Thread: "Order-2 Klein's Quartic permutation count"

From: "djs314djs314" <djs314djs314@yahoo.com>
Date: Sat, 30 Apr 2011 15:35:30 -0000
Subject: Order-2 Klein's Quartic permutation count



Hello everyone,

I've calculated the number of permutations of the order-2 Klein's Quartic f=
rom MagicTile and got the following:

209!/(2*((7!)^29)*(6!)) =3D=20

149112918741470338254592050364360405502211555755239620088938217626100291817=
023191892506093747359259502835790155080391360482757810381809070108373294909=
299122149317278822861377950180063964883966622624746304806476578418430711794=
2283643839214638774130151970617008167321600000000000000000000

I have done this calculation in half an hour, so there is a possibility of =
error. I'm leaving for some appointments soon, but still wanted to get thi=
s post in before I leave. :) I apologize if this value needs to be correct=
ed.

I'm planning to calculate all of the MagicTile puzzles' permutations, and a=
ll of the MagicCube4D puzzles' permutations. Perhaps I'll even try the 'in=
vent your own' option which would cover all of the puzzles and an infinitud=
e of others, if I grow two more brains by then. ;) Also, I'm going to calc=
ulate the specific values for each puzzle (e.g. the order-2 Klein's Quartic=
, the order-3 Klein's quartic, etc.) Then, I will generalize them into a g=
eneral formula for each type of puzzle for arbitrary order.

It's up to Melinda, Don and Roice if they want to use my results at all. T=
here is obviously no pressure to make use of them in any way. I'm just doi=
ng it in my spare time for enjoyment, and to try to contribute to this mail=
ing list in any small way I can. :) There are going to be many results, an=
d who needs all of these formulas anyway? :)

I should mention that the above count is the number of visually distinguish=
able permutations of Klein's Quartic. Note that this number is different f=
rom the number of visually distinguishable permutations we see on the scree=
n of MagicTile, because in MagicTile the center is fixed in both position a=
nd rotation. (The fact that there are a finite number of pieces would actua=
lly not come into play from this point of view.) The goal of the permutati=
on count is to count the number of actual Klein's Quartic positions that ca=
n occur in the hyperbolic plane, not what we can see on the screen. ('Visua=
lly distinguishable' is from the point of view of being inside the hyperbol=
ic 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 hyperb=
olic plane as a unit disc, and some pieces would appear larger than others.=
)

I would like to once again thank Melinda, Don, Jay, Roice, Audrey and anyon=
e else I have missed for creating these excellent puzzles! :) They have pr=
ovided much enjoyment for myself and so many others.

(It looks like my other message is going to be duplicated. My apologies.)

All the best,
David



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



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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

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Content-Transfer-Encoding: quoted-printable

Thanks David,


I've always been interested to know KQ=
permutation counts, and I'll definitely look forward to seeing the num=
ber for the order-3 puzzle! =A0I have to admit I never intended for the ord=
er-2 puzzle to be twistable. =A0How it is right now happened at some point =
without me noticing. =A0I left it in when I noticed though, because it is a=
workable puzzle even if material overlaps during a twist. =A0However, it d=
oesn't really fit the mold of puzzles I had intended for the program.div>

=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
It's up to Melinda, Don and Roice if they want to use my results at all=
. =A0There is obviously no pressure to make use of them in any way. =A0I=
9;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. :) =A0There are going to be ma=
ny results, and who needs all of these formulas anyway? :)


I think a great use for any numbers you fe=
el motivated to calculate would be to list them in the ki.superliminal.com/wiki/Main_Page">wiki=A0:)

kquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px #cc=
c solid;padding-left:1ex;">
I should mention that the above count is the number of visually distinguish=
able permutations of Klein's Quartic. =A0Note that this number is diffe=
rent from the number of visually distinguishable permutations we see on the=
screen of MagicTile, because in MagicTile the center is fixed in both posi=
tion and rotation. (The fact that there are a finite number of pieces would=
actually not come into play from this point of view.) =A0The goal of the p=
ermutation count is to count the number of actual Klein's Quartic posit=
ions that can occur in the hyperbolic plane, not what we can see on the scr=
een. ('Visually distinguishable' is from the point of view of being=
inside the hyperbolic plane, as we always count visually distinguishable p=
ermutations from the point of view of the space it resides in, just as I di=
d for the n^4 and n^d cubes. =A0Otherwise, from a Euclidean point of view w=
e would see the hyperbolic plane as a unit disc, and some pieces would appe=
ar larger than others.)


I'm not sure I'm completely follow=
ing the distinction you are pointing out. =A0Is it just that the projection=
from the hyperbolic plane to the disk model warps the pieces, and this sho=
uld be ignored? =A0The next version, which I've made some good progress=
on, supports both rotating and hyperbolic panning (allowing arbitrary reor=
ientations), so equivalence of piece shapes should be more directly obvious=
. =A0As with the cubes, I would think reorientations of the puzzle should n=
ot affect the permutation counts, as I'm sure you're doing. =A0Plea=
se let me know if I missed what you intended to point out, since I probably=
did.

