Thread: "new {5,4} puzzles"

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Hi all,

I added some fun {5,4} puzzles. With cells identified, the tiling for
these new puzzles has a checkerboard pattern with *only two* colors.

The FT version is trivial because a twist does not change the puzzle state,
so I didn't include it.

The ET version is enjoyable, even if not very difficult. A
checkerboard
of
the stickers is easy to make on this puzzle.

I haven't figured out the VT versions, but they seem pretty coupled and
should be more difficult. Here is something unusual about them... Each
slicing circle is "double covered". The identified vertices of two adjacent
cells (of a given color) are at the same location, so there are really two
overlapping circles there. The puzzle still makes sense when twisting
because of the checkerboard color pattern around vertices.

For some time now, there have already been 6-color and 12-color puzzles
based on the {5,4} tiling. The 6-color is non-orientable, the 12-color
orientable, and the patterns of both are strange. I don't think those
colorings can be used as a basis for a {5,3,4} periodic painting, which we've
wondered about in the past
here,
because
the coloring does not remain the same after a 1/5th rotation of the entire
plane. I hope I'm wrong, but I suspect that the maximal coloring for the
{5,3,4} has just two colors. However, VT and ET puzzle versions of that
honeycomb would still be very neat to see.

Cheers,
Roice

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Hi Roice,
it's very interesting. I haven't check colorings of {5,4} yet, but it sho=
uldn'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 s=
ame color), and much more others. Finite algebaric fields technique should =
work fine for this honeycomb.

Andrey

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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
.

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 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
>

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Thank you for your thoughts, Andrey.=A0 I was glad to read you know there a=
re more possibilities for the {5,3,4}, and this encouraged me to further ex=
periment with {5,4} cell identifications. =A0I found some more that work, s=
o there are more puzzles now :)

• 4-Color, Orientable
• 12-Color, Non-Orientable *
• 16-Color=
  , Non-Orientable
• 24-Color, Orientable *