Message #2371

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] More {3,3,8} pics and new {3,8} MagicTile puzzles [1 Attachment]
Date: Sun, 12 Aug 2012 18:16:17 -0500

Nice!

Sorry for the long delay in responding. I’m not able to describe the
construction as well as you, because the MagicTile code is doing most of
the work for me. The way it works is that you construct a set of
"identifications". You do this by configuring edges to reflect across to
go from the central white tile to an identified copy. It internally
calculates the associated isometries, and uses those to recursively copy
tiles all over the plane. One consequence is that you can take some tile
to one of its copies, and the whole coloring will remain unchanged. So
while your coloring has multiple 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 reproduce the coloring you’ve found (without setting
multiple tiles to the same color). 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
identifications to fill in the two areas to either side of this h-line, and
those resulted in the stripe of 3 colors. Though I’ve gotten an intuitive
feel 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 files, 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
color background with an overlaid black & white lace doily with windows
through 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} tiling 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

seeya,
Roice

On Thu, Aug 2, 2012 at 1:00 PM, Don Hatch <hatch@plunk.org> 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, counterclockwise.
>
> ((i+6) mod 8) could be used instead of ((i+2) mod 8).
>
> Don
>
>