=A0
seeya,
Roice


--0015175cd264f7236804a22896fe--



From: David Smith <djs314djs314@yahoo.com>
Date: Sat, 30 Apr 2011 14:21:29 -0700 (PDT)
Subject: Re: [MC4D] Order-2 Klein's Quartic permutation count








=C2=A0



=20=20


=20=20=20=20
=20=20=20=20=20=20
=20=20=20=20=20=20
Thanks David,
I've always been interested to know KQ permutation counts, and I'll definit=
ely look forward to seeing the number for the order-3 puzzle! =C2=A0I have =
to admit I never intended for the order-2 puzzle to be twistable. =C2=A0How=
it is right now happened at some point without me noticing. =C2=A0I left i=
t in when I noticed though, because it is a workable puzzle even if materia=
l overlaps during a twist. =C2=A0However, 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. =
=C2=A0There is obviously no pressure to make use of them in any way. =C2=A0=
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. :) =C2=A0There 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 b=
e to list them in the wiki=C2=A0:)

I should mention that the above count is the number of visually distinguish=
able permutations of Klein's Quartic. =C2=A0Note that this number is differ=
ent from the number of visually distinguishable permutations we see on the =
screen of MagicTile, because in MagicTile the center is fixed in both posit=
ion and rotation. (The fact that there are a finite number of pieces would =
actually not come into play from this point of view.) =C2=A0The goal of the=
permutation count is to count the number of actual Klein's Quartic positio=
ns that can occur in the hyperbolic plane, not what we can see on the scree=
n. ('Visually distinguishable' is from the point of view of being inside th=
e hyperbolic plane, as we always count visually distinguishable permutation=
s from the point of view of the space it resides in, just as I did for the =
n^4 and n^d cubes. =C2=A0Otherwise, from a Euclidean point of view we would=
see the hyperbolic plane as a unit disc, and some pieces would appear larg=
er
than others.)


I'm not sure I'm completely following the distinction you are pointing out.=
=C2=A0Is it just that the projection from the hyperbolic plane to the disk=
model warps the pieces, and this should be ignored? =C2=A0The next version=
, which I've made some good progress on, supports both rotating and hyperbo=
lic panning (allowing arbitrary reorientations), so equivalence of piece sh=
apes should be more directly obvious. =C2=A0As with the cubes, I would thin=
k reorientations of the puzzle should not affect the permutation counts, as=
I'm sure you're doing. =C2=A0Please let me know if I missed what you inten=
ded to point out, since I probably did.
=C2=A0seeya,Roice


=20=20=20=20
=20=20=20=20=20

=20=20=20=20
=20=20=20=20


=20



=20=20




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Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable

top" style=3D"font: inherit;">No problem!  I thought you would like to=
see the Klein's Quartic permutation counts.  Yes, I would love to joi=
n the wiki (again!); please sign me up! (The page instructs me to contact e=
ither you or Melinda.)  Should I post every time I complete a calculat=
ion?  That would almost certainly be unnecessary and flood the message=
board.  But, I'll certainly let at least you know Roice, as the creat=
or 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 Magic120Cel=
l and the original MagicCube4D.  When I complete the MagicTile and Mag=
icCube4D 4.0 formulas, I'll create a file for each with all of my results a=
nd 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 wa=
s that as MagicTile displays the puzzles now; the number of visually differ=
ent configurations of the puzzle itself is different (smaller) than the num=
ber of visually different screenshots one could take of MagicTile.  Th=
is 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, a=
s 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 Kle=
in'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 co=
rrect, reorientations should not affect the permutation counts, and these r=
eorientations cannot be performed in MagicTile, thus artificially raising t=
he number of visually (screen-wise) different positions, as two differently=
appearing positions would in fact be the same position once a reorientatio=
n is applied.  I think I now see why you became confused; at first gla=
nce saying 'reorientations do not affect the permutation counts' and 'the l=
ack of the ability to reorient the position makes the two counts different'=
appears to be nonsensical.  If the faces had fixed centers, then it w=
ould be nonsensical. :)

I still believe that my count is correct, de=
spite 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 pu=
zzle.  I appreciate that you are genuinely interested in my
results!  That will keep me motivated to continue, despite the fact t=
hat 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 th=
e actual cubes.  Then I will be convinced that I haven't made some sub=
tle error with the Klein's Quartic counts.

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

All the best,
Dav=
id




--- On Sat, 4/30/11, Roice Nelson <roice3@gm=
ail.com> wrote:
gb(16, 16, 255); margin-left: 5px; padding-left: 5px;">
From: Roice Nels=
on <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







 




=20=20=20=20=20=20
=20=20=20=20=20=20

Thanks David,


I've always been interested t=
o know KQ permutation counts, and I'll definitely look forward to seeing th=
e 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, b=
ecause it is a workable puzzle even if material overlaps during a twist. &n=
bsp;However, it doesn't really fit the mold of puzzles I had intended for t=
he program.


8375gmail_quote" style=3D"border-left: 1px solid rgb(204, 204, 204);">
It's up to Melinda, Don and Roice if they want to use my results at all. &n=
bsp;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 m=
any results, and who needs all of these formulas anyway? :)


I think a great use for any numbers you fe=
el motivated to calculate would be to list them in the target=3D"_blank" href=3D"http://wiki.superliminal.com/wiki/Main_Page">wiki=
 :)

e" style=3D"border-left: 1px solid rgb(204, 204, 204);">
I should mention that the above count is the number of visually distinguish=
able permutations of Klein's Quartic.  Note that this number is differ=
ent from the number of visually distinguishable permutations we see on the =
screen of MagicTile, because in MagicTile the center is fixed in both posit=
ion 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 positio=
ns that can occur in the hyperbolic plane, not what we can see on the scree=
n. ('Visually distinguishable' is from the point of view of being inside th=
e hyperbolic plane, as we always count visually distinguishable permutation=
s 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 b=
e ignored?  The next version, which I've made some good progress on, s=
upports both rotating and hyperbolic panning (allowing arbitrary reorientat=
ions), so equivalence of piece shapes should be more directly obvious. &nbs=
p;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 m=
e know if I missed what you intended to point out, since I probably did.iv>
 
seeya,
Roice




=20=20=20=20=20



=20



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