# 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

> >

>