Message #2307

From: Andrey <andreyastrelin@yahoo.com>
Subject: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Wed, 04 Jul 2012 12:01:57 -0000

Hi, Don
Let’s start with triangle with vertices "beyond infinity". Two its edges are "ultraparallel" and they have common perpendicular. If you cut triange by three such perpendiculars, you’ll get irregular hexagon with 6 right angles. Now you can reflect it about its short sides and continue this process to infinity. In result you’ll get very strange object - it is convex, it’s infinite, all its boundaries are straight lines and it’s regular - you for every two edges there is a movement of H2 that moves one edge to another and moves the whole object to itself.
Something like this:
http://groups.yahoo.com/group/4D_Cubing/photos/album/772706687/pic/856386394/view
Red lines are sides of hexagons.
You see that each hexagon is connected to three another and they make an acyclic graph (tree) together.

The same is for tetrahedra in H3. You have a convex objects, bounded by planes (and faces of this object are exactly like H2 pattern), each truncated tetrahedron is connected to 4 others and together they make regular polyhedron with tree structure (all nodes of graph are equivalent).

Points on edges of tetrahedron with the minimal distance are not infinite - they are inside H3, so whole object is "real".

Andrey

— In 4D_Cubing@yahoogroups.com, Don Hatch <hatch@…> wrote:
>
> Hi Andrey,
>
> Maybe I finally get what you mean now…
> there is a unique plane that contains
> the three points at infinity where three edges of the tet come closest to meeting;
> I didn’t see that before.
> But are you sure the tet faces meet that plane
> at a right angle as you claimed?
> I believe the tet faces meet the sphere-at-infinity at right angles;
> I don’t think both can be true.
>
> And I don’t understand what you mean by "a structure of regular infinite
> 4-graph without loops" at all.
> By "4-graph", do you mean every node has degree 4,
> and by "without loops", do you mean a tree?
> But I don’t see any such tree in what we’re talking about, so I’m lost :-(
>
> Don
>
> On Wed, Jul 04, 2012 at 04:03:38AM -0000, Andrey wrote:
> >
> >
> > And cutting triangles are planar (i.e. H2), not sperical.
> > Their position is selected so that triangles have minimal possible size
> > (in the narrowest point of the "vertex"). I’ll try to draw one face of the
> > object, but it will be not easy.
> > Looks like this combination of truncated tetrahedra will be convex in H3
> > (and have a structure of regular infinite 4-graph without loops).
> >
> > Andrey
> >
> > — In 4D_Cubing@yahoogroups.com, Don Hatch <hatch@> wrote:
> > >
> > > On Tue, Jul 03, 2012 at 03:29:38PM -0400, I wrote:
> > > > But each face formed by truncation…
> > > > it’s a triangle, not a hexagon, right?
> > >
> > > Sorry, mental lapse on my part!
> > > The hexagons you’re talking about
> > > are what’s left of the *original* faces when you truncate
> > > a tetrahedron.
> > > Don’t know where my mind went.
> > >
> > > Don
> > >
> >
> >
>
> –
> Don Hatch
> hatch@…
> http://www.plunk.org/~hatch/
>