Thread: "8Colors solved"

Yes, 3-cycle on 4C is not a bug, it's really possible situation. But may wa=
y to resolve it was too long (more than 1500 twists?).

Andrey

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Very impressive, Andrey! Of course we have all come to expect greatness
from you, and with regularity, so I would like to suggest that we take a
step back for a moment and get a sense of what this accomplishment
means. In many ways the higher-dimensional puzzles we have been getting
used to can be understood well enough through dimensional analogy but
these hyperbolic puzzles require you to wrap your mind around warped
spaces which seems like a whole different story. In some ways there are
direct connections between hyperbolic and higher dimensional spaces as
we've seen with the Roice's embedding of the 3D cube and Klien's quartic
into hyperbolic 2D space, but hyperbolic 3D spaces are literally
mind-bending because we live in a 3D space and it seems impossible to
see these new spaces from the outside. It feels to me like the
difference between solving a 2D maze on paper versus solving a hedge
maze from the inside. I have no comfortable frame of reference from
which to solve it or even to fully grasp what I am seeing. Without
having seen Roice's MagicTile in action I would have a lot more
difficulty understanding this display.

The one thing that is clear is that these are some amazingly beautiful
objects, and the mechanisms behind their function are even more
impressive. Don, Roice and Andrey are programming gods who have breathed
life into these clockwork universes. They give them to us and then dance
within them. Andrey's new solution belongs in a shrine. It begs the
question of whether he is god enough to create a puzzle so difficult
that even he can not solve it. I think that the answer is no but
something tells me that he will continue trying to show me that I am wrong.

I love that he implemented twist animations. That really helps to show
its finite nature. I notice that in some cases only the clicked face
animates and other affected faces only change state at the end. I don't
think that is a bug because it only seems to happen at the edges where
the tiling stops. Other than that the puzzle now seems very complete. I
particularly like the contiguous-cubies mode when setting the
face-shrink slider to its max. That mode never worked very well in MC4D
because it was never possible to see through the clutter, but the extra
room in hyperbolic space allows exactly that and in many ways seems like
it could be as useful as it is pretty. It makes the pieces look like so
much candy!

A couple of embellishments that I would like to put on the wish list
include

* A cross-eyed stereo view. A stereo view is the most important
thing that I intend to add to MC4D. I had it sort of working but
not perfectly so I took it out for the moment. I think that it
will be very helpful when solving any of these puzzles when more
than a few hundred stickers are visible because the added depth
makes the view much easier to grok.
* A slider to control how much of the hyperbolic space to draw. The
animation is so fast on my machine that it can clearly handle a
lot more pieces.
* Highlighting by sticker or piece. This will look far more
impressive than it does in MC4D since stickers and pieces will
light up all over the puzzle. it will also help when demonstrating
the finite nature of the puzzle to others. Currently only the
twist animations make that clear but that also changes the puzzle
state.

Well it's time to tear myself from this magical land and be social for a
few days. I wish that I had better ways to show all this beauty to
people without frightening them but I get an incredible amount of
pleasure just knowing that these things exist and that a few people can
grapple with them just fine. Thanks again Andrey and congratulations on
achieving the first ever solution of a 3D hyperbolic twisty puzzle, and
thank you for this wonderful holiday gift.

-Melinda

On 12/24/2010 1:29 PM, Andrey wrote:
> Yes, 3-cycle on 4C is not a bug, it's really possible situation. But may way to resolve it was too long (more than 1500 twists?).
>
> Andrey


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Very impressive, Andrey! Of course we have all come to expect
greatness from you, and with regularity, so I would like to suggest
that we take a step back for a moment and get a sense of what this
accomplishment means. In many ways the higher-dimensional puzzles we
have been getting used to can be understood well enough through
dimensional analogy but these hyperbolic puzzles require you to wrap
your mind around warped spaces which seems like a whole different
story. In some ways there are direct connections between hyperbolic
and higher dimensional spaces as we've seen with the Roice's
embedding of the 3D cube and Klien's quartic into hyperbolic 2D
space, but hyperbolic 3D spaces are literally mind-bending because
we live in a 3D space and it seems impossible to see these new
spaces from the outside. It feels to me like the difference between
solving a 2D maze on paper versus solving a hedge maze from the
inside. I have no comfortable frame of reference from which to solve
it or even to fully grasp what I am seeing. Without having seen
Roice's MagicTile in action I would have a lot more difficulty
understanding this display.



