Message #2310

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Wed, 04 Jul 2012 14:48:31 -0500

Hi Don,

I’m loving this whole thread. Lots of cool ideas being thrown out by you
and Andrey!

I really like the fractal image you’re envisioning on the sphere at
infinity. This past weekend, I was playing with recursive circle
inversions<http://roice3.blogspot.com/2012/07/recursive-circle-inversions.html>,
and I had no idea it would apply to a discussion like this at the time. Here
is an image<http://www.gravitation3d.com/roice/math/%7Binf,3,3%7D_sphere_at_inf.png>close
to the the picture you describe. The difference is that each circle
in the gasket is filled with a {3,inf} rather than a {3,7}.

I’m actually wondering, is the Apollonian gasket the result for {inf,3,3}?
For {7,3,3}, I’m thinking the initial 4 circles in the circle packing
would be smaller (not tangent), and would approach the Apollonian as p ->
inf. In the {inf,3,3} case, the 4 cells that meet at the origin in Nan’s
applet could represent the 4 initial circles in the gasket. So the cells
that meet at the origin meet again at the sphere at infinity! (although not
all at the one location this time) For the {7,3,3}, my intuition says the
initial 4 cells don’t meet again at the sphere at infinity. I’m curious
what you think about these speculations.

Aside: as traditionally shown (e.g. on
wikipedia<http://en.wikipedia.org/wiki/File:Apollonian_gasket.svg>),
the Apollonian gasket is a stereographic projection of the pattern on the
sphere at infinity we are discussing here, which I think is neat. 3 of the
4 cells jump out visually, and the 4th is inverted - the outside of the
whole pattern.

Best,
Roice

On Tue, Jul 3, 2012 at 2:51 PM, Don Hatch <hatch@plunk.org> wrote:

> Oh wait!
> I realized I got last part wrong, just after I hit the "send" button :-)
>
> The picture would start with an Apollonian gasket (see wikipedia article)
> of circles on the sphere;
> this is the intersection of the {7,3,3} with the sphere at infinity.
> Then each circle in the gasket is filled with a {3,7},
> the final result being the intersection of the {3,3,7} with the sphere
> at infinity.
> So, it isn’t true that there are isolated cluster points
> in the *center* of each circle; the clustering is
> towards the *boundary* of each circle. I think I have
> a clear picture in my head of what this looks like now.
>
> A cell of the {3,3,7} would touch the sphere
> in 4 spherical triangles (its "feet"),
> each foot in a different one of four mutually kissing circles
> of the gasket, I think.
>
> Don
>
>
> On Tue, Jul 03, 2012 at 03:29:38PM -0400, Don Hatch wrote:
> >
> >
> > Okay I think maybe I follow you now…
> > But each face formed by truncation…
> > it’s a triangle, not a hexagon, right?
> > In fact it’s a spherical triangle, on the sphere at infinity, right?
> > All of these spherical triangles, of different apparent sizes,
> > would tile the sphere, 7 at each vertex…
> > but with some "cluster points" which are the limit points
> > of infinitely many of these triangles of decreasing size.
> > I’d like to see a picture of this– it shouldn’t be too hard to
> generate
> > (together with the spherical circles
> > that are the intersection of the dual {7,3,3} with the sphere
> > at infinity, in a different color… I think each cluster point
> > would be the center of one of these circles).
> >
> > Don
> >
> > On Tue, Jul 03, 2012 at 02:43:52PM -0000, Andrey wrote:
> > >
> > >
> > > Yes, dihedral… I mean angle between hexagonal and triangular
> faces of
> > > truncated tetrahedron. By the selection of truncating planes it
> will be
> > > pi/2. And angles between hexagonal faces are 2*pi/7.
> > >
> > > Andrey
> > >
> > > — In 4D_Cubing@yahoogroups.com, Don Hatch <hatch@…> wrote:
> > > >
> > > > 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@…
> > > > http://www.plunk.org/~hatch/
> > > >
> > >
> > >
> >
> > –
> > Don Hatch
> > hatch@plunk.org
> > http://www.plunk.org/~hatch/
> >
> >
>
> –
> Don Hatch
> hatch@plunk.org
> http://www.plunk.org/~hatch/
>
>
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