Message #2297

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: Hyperbolic Honeycomb {7,3,3}
Date: Tue, 03 Jul 2012 05:14:47 -0000

Hi all,
About {3,3,7} I had some idea (but it was long ago…). We know that its cell is a tetragedron that expands infinitely beyond "vertices". For each 3 its faces we have a plane perpendicular to them (that cuts them is the narrowest place). It we cut {3,3,7} cell by these planes, we get truncated tetrahedron with behedral angles = 180 deg. What happens if we reflect it about triangle faces, and continue this process to infinity? It will be some "fractal-like" network inscribed in the cell of {3,3,7} - regular polyhedron with infinite numer of infinite faces but with no vertices. I’m sure that it has enough regular patterns of face coloring, and it may be a good base for 3D puzzle.

Andrey

— In 4D_Cubing@yahoogroups.com, Don Hatch <hatch@…> wrote:
>
> Hi Nan,
>
> Heh, I try not to make judgements…
> I think {3,3,7} is as legit as {7,3,3}
> (in fact I’d go so far as to say they are the same object,
> with different names given to the components).
> But pragmatically, {7,3,3} seems easier to get a grip on,
> in a viewer program such as yours which focuses naturally
> on the vertices and edges.
>
> Perhaps the best way to get a feeling for {3,3,7}
> would be to view it together with the {7,3,3}?
> Maybe one color for the {7,3,3} edges,
> another color for the {3,3,7} edges,
> and a third color for the edges formed where
> the faces of one intersect the faces of the other.
> And then perhaps, optionally,
> the full outlines of the characteristic tetrahedra?
> There are 6 types of edges in all (6 edges of a characteristic tet);
> I wonder if there’s a natural coloring scheme
> using the 6 primary and secondary colors.
>
> Don
>
>
> On Sat, Jun 30, 2012 at 11:29:32PM -0000, schuma wrote:
> >
> >
> > Hi Don,
> >
> > Nice to see you here. Here are my thoughts about {3,3,7} and the things
> > similar to it.
> >
> > Just like when we constructed {7,3,3} we were not able locate the cell
> > centers, when we consider {3,3,7} we have to sacrifice the vertices. Let’s
> > start by considering something simpler in lower dimensions.
> >
> > For example, in 2D, we could consider a hyperbolic "triangle" for which
> > the sides don’t meet even at the circle of infinity. The sides are
> > ultraparallel. Since there’s no "angle", the name "triangle" is not
> > appropriate any more. I’ll call it a "trilateral", because it does have
> > three sides (the common triangle is also a trilateral in my notation).
> > Here’s a tessellation of H2 using trilaterals, in which different colors
> > indicate different trilaterals.
> >
> > http://games.groups.yahoo.com/group/4D_Cubing/files/Nan%20Ma/figure3.gif
> >
> > I constructed it as follows:
> >
> > In R2, a hexagon can be regarded as a truncated triangle, that is, when
> > you extend the first, third, and fifth side of a hexagon, you get a
> > triangle. In H3, when you extend the sides of a hexagon, you don’t always
> > get a triangle in the common sense: sometimes the extensions don’t meet.
> > But I claim you always get a trilateral. So I started with a regular {6,4}
> > tiling, and applied the extensions to get the tiling of trilaterals.
> >
> > And I believe we can do similar things in H3: extend a properly scaled
> > truncated tetrahedron to construct a "tetrahedron" with no vertices.
> > Fortunately the name "tetrahedron" remains valid because hedron means face
> > rather than vertices. But I have never done an illustration of it yet.
> > Then, maybe we can go ahead and put seven of them around an edge and make
> > a {3,3,7}.
> >
> > I agree that these objects are not conventional at all. We lost something
> > like the vertices. But just like the above image, they do have nice
> > patterns and are something worth considering.
> >
> > So, what do you think about them? Do they sound more legit now?
> >
> > Nan
> >
> > — In 4D_Cubing@yahoogroups.com, Don Hatch <hatch@> wrote:
> > > {3,3,7} less so… its vertices are not simply at infinity (as in
> > {3,3,6}),
> > > they are "beyond infinity"…
> > > If you try to draw this one, none of the edges will meet at all (not
> > even at
> > > infinity)… they all diverge! You’ll see each edge
> > > emerging from somewhere on the horizon (although there’s no vertex
> > > there) and leaving somewhere else on the horizon…
> > > so nothing meets up, which kind of makes the picture less satisfying.
> > > If you run the formula for edge length or cell circumradius, you’ll get,
> > not infinity,
> > > but an imaginary or complex number (although the cell in-radius is
> > finite, of
> > > course, being equal to the half-edge-length of the dual {7,3,3}).
> >
> >
>
> –
> Don Hatch
> hatch@…
> http://www.plunk.org/~hatch/
>