The one thing that is clear is that these are some amazingly
beautiful objects, and the mechanisms behind their function are even
more impressive. Don, Roice and Andrey are programming gods who have
breathed life into these clockwork universes. They give them to us
and then dance within them. Andrey's new solution belongs in a
shrine. It begs the question of whether he is god enough to create a
puzzle so difficult that even he can not solve it. I think that the
answer is no but something tells me that he will continue trying to
show me that I am wrong.



I love that he implemented twist animations. That really helps to
show its finite nature. I notice that in some cases only the clicked
face animates and other affected faces only change state at the end.
I don't think that is a bug because it only seems to happen at the
edges where the tiling stops. Other than that the puzzle now seems
very complete. I particularly like the contiguous-cubies mode when
setting the face-shrink slider to its max. That mode never worked
very well in MC4D because it was never possible to see through the
clutter, but the extra room in hyperbolic space allows exactly that
and in many ways seems like it could be as useful as it is pretty.
It makes the pieces look like so much candy!



A couple of embellishments that I would like to put on the wish list
include

• A cross-eyed stereo view. A stereo view is the most important
  thing that I intend to add to MC4D. I had it sort of working but
  not perfectly so I took it out for the moment. I think that it
  will be very helpful when solving any of these puzzles when more
  than a few hundred stickers are visible because the added depth
  makes the view much easier to grok.


• A slider to control how much of the hyperbolic space to draw.
  The animation is so fast on my machine that it can clearly
  handle a lot more pieces.
  
  


• Highlighting by sticker or piece. This will look far more
  impressive than it does in MC4D since stickers and pieces will
  light up all over the puzzle. it will also help when
  demonstrating the finite nature of the puzzle to others.
  Currently only the twist animations make that clear but that
  also changes the puzzle state.
  
  

Yes, 3-cycle on 4C is not a bug, it's really possible situation. But may way to resolve it was too long (more than 1500 twists?).

Andrey

Melinda,
Thank you very much for your words. Yes, it's difficult to find something=
familiar in this H3 world, so for 8C I took the most simple way: I worked =
only in contiguous-cubies mode, and most twists were of one face and of "op=
posite" faces - only around 2C stickers adjacent to the first face. There w=
as easy to build 6-twists macro for 3-loop of 3C, 14-twists for the pair of=
4C transpositions (like 3^3 (FRF'R')^3 operation) and 30-twists for rotati=
on of two 4Cs (hmm... it could be 14-twist? need to check). Everything went=
smoothly and I was very surprised when twist counter showed 1400 twists, t=
hen 2300, 3300...

To create puzzle that I can't solve? I think it's easy. You see, even simpl=
est 4D douprizms are too hard for me now :) So something deep-cut and 4D ea=
sily may be out of my reach...

BTW, next game will probably be 5D Pacman - just for fun and to take a rest=
of puzzles :) I think that I see good interface for it.

Yes, some stickers are not animated. It happens if there is no 2C sticker c=
onnected to the moving face. Unfortunately, mathematics in the geometry par=
t is not as elegant as I hoped, so some internal navigation in data structu=
res is almost impossible witout risk of serious bugs in the code.

As for the wish list:
- I'm not going to implement stereo view in the nearest future: screen is a=
lready too small for this puzzle.
- Slider with the computed area? May be, I'll add one or two layers, but nu=
mber of stickers increases exponentionally with the radius of the area. And=
recalculation of geometry will be slow and probably you'll loose position =
of camera during it. So it will be operation that you want to do once in a =
game. May be, it's better to put it in the menu.
- sticker highlighting is implemented. It is shift-click or shift-right-cli=
ck. One of them highlights a piece and a place for it, and another highligh=
ts a piese that should go there (like in 120Cell program).

