Thread: "More {3,3,8} pics and new {3,8} MagicTile puzzles"

--20cf307c9b8cee04e604c6410073
Content-Type: text/plain; charset=ISO-8859-1

I see where you were coming from now too. You are right, I was focused on
the {3,3,8} edges (the points that look like vertices in the {3,8}s). I
found a {3,8} 8-color painting with no repeat colors around vertices this
past weekend. As you expected, it feels better for finding cells.

Here are pictures with this 8-coloring. I find it interesting how the the
intersection of cells with the sphere-at-infinity relate to lunes.

gravitation3d.com/roice/math/ultrainf/338/38_8C.png
gravitation3d.com/roice/math/ultrainf/338/338_neighbors_8C.png
gravitation3d.com/roice/math/ultrainf/338/338_neighbors_8C_half_plane.png

I did start working on a second {3,3,8} video to cover the other
suggestions this weekend as well. I generated 800 frames, but am having
some numerical stability issues with a couple dozen of them.
Unfortunately, I've had some unexpected life changing things come
up and need to table the project for at least a few weeks, but hopefully I
can get it made relatively soon.

To pull in some puzzling, I went ahead and configured new {3,8} MagicTile
puzzles using the two colorings. They are available in the latest download.

gravitation3d.com/magictile/downloads/MagicTile_v2.zip

seeya,
Roice


On Sat, Jul 28, 2012 at 10:34 AM, Don Hatch wrote:

> Ah, I think I see your point...
> I'm now looking at the two pictures side-by-side as you suggested
> (interesting!)
> and the checkerboard regions do help me get my bearings
> as I correlate the two pictures. They are helpful
> in locating a particular *edge* of the {3,3,8}.
>
> They confuse me as I try to locate a particular cell, though.
>
> Don
>
> On Fri, Jul 27, 2012 at 01:26:39PM -0400, Don Hatch wrote:
> >
> >
> > Hmm, I don't know about the "help ground oneself" part...
> > I feel like the checkerboard areas are confusing me, more than
> helping,
> > in my effort to visually locate cells.
> > I really think no-two-of-same-color-at-a-vertex would be good.
> >
> > One other suggestion I think I forgot to mention before...
> > it would be nice to see one animation
> > with the "stationary" {3,n} and its neighbors colored,
> > and another with the initially inverted {3,n} and its neighbors
> colored.
> >
> > Don
> >
> > On Thu, Jul 26, 2012 at 08:23:53PM -0500, Roice Nelson wrote:
> > >
> > >
> > > I found a nice periodic (though irregular) 10-color painting of the
> > {3,8}
> > > using MagicTile. (aside: I think I can turn this into a
> vertex-turning
> > > puzzle, so I'll plan on that :D)
> > > http://gravitation3d.com/roice/math/ultrainf/338/38_10C.png
> > > Here is the {3,3,8} where the cells attached to the outer circle use
> > this
> > > coloring. It's cool to look at it side-by-side with the one above.
> > >
> http://gravitation3d.com/roice/math/ultrainf/338/338_neighbors_10C.png
> > > The 7C vertices make it easy to distinguish individual cells, and
> the
> > > checkerboard vertices give salient areas to help ground oneself, so
> I
> > > think this coloring would work quite well for the next animation.
> > > Roice
> > >
> > > On Thu, Jul 26, 2012 at 1:41 AM, Don Hatch wrote:
> > >
> > > As for coloring...
> > > yeah it won't be periodic,
> > > but I think it would be really helpful
> > > to get a coloring of the outer {3,n}
> > > in which the n tris around any vertex are n different colors.
> > > That would accomplish the goal of getting sufficient separation
> > > between any two cells of the same color in the {3,3,n},
> > > so that it's easier to tell which tris are from a common cell.
> > > (a 2-coloring of the {3,8} wouldn't accomplish this)
> > >
> > > I think the following coloring algorithm works:
> > > color each tri in order of increasing distance (of tri center,
> > > in hyperbolic space) from some fixed
> > > starting point, breaking ties arbitrarily.
> > > When choosing a color for a tri,
> > > at most n-1 of its 3*(n-2) "neighbor" tris have already been colored
> > > (I haven't proved this, but it seems to hold,
> > > from looking at a {3,7} and {3,8}).
> > > So color the new tri with any color other than
> > > the at-most-(n-1) colors used by its already-colored neighbors.
> > > Don
> > >
> > >
> >
> > --
> > Don Hatch
> > hatch@plunk.org
> > http://www.plunk.org/~hatch/
> >
> >
>
> --
> Don Hatch
> hatch@plunk.org
> http://www.plunk.org/~hatch/
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

