Message #1945

From: Andrey <andreyastrelin@yahoo.com>
Subject: [MC4D] Re: new {5,4} puzzles
Date: Fri, 02 Dec 2011 20:18:23 -0000

Hi Roice,
Something was wrong in my mathematics. Now my calculations give different result:

Field F_25 gives 24-color {5,4} and 130-color {5,3,4}
Field F_81 gives 72-color {5,4} and 4428-color {5,3,4}
Field Z_11 gives 264-color {5,4}
Field Z_19 gives 1368-color {5,4}
Field Z_29 gives 406 colors for both {5,4} and {5,3,4}

So probably there is no good puzzles in {5,3,4}. Only hope is that thee will be good subgroup of 26 elements for F_25 or 29 or 58 elements for Z_29.

Andrey

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:
>
> 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 :)
>
> - 4-Color, Orientable
> - 12-Color, Non-Orientable *
> - 16-Color, Non-Orientable
> - 24-Color, Orientable *
>
> 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@…> 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
> >
>