Thread: "MagicTile Coloring"

Hi Roice,

Have you seen my description of the organisation of "MT skew {4,6|3} 30 v02=
0" ?
Each 3-prismatique edge can be untwisted or twisted by +60=B0 or -60=B0
(separated from tetrahedron-vertex and reglued). So 10^3 different coloring=
s can be constructed. That's a lot. Are some of them equivalent?
Is it difficult to find the corresponding "edge-sets"?

Kind regards
Ed

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cool, I made checkerboards on both the 2^3 and 4^3 "Mirror & Twist" cubes!
I remember probably 15 years ago setting out to try that on a physical
4^3, and giving up an hour or so later, mostly convinced it was impossible.
And of course, what I was trying is impossible. But not here :)

Happy New Year all,
Roice


On Sun, Dec 30, 2012 at 6:01 PM, schuma wrote:

> Hi RefleCube solvers. Thank you all for your support.
>
> Since you guys find these puzzles interesting and have solved them
> quickly, I just added several sizes: 2x2, 4x4 and 5x5. For each size all
> the mirroring styles are supported. Use shift+click and alt+click to turn
> the deeper layers.
>
> Imagine what kind of weird parities you'll see on the 4x4. Have fun!
>
> Nan
>
>

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cool, I made checkerboards on both the 2^3 and 4^3 "Mirror & Twist=
" cubes! =A0I remember probably 15 years ago setting out to try that o=
n a physical 4^3, and giving up an hour or so later, mostly convinced it wa=
s impossible. =A0And of course, what I was trying is impossible. =A0But not=
here :)

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Hi Ed,

Yep, I've been following your posts and progress. Nice job on all your
solves btw! Getting to the half-way mark would be a big milestone, and I
hope you make it.

I may be missing something, but it seems that if you recolor one of the
triangular prisms by cycling the 3 colors on it, the puzzle hasn't really
changed. So it seems to me that all of the colorings you are describing
are equivalent. It is still a 30-faced puzzle with 30 colors, connected up
with the same global topology. The edge sets in this puzzle had to be made
to fit the topology of the {4,6|3} skew polyhedron, and changing the edge
sets would change the topology (resulting in some other shape).

But maybe you are thinking something else. Are you talking about twisting
up one of the triangular prisms and re-gluing, such that one triangle base
remains unchanged and the other is rotated 60 degrees? If so, that would
indeed be different, but the resulting shape wouldn't be this skew
polyhedron, and MagicTile can't currently support something like this.

Here's some links I used when making these two puzzles. They might be
helpful for further study.

- For the {4,6|3}, the wikipedia page on the runcinated
5-cell
.
- For the {6,4|3}, the wikipedia page on the bitruncated
5-cell
.
- Also, see the section 'Finite regular skew polyhedra of
4-space_skew_polyhedra_of_4-space>'
for other topology possibilities (unfortunately, most would have too man=
y
faces to make good puzzles).

Let me know if I'm on track with my understanding.

seeya,
Roice


On Mon, Dec 31, 2012 at 1:03 PM, Eduard wrote:

> Hi Roice,
>
> Have you seen my description of the organisation of "MT skew {4,6|3} 30
> v020" ?
> Each 3-prismatique edge can be untwisted or twisted by +60=B0 or -60=B0
> (separated from tetrahedron-vertex and reglued). So 10^3 different
> colorings can be constructed. That's a lot. Are some of them equivalent?
> Is it difficult to find the corresponding "edge-sets"?
>
> Kind regards
> Ed
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

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Hi Ed,

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Thanks very much.

Yes, I was aware that simply turning the prismes has no impact.
I didn't read the articles about skew polyhedra before. Im glad that my de=
sriptions of the "organisation" is correct (but evidently not complete).=20

For new colorings: it is now evident for me that finding such colorings is =
a very deep and delicate process.

I have finished now the deep cut runcinated 5-cell v020 for which the "theo=
rem of Baumann" applies (no pieces stay at home). Counting more carefully t=
he orbits (not an easy job) I had to reformulate the theorem.
The "theorem of Astrelin" applies to bitruncated 5-cell v200 where no piece=
s stay at home.

Theorem of Baumann old, erroneous

=20

A 90-deg rotation of the whole puzzle "MT skew {4,6|3} 30 v020" (Roice Nels=
on) around some face gives odd permutation of the edges (15 4-loops) and ev=
en permutation of centers (6 4-loops + 1 fixed points). This is reducible t=
o a single edge swap.

Theorem of Baumann new, corrected

=20

A 90-deg rotation of the whole puzzle "MT skew {4,6|3} 30 v020" (Roice Nels=
on) around some face gives even permutation of the edges (14 4-loops and 2 =
2-loops) and odd permutation of centers (7 4-loops + 2 fixed points). This =
is reducible to a single center swap.

The edge 2-loops are about the two opposite colors of the rotation axis (he=
re white and (192,192,0)).

Conclusion of the theorems: after scrambling avoid turning of the whole bef=
ore you have fixed the home positions.

Best regards
Ed

----- Original Message -----=20
From: Roice Nelson=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Tuesday, January 01, 2013 10:47 PM
Subject: Re: [MC4D] MagicTile Coloring


=20=20=20=20
Hi Ed,



Yep, I've been following your posts and progress. Nice job on all your s=
olves btw! Getting to the half-way mark would be a big milestone, and I ho=
pe you make it.


I may be missing something, but it seems that if you recolor one of the t=
riangular prisms by cycling the 3 colors on it, the puzzle hasn't really ch=
anged. So it seems to me that all of the colorings you are describing are =
equivalent. It is still a 30-faced puzzle with 30 colors, connected up wit=
h the same global topology. The edge sets in this puzzle had to be made to=
fit the topology of the {4,6|3} skew polyhedron, and changing the edge set=
s would change the topology (resulting in some other shape).


But maybe you are thinking something else. Are you talking about twistin=
g up one of the triangular prisms and re-gluing, such that one triangle bas=
e remains unchanged and the other is rotated 60 degrees? If so, that would=
indeed be different, but the resulting shape wouldn't be this skew polyhed=
ron, and MagicTile can't currently support something like this.


Here's some links I used when making these two puzzles. They might be he=
lpful for further study.
a.. For the {4,6|3}, the wikipedia page on the runcinated 5-cell.=20
b.. For the {6,4|3}, the wikipedia page on the bitruncated 5-cell.
c.. Also, see the section 'Finite regular skew polyhedra of 4-space' fo=
r other topology possibilities (unfortunately, most would have too many fac=
es to make good puzzles).=20
Let me know if I'm on track with my understanding.


seeya,
Roice




On Mon, Dec 31, 2012 at 1:03 PM, Eduard wrote:

Hi Roice,

Have you seen my description of the organisation of "MT skew {4,6|3} 30=
v020" ?
Each 3-prismatique edge can be untwisted or twisted by +60=B0 or -60=B0
(separated from tetrahedron-vertex and reglued). So 10^3 different colo=
rings can be constructed. That's a lot. Are some of them equivalent?
Is it difficult to find the corresponding "edge-sets"?

Kind regards
Ed



------------------------------------

Yahoo! Groups Links







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