Message #1214

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: [MC4D] MHT633 v0.1 uploaded
Date: Wed, 27 Oct 2010 19:24:05 -0000

Roice,
by the way, there exists {infinity,3} tiling of the hyperbolic plane ;) What about including it in the 2D Magic Tiles program?

Andrey

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:
>
> Thanks for the explanations Andrey! They help a lot, and it’s so cool that
> the infinite cells are horospheres. Though they are spheres in the Poincare
> model after all, at least I was right about them not living in a hyperbolic
> plane, on sphere sections orthogonal to the model boundary :)
>
> You have really produced just about the ultimate analogue puzzle in my
> opinion. The fact that the "face" shape, the dimension, and the geometry
> are all three relaxed makes it such a lovely abstraction. I am quite
> excited to study and think about this more, and to actually spend a little
> time playing with the puzzle too!
>
> Cheers,
> Roice
> P.S. No pressure on the autorotation/autosliding of course (though I think
> it would be neat). I am able to get a good feel with the mouse alone.
>
> P.P.S. If it was not too difficult, it would be amazing if the puzzle could
> also be viewed in the Poincare model. I’m not sure how easy it would be to
> transform the mouse controls, or what other issues might arise. And in any
> case, it does feel like the half-space view is the best choice for ease of
> working with the puzzle.
>
>
> On Wed, Oct 27, 2010 at 1:09 PM, Andrey <andreyastrelin@…> wrote:
>
> > Roice,
> > Funny thing about the projection - that it’s not the model! It’s real view
> > of H3 from inside, central projections of points to the almost planar sensor
> > of the small camera. So it was not me who selected the shape and angles of
> > infinite polyhedra, it’s their real images (unless you use FishEye slider).
> > I thought that we’ll see more of the surface is we’ll take a look from
> > large distance, but it looks like not the case. And I almost know why :)
> > In the Poincaré models (both half-plane and disk) cells are going by
> > spheres that are tangent to the boundary plane/sphere of the model.
> >
> > Andrey
> >
> >
> >
> >
> > — In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@> wrote:
> > >
> > > Andrey,
> > >
> > > I’m trying to understand how the infinite {6,3} cells appear to wrap
> > around
> > > on themselves. You did a really nice job making them look like convex
> > > polyhedra…so much so, that when I first looked at the program, I
> > thought
> > > they were dodecahedra!
> > >
> > > Would you mind describing the projection to Euclidean space you’re using?
> > > Beltrami-Klein model, Poincare disk model, something else? If you showed
> > > more of the {6,3} cells, would the projection cause these infinite cells
> > to
> > > visually intersect with themselves? (It appears like it would.) More
> > > generally, I’d like to answer the question of what an entire cell would
> > look
> > > like in your projection and in other models. (I think a cell does not
> > live
> > > on a hyperbolic plane, so I’m betting a cell would not be a portion of a
> > > sphere in the Poincare model). Thanks for any insight or references on
> > this
> > > topic you can provide!
> > >
> > > Take Care,
> > > Roice
> >
> >
> >
> >
> > ————————————
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
>