Message #1314

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] 8Colors solved
Date: Thu, 30 Dec 2010 12:14:31 -0600

@andrey
Thanks for correcting my thoughts about the chirality of the pattern.
Sounds like I was on the right track by observing the two unique pieces
during a twist, but jumped to conclusions as to why they were behaving
differently (classic case of trying to extrapolate using too few examples).

About ctrl-left clicking, the same frustrating thing was happening to me in
the beta, so I’m glad you decided to turn it off. I would recommend
updating the help text to be current though, and maybe using a different key
sequence to still offer recentering functionality.

@melinda
For choosing the targets, I just tried to look at all the cells close by.
Since there can be non-orientable topologies where some nearby copies are
mirrored and others are not (an example in the 2D case is a hexagonal
periodic painting that is topologically a Klein bottle), I was aware that
looking at only a few nearby cells wouldn’t prove things are orientable. So
in truth, I didn’t have 100% confidence in my claim that the 8Colors puzzle
was orientable. Perhaps there is some rule by which you could guarantee
you’ve looked at enough copies, though I don’t know what that might be. In
any case, I agree with your guess :)

In thinking about this, I noticed the 3-color hexagonal puzzle in MagicTile
is strange in that the state after a twist would appear the same whether
copies were twisted clockwise or counterclockwise. So it appeared to be
able to simultaneously behave in both an orientable and non-orientable
fashion. It was fun to think about, and I figure the reason it works either
way is that each face already has both left-handed and right-handed versions
of 3C pieces.

All the best,
Roice

On Tue, Dec 28, 2010 at 3:53 PM, Melinda Green <melinda@superliminal.com>wrote:

>
>
> Hey, that is a really clever idea, Roice! I don’t know if I would have
> thought of that method but now that you point it out I see that it is
> definitely the most natural way to answer that question. One thing you
> didn’t mention was your method of choosing your target copy. I think that
> you would need to choose a face that appears to twist in the opposite
> direction to the starting face of the 4C piece that you then send on its
> journey to the copy face. I’m going to guess that puzzles are always
> orientable when twists of face copies are all in the same direction as the
> clicked face. I’m a little surprised that the inverse is not always true.
>
> -Melinda
>
> On 12/28/2010 9:02 AM, Roice Nelson wrote:
>
> […]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.
>
>
>
>