I'll add to the list such feature as the checkbox "reverse twist direction"=
(very useful when you work in contiguous-cubies mode) and something that h=
elps to see 3C cubies that are hidden below 4C. May be it will be mask "sho=
w/hide stickers of particular type" or just a flag/bar "shrink corners".

Andrey

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A single twirled 4C piece was very surprising to me! At first blush, I
would expect things to behave exactly as described in section 6d of
the Keane/Kamack
paper .

We've seen single twirled corners on the 5D puzzle, and the allowance for
that has to do with being able to mirror pieces through four dimensions
(using the 5th dimension), as described by
Don.
So my initial guess was that the 8-Colors puzzle must therefore have a
non-orientable topology, which
would allow mirrored pieces to show up, and that a combination of
reflections could give the observed result. But the topology is orientable.
(For any interested, to check to see if the puzzle was non-orientable, I
traced paths from a cell to its copies, then looked to see if any of the 4C
pieces in the copy appeared mirrored relative to the parent cell.)

At a loss for an explanation, what I tried next was to count the number of
4C pieces connected to a given cell (fourteen), and in doing so, I think I
stumbled onto the cause of this. If you look at how these 4C pieces move
during a 2C twist of this cell, there are two sets of 6 that change
positions in 6-cycles. But the remaining two 4C pieces change positions in
2-cycles, with their orientations twirling a third after each two cycle.
This seems to be the root of the unusual behavior, though I still haven't
nailed down exactly which part of Keane/Kamack's analysis no longer applies.

Why are there two 4C pieces that have this unique behavior during each 2C
twist? I think the cause of that has to do with the fact that the periodic
painting on the 8-colored puzzle is
chiral.
You can get a sense of this by studying the pattern of colors on a pristine
puzzle from within one of the cells (such that you can easily see the color
pattern on the hexagonal grid). Contrast with the 12-Colors puzzle, where
the periodic painting is not chiral. In the latter, you have eighteen 4C
pieces connected to every face, which breaks down into 3 sets that each
permute positions in 6-cycles, so there are no uniquely behaving pieces.
I'm placing my bet that a single twirled corner on the 12-Colors puzzle is
impossible.

I would of course appreciate further insights or corrections to my thoughts
about this unusual behavior! One last comment on the topic though - the
single twirled piece is noteworthy because, unlike the other kinds of
impossible-looking states usually encountered, this is not due to an
even/odd (binary) phenomenon, but instead a trinary one. So "parity" is
probably not a word that should even enter the discussion :D

I want to add to the chorus and also say* thank you to Andrey* very much for
the new goodies in MHT633. I really appreciate the Poincare Ball view in
particular. It's very nice, and I consider it the closest possible visual
analogue to MC4D in terms of display and controls. I like that you've
matched the controls so well, though to be as close as possible, I would
recommend switching the directions in the left-click dragging (so that the
dragging pulls the front of the ball model, rather than the rear). I'd also
like to echo others thoughts on how nice the animated twisting is for
understanding.

Here is some *very minor* feedback for things I ran across while playing
with the latest version (some of it probably not even worth fixing):

- The ctrl-left-click to set a new center feature looks to no longer work.

- If you left-click to twist while it is still animating the previous
twist, both twists are instantaneously applied.

- Ctrl-z to undo does not always undo relative to the same clicked sticker
when animating, but instead always undoes relative to some fundamental
location. This only looked noticeable when you are viewing the puzzle from
within a cell, and the end result of the state is the same regardless, so I
don't think it is worth spending time on. In fact, it was a nice insight
for me for further possibilities in MagicTile (I didn't realize things fit
together so well in a hexagonal periodic painting, so I see now there are
some additional slice-masked twists I could allow).

Thanks again Andrey! MHT633 is the ultimate Rubik analogue!

All the best,
Roice



On Fri, Dec 24, 2010 at 3:29 PM, Andrey wrote:

> Yes, 3-cycle on 4C is not a bug, it's really possible situation. But may
> way to resolve it was too long (more than 1500 twists?).
>
> Andrey
>

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