Message #1942

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: new {5,4} puzzles
Date: Mon, 28 Nov 2011 21:59:01 -0600

Thank you for your thoughts, Andrey. I was glad to read you know there are
more possibilities for the {5,3,4}, and this encouraged me to further
experiment with {5,4} cell identifications. I found some more that work,
so there are more puzzles now :)

I think the two paintings I’ve starred could possibly be used as a basis
for a {5,3,4} painting, since they have the following property: Center any
color, and a 1/5th rotation of the whole plane will take copies of that
color to copies. (The existence of a non-orientable coloring with this
property surprised me a little, since in the past we tried and failed to
find a non-orientable {6,3} tiling that could do this with a 1/6th
rotation).

Both the 12-Color and 24-Color patterns do not have identified cells that
are adjacent, which allowed me to deepen the FT cuts. The 12-Color is an
especially lovely puzzle, and I uploaded a picture of it in the pristine
state<http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/1983285309/view>
.

If there is a 22-color {5,3,4}, I wonder if there is an orientable {5,4}
painting with less than 24 colors and that special property that copies
will get rotated to copies during a 1/5th rotation of the plane. I haven’t
found one yet though. Your algebraic fields techniques may be a faster way
to track paintings down than my experiments with MagicTile configurations…

seeya,
Roice


On Mon, Nov 28, 2011 at 2:56 AM, Andrey <andreyastrelin@yahoo.com> wrote:


> Hi Roice,
> it’s very interesting. I haven’t check colorings of {5,4} yet, but it
> shouldn’t be very difficult. As for {5,3,4}, I’m sure that there is
> 22-colors pattern (when dodecahedra on the opposite sides of some
> dodecahedron have same color), and much more others. Finite algebaric
> fields technique should work fine for this honeycomb.
>
> Andrey
>