http://youtu.be/xkkJcNwrDVM
http://youtu.be/4Me94JK09uk

> 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.
>
>
You were right at the start - these are artifacts of not-enough filling in.
In these latest images, I’m not drawing any edges, just triangles over a
black background. So the parts that look black but should be white are
because they are not filled in. I can make these slightly better by using
double the number of triangles, but am running into some scaling issues
after that point. The true picture of things would be brighter in all the
interstitial spaces (I’m culling all triangles with vertex circles smaller
than a certain threshold).

Thanks for this thought-out email. I enjoyed reading it more than once,
and my understanding continues to deepen from all your thoughts. I’m still
very much enjoying this topic!

Best,
Roice