# Message #2299

From: Don Hatch <hatch@plunk.org>

Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}

Date: Tue, 03 Jul 2012 01:58:56 -0400

Hi Andrey,

I’m not sure if I’m understanding correctly…

is "behedral angle" the same as "dihedral angle"?

If so, isn’t the dihedral angle going to be 2*pi/7,

since, by definition, 7 tetrahedra surround each edge?

Don

On Tue, Jul 03, 2012 at 05:14:47AM -0000, Andrey wrote:

>

>

> 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/

> >

>

>

–

Don Hatch

hatch@plunk.org

http://www.plunk.org/~hatch/