# Message #1631

From: David Smith <djs314djs314@yahoo.com>

Subject: Re: [MC4D] Order-2 Klein’s Quartic permutation count

Date: Sat, 30 Apr 2011 14:21:29 -0700

No problem! I thought you would like to see the Klein’s Quartic permutation counts. Yes, I would love to join the wiki (again!); please sign me up! (The page instructs me to contact either you or Melinda.) Should I post every time I complete a calculation? That would almost certainly be unnecessary and flood the message board. But, I’ll certainly let at least you know Roice, as the creator of MagicTile, and also update the wiki. Also, I’ve just created a new folder in the files section with all of my relevant work on Magic120Cell and the original MagicCube4D. When I complete the MagicTile and MagicCube4D 4.0 formulas, I’ll create a file for each with all of my results and include them there. Then, maybe I could research some of Andrey’s programs; that would be fun.

Congratulations on working on a new MagicTile version! I’m looking forward to it! My main point was that as MagicTile displays the puzzles now; the number of visually different configurations of the puzzle itself is different (smaller) than the number of visually different screenshots one could take of MagicTile. This is because of the fixed position and orientation of the tiles. If we could switch the center tile for another and rotate the entire puzzle, as it will be in your future version, then the two counts would be identical, despite the finite number of tiles displayed. I then clarified what should be meant by ‘visually distinguishable’ as it relates to the actual permutations of Klein’s Quartic, since I had just used it in two different ways, and gave an example making clear that we technically need to view Klein’s Quartic from within the hyperbolic plane to correctly count the number of permutations of the structure itself (your future

version will artificially simulate this from a Euclidean perspective). You are correct, reorientations should not affect the permutation counts, and these reorientations cannot be performed in MagicTile, thus artificially raising the number of visually (screen-wise) different positions, as two differently appearing positions would in fact be the same position once a reorientation is applied. I think I now see why you became confused; at first glance saying ‘reorientations do not affect the permutation counts’ and ‘the lack of the ability to reorient the position makes the two counts different’ appears to be nonsensical. If the faces had fixed centers, then it would be nonsensical. :)

I still believe that my count is correct, despite the fact that I am new to counting MagicTile permutations. I’ll continue to check it for a few minutes, and then move on to the order-3 puzzle. I appreciate that you are genuinely interested in my results! That will keep me motivated to continue, despite the fact that I already find it fun. :) I look forward to tackling the Rubik’s Cubes next, and verifying that they are the same as the known counts for the actual cubes. Then I will be convinced that I haven’t made some subtle error with the Klein’s Quartic counts.

Thanks again to everyone! I look forward to continuing my work. :)

All the best,

David

— On Sat, 4/30/11, Roice Nelson <roice3@gmail.com> wrote:

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Order-2 Klein’s Quartic permutation count

To: 4D_Cubing@yahoogroups.com

Date: Saturday, April 30, 2011, 4:25 PM

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 :)

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