Message #1630

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Order-2 Klein’s Quartic permutation count
Date: Sat, 30 Apr 2011 15:25:21 -0500

Thanks David,

I’ve always been interested to know KQ permutation counts, and I’ll
definitely look forward to seeing the number for the order-3 puzzle! I have
to admit I never intended for the order-2 puzzle to be twistable. How it is
right now happened at some point without me noticing. I left it in when I
noticed though, because it is a workable puzzle even if material overlaps
during a twist. However, it doesn’t really fit the mold of puzzles I had
intended for the program.

It’s up to Melinda, Don and Roice if they want to use my results at all.
> There is obviously no pressure to make use of them in any way. I’m just
> doing it in my spare time for enjoyment, and to try to contribute to this
> mailing list in any small way I can. :) There are going to be many results,
> and who needs all of these formulas anyway? :)
>

I think a great use for any numbers you feel motivated to calculate would be
to list them in the wiki <http://wiki.superliminal.com/wiki/Main_Page> :)

I should mention that the above count is the number of visually
> distinguishable permutations of Klein’s Quartic. Note that this number is
> different from the number of visually distinguishable permutations we see on
> the screen of MagicTile, because in MagicTile the center is fixed in both
> position and rotation. (The fact that there are a finite number of pieces
> would actually not come into play from this point of view.) The goal of the
> permutation count is to count the number of actual Klein’s Quartic positions
> that can occur in the hyperbolic plane, not what we can see on the screen.
> (‘Visually distinguishable’ is from the point of view of being inside the
> hyperbolic plane, as we always count visually distinguishable permutations
> from the point of view of the space it resides in, just as I did for the n^4
> and n^d cubes. Otherwise, from a Euclidean point of view we would see the
> hyperbolic plane as a unit disc, and some pieces would appear larger than
> others.)
>

I’m not sure I’m completely following the distinction you are pointing out.
Is it just that the projection from the hyperbolic plane to the disk model
warps the pieces, and this should be ignored? The next version, which I’ve
made some good progress on, supports both rotating and hyperbolic panning
(allowing arbitrary reorientations), so equivalence of piece shapes should
be more directly obvious. As with the cubes, I would think reorientations
of the puzzle should not affect the permutation counts, as I’m sure you’re
doing. Please let me know if I missed what you intended to point out, since
I probably did.

seeya,
Roice