--20cf307c9b8cee04e604c6410073
Content-Type: text/html; charset=ISO-8859-1
Content-Transfer-Encoding: quoted-printable

I see where you were coming from now too. =A0You are right, I was focused o=
n the {3,3,8} edges (the points that look like vertices in the {3,8}s). =A0=
I found a {3,8} 8-color painting with no repeat colors around vertices this=
past weekend. =A0As you expected, it feels better for finding cells.=A0v>

--X1bOJ3K7DJ5YkBrT
Content-Type: text/plain; charset=us-ascii
Content-Disposition: inline

Hey Roice,

These are looking great. I get a good sense of where the cells are
now, especially in the half-plane one.

How did you construct your 8-coloring for the {3,8}?

I came up with a periodic coloring, but it's not the same as yours--
I see that yours has stripes of 3 colors (grey,yellow,blue)
going through it, but mine doesn't have any such stripes.

I'm attaching an image of mine (not sure whether this will work).

Here's how I construct it...
Imagine the {3,8} partitioned into an {8,4}
(8 triangles of the {3,8} in each octagon of the {8,4}).
Call half of the octagons "even" and the other half "odd",
in a checkerboard pattern.
Start with any even octagon, and color its 8 triangles
counterclockwise: 0 1 2 3 4 5 6 7.
Then for each of the eight even octagons
"diagonal" to the first even octagon,
again color it CCW with 0 1 2 3 4 5 6 7,
in the same orientation as the first even octagon
(so, for example, one of them will have its 4 5
sharing a vertex with the first octagon's 0 1).
Continue in this way, coloring all even octagons.

Finally, for each not-yet-colored triangle (in an odd octagon),
color it ((i+2) mod 8) where i is the color of its
already-colored neighbor triangle (in an even octagon).
This gives each odd octagon the colors 0 3 6 1 4 7 2 5, counterclockwise.

((i+6) mod 8) could be used instead of ((i+2) mod 8).

