# Message #2367

From: Don Hatch <hatch@plunk.org>

Subject: Re: [MC4D] More {3,3,8} pics and new {3,8} MagicTile puzzles

Date: Thu, 02 Aug 2012 14:00:43 -0400

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 <hatch@plunk.org> 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/

>

> ————————————

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–

Don Hatch

hatch@plunk.org

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