Message #2349

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Sat, 21 Jul 2012 17:58:45 -0500

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.

• 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.

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).

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
>
>