Don

On Wed, Aug 01, 2012 at 11:39:57PM -0500, Roice Nelson wrote:
>
>
> I see where you were coming from now too. You are right, I was focused on
> the {3,3,8} edges (the points that look like vertices in the {3,8}s). I
> found a {3,8} 8-color painting with no repeat colors around vertices this
> past weekend. As you expected, it feels better for finding cells.
> Here are pictures with this 8-coloring. I find it interesting how the the
> intersection of cells with the sphere-at-infinity relate to lunes.
> gravitation3d.com/roice/math/ultrainf/338/38_8C.png
> gravitation3d.com/roice/math/ultrainf/338/338_neighbors_8C.png
> gravitation3d.com/roice/math/ultrainf/338/338_neighbors_8C_half_plane.png
> I did start working on a second {3,3,8} video to cover the other
> suggestions this weekend as well. I generated 800 frames, but am having
> some numerical stability issues with a couple dozen of them.
> Unfortunately, I've had some unexpected life changing things come
> up and need to table the project for at least a few weeks, but hopefully I
> can get it made relatively soon.
> To pull in some puzzling, I went ahead and configured new {3,8} MagicTile
> puzzles using the two colorings. They are available in the latest
> download.
> gravitation3d.com/magictile/downloads/MagicTile_v2.zip
> seeya,
> Roice
> On Sat, Jul 28, 2012 at 10:34 AM, Don Hatch wrote:
>
> Ah, I think I see your point...
> I'm now looking at the two pictures side-by-side as you suggested
> (interesting!)
> and the checkerboard regions do help me get my bearings
> as I correlate the two pictures. They are helpful
> in locating a particular *edge* of the {3,3,8}.
>
> They confuse me as I try to locate a particular cell, though.
> Don
> On Fri, Jul 27, 2012 at 01:26:39PM -0400, Don Hatch wrote:
> >
> >
> > Hmm, I don't know about the "help ground oneself" part...
> > I feel like the checkerboard areas are confusing me, more than
> helping,
> > in my effort to visually locate cells.
> > I really think no-two-of-same-color-at-a-vertex would be good.
> >
> > One other suggestion I think I forgot to mention before...
> > it would be nice to see one animation
> > with the "stationary" {3,n} and its neighbors colored,
> > and another with the initially inverted {3,n} and its neighbors
> colored.
> >
> > Don
> >
> > On Thu, Jul 26, 2012 at 08:23:53PM -0500, Roice Nelson wrote:
> > >
> > >
> > > I found a nice periodic (though irregular) 10-color painting of
> the
> > {3,8}
> > > using MagicTile. (aside: I think I can turn this into a
> vertex-turning
> > > puzzle, so I'll plan on that :D)
> > > http://gravitation3d.com/roice/math/ultrainf/338/38_10C.png
> > > Here is the {3,3,8} where the cells attached to the outer circle
> use
> > this
> > > coloring. It's cool to look at it side-by-side with the one
> above.
> > >
> http://gravitation3d.com/roice/math/ultrainf/338/338_neighbors_10C.png
> > > The 7C vertices make it easy to distinguish individual cells, and
> the
> > > checkerboard vertices give salient areas to help ground oneself,
> so I
> > > think this coloring would work quite well for the next animation.
> > > Roice
> > >
> > > On Thu, Jul 26, 2012 at 1:41 AM, Don Hatch wrote:
> > >
> > > As for coloring...
> > > yeah it won't be periodic,
> > > but I think it would be really helpful
> > > to get a coloring of the outer {3,n}
> > > in which the n tris around any vertex are n different colors.
> > > That would accomplish the goal of getting sufficient separation
> > > between any two cells of the same color in the {3,3,n},
> > > so that it's easier to tell which tris are from a common cell.
> > > (a 2-coloring of the {3,8} wouldn't accomplish this)
> > >
> > > I think the following coloring algorithm works:
> > > color each tri in order of increasing distance (of tri center,
> > > in hyperbolic space) from some fixed
> > > starting point, breaking ties arbitrarily.
> > > When choosing a color for a tri,
> > > at most n-1 of its 3*(n-2) "neighbor" tris have already been
> colored
> > > (I haven't proved this, but it seems to hold,
> > > from looking at a {3,7} and {3,8}).
> > > So color the new tri with any color other than
> > > the at-most-(n-1) colors used by its already-colored neighbors.
> > > Don
> > >
> > >
> >
> > --
> > Don Hatch
> > hatch@plunk.org
> > http://www.plunk.org/~hatch/
> >
> >
>
> --
> Don Hatch
> hatch@plunk.org
> http://www.plunk.org/~hatch/
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

--
Don Hatch
hatch@plunk.org
http://www.plunk.org/~hatch/

--X1bOJ3K7DJ5YkBrT
Content-Type: application/x-ygp-stripped
Content-Transfer-Encoding: 7bit

Content-Type: image/png
Content-Disposition: attachment; filename="38_8color_don.png"
Content-Transfer-Encoding: base64

--X1bOJ3K7DJ5YkBrT--

------=_NextPart_000_0018_01CD7931.C7708190
Content-Type: text/plain;
charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable

Hi Nelson,

Your interaction with the program MT is fascinating. It is typical for our =
modern time and new since Madelbrot has looked at his first prints.
"Computer aided", "constructive" math allows very new and possibly deep ins=
ights.

Ed

----- Original Message -----=20
From: Roice Nelson=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Monday, August 13, 2012 1:16 AM
Subject: Re: [MC4D] More {3,3,8} pics and new {3,8} MagicTile puzzles


=20=20=20=20
Nice!



Sorry for the long delay in responding. I'm not able to describe the con=
struction as well as you, because the MagicTile code is doing most of the w=
ork for me. The way it works is that you construct a set of "identificatio=
ns". You do this by configuring edges to reflect across to go from the cen=
tral white tile to an identified copy. It internally calculates the associ=
ated isometries, and uses those to recursively copy tiles all over the plan=
e. One consequence is that you can take some tile to one of its copies, an=
d the whole coloring will remain unchanged. So while your coloring has mul=
tiple kinds of red tiles, every red tile in my coloring is the same - each =
is surrounded by the purple/cyan/green tiles. MagicTile could not reproduc=
e the coloring you've found (without setting multiple tiles to the same col=
or). I think your coloring might be a 16-coloring in disguise.


Anyway, here's how I went. I started by mentally grouping together a set=
of 8 triangles into an octagon, then picked an identification which would =
make copies of the 8-color pattern along an h-line. This gave me a line of=
stacked octagons where every other octagon was mirrored. The h-line goes =
through the center of the white/green tiles. Then I picked additional iden=
tifications to fill in the two areas to either side of this h-line, and tho=
se resulted in the stripe of 3 colors. Though I've gotten an intuitive fee=
l for configuring these, it still involves trial and error for me, and it's=
magical when things "click" together, filling in the whole plane. Let me =
know if you'd like more info on the MagicTile configuration (they are xml f=
iles, editable by hand). The format is not perfect for sure, but I did try=
to make it clear.


Btw, Melinda described the {3,3,8} image to me as consisting "of a full c=
olor background with an overlaid black & white lace doily with windows thro=
ugh which you can see parts of the background". I liked that mental image.=
The "full color background" without the lace looks like a colored {3,8} t=
iling in the disk, plus its inversion in the disk boundary, like this:

www.gravitation3d.com/roice/math/ultrainf/338/38_8C_with_inverse.png

Here is the {3,3,8} image again, for reference:


http://gravitation3d.com/roice/math/ultrainf/338/338_neighbors_8C.png=20

seeya,
Roice






On Thu, Aug 2, 2012 at 1:00 PM, Don Hatch wrote:

Hey Roice,

These are looking great. I get a good sense of where the cells are
now, especially in the half-plane one.

How did you construct your 8-coloring for the {3,8}?

I came up with a periodic coloring, but it's not the same as yours--
I see that yours has stripes of 3 colors (grey,yellow,blue)
going through it, but mine doesn't have any such stripes.

I'm attaching an image of mine (not sure whether this will work).

Here's how I construct it...
Imagine the {3,8} partitioned into an {8,4}
(8 triangles of the {3,8} in each octagon of the {8,4}).
Call half of the octagons "even" and the other half "odd",
in a checkerboard pattern.
Start with any even octagon, and color its 8 triangles
counterclockwise: 0 1 2 3 4 5 6 7.
Then for each of the eight even octagons
"diagonal" to the first even octagon,
again color it CCW with 0 1 2 3 4 5 6 7,
in the same orientation as the first even octagon
(so, for example, one of them will have its 4 5
sharing a vertex with the first octagon's 0 1).
Continue in this way, coloring all even octagons.

Finally, for each not-yet-colored triangle (in an odd octagon),
color it ((i+2) mod 8) where i is the color of its
already-colored neighbor triangle (in an even octagon).
This gives each odd octagon the colors 0 3 6 1 4 7 2 5, counterclockwis=
e.

((i+6) mod 8) could be used instead of ((i+2) mod 8).

Don



=20=20
------=_NextPart_000_0018_01CD7931.C7708190
Content-Type: text/html;
charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable



>