Thread: "Notation"

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I prefer the first of these two.

The standard convention in mathematics is
y-right, z-up, x-toward yourself

Kind regards
Ed



----- Original Message -----=20
From: Joel Karlsson joelkarlsson97@gmail.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Tuesday, January 02, 2018 5:46 PM
Subject: [MC4D] Notation


=20=20=20=20
Hello,

I'm planning to post a more detailed solution of the physical 2^4 but
to do so I need some notation. I understand that my previous post on
notation was too long and too complicated. Luckily, I have since
realised that a simpler notation is sufficient but before I introduce
it I need some input from you. What coordinate system do you prefer?

It would be great if we could decide on one coordinate system and then
use that as a convention. Personally, I think that the coordinate
system should be right-handed but besides that, I can use pretty much
any. Two great alternatives are (for the positive half axes):
x-right, y-away from yourself, z-up
x-right, y-up, z-toward yourself

What do you think?

Best regards,
Joel


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Can I get clearer definitions of 4, 5 and 7?

~Luna

On 4 Jan 2018 23:28, "Melinda Green melinda@superliminal.com [4D_Cubing]" <
4D_Cubing@yahoogroups.com> wrote:

>
>
> First off, please check out Zander Bolgar's lovely solution video
> that he invited me to
> share. It's very cool to see someone developing something like finger
> tricks and blasting through a solution. It's very much like Bob's
>
> and Joel's
> 4>
> solutions as well as Marc's Y>
> approach.
>
> This makes for a great launching point for questions about which moves
> should be included in a canonical set. Of course any move that results in=
a
> reachable state can be justified in a solution, but there's such a spectr=
um
> from "obviously fine" to "obviously not". Now that we've gotten some
> experience with this puzzle and the practicalities of solving it, I feel
> it's time to see if we can find some sort of natural canonical set, so I'=
d
> love to hear your thoughts.
>
> Here is the list of moves I know about, loosely ordered as described abov=
e:
>
> 1. Simple rotations
> 2. 90 degree twists of outer face
> 3. 180 degree twists of side face
> 4. Center face axial twist
> 5. Arbitrary half-puzzle juxtapositions
> 6. Clamshell move
> 7. Whole-puzzle reorientations
> 8. 90 degree twist of side face (each 2x2x1 square rotate in opposite
> directions)
> 9. Single end cap twist (with parity restrictions?) [fine for
> scrambling]
> 10. Restacking moves [fine for scrambling]
> 11. Single piece flip
> 12. Reassemble entire puzzle
>
> I suspect the trickiest part has to do with #9 which is the one I would
> most like to nail down.
>
> I intend to create a follow-up video to talk about all of these and any
> others you can think of. The way you can help is to offer additions and
> corrections to the above list, and especially in suggesting ways to reord=
er
> it. Then please suggest where you'd draw three lines:
>
> - Everything above is primitive (Or "basic" or "elementary" as Joel
> calls them)
> - Everything above is canonical. IE always acceptable in solutions
> - Nothing below is acceptable in solutions.
>
> Thanks all!
> -Melinda
>=20
>

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From Wikipedia:

"a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.

In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions)."

We're using the term in the second sense; specifically, the components of N-1 dimensions. Note that a 2x2x2 half puzzle is much more than a face but we are typically referring to the face that occupies the outer 8 corners of such a half. We'll sometimes be loosely referring to that entire half puzzle when we should instead say something like "purple half", but it's usually clear in context.

-Melinda

On 1/5/2018 9:26 AM, emil.indjev@gmail.com [4D_Cubing] wrote:
> Great post, though I don't see how to notate a "whole cube reorientation". BTW you keep calling them faces, but the are whole cubes and are called cells.


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From Wikipedia:

"a face is a flat (planar) surface that forms part
of the boundary of a solid object; a three-dimensional solid
bounded exclusively by flat faces is a polyhedron.



In more technical treatments of the geometry of polyhedra
and higher-dimensional polytopes, the term is also used to mean
an element of any dimension of a more general polytope (in any
number of dimensions)."

Great post, though I
don't see how to notate a "whole cube reorientation". BTW you keep
calling them faces, but the are whole cubes and are called cells.

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Hello Joel,

Thank you for another interesting post. This time I understand most of it. =
:-)

I like the idea of 'G' for the whole-puzzle reorientations, which is a mout=
hful. I would suggest calling it "gyro" rather than "girabit". Gyro is the =
Greek root for circle, which is appropriate, and people can associate it wi=
th gyroscopes. I mainly think of the move as a "big" rotation but we do nee=
d a good name and notation for it, and I think 'G' will do nicely.

The only other thing I'll mention is that I thought we had decided to call =
the two overall puzzle forms "projections" instead of "representations", ri=
ght? So abbreviated "proj" instead of "rep". Please correct me if I'm wrong=
. I forget everything. Maybe an even better term will appear once we really=
begin to understand exactly what these two forms mean and how best to thin=
k about them.

Everything else seems fine as far as I understand it. I'd love to see a vid=
eo of your solution as well as one or more showing these sequences you are =
describing and notating. Without some sort of pictorial form, it's difficul=
t to know when I'm fully understanding your text.

Best,
-Melinda

On 1/9/2018 1:02 PM, Joel Karlsson joelkarlsson97@gmail.com [4D_Cubing] wro=
te:
>
>
> A third post, revising the notation:
>
> Vocabulary:
> elementary twist: a move rotating the set of pieces which all have a stic=
ker in a specific cell (the only twists possible in MC4D)
> rotation: a move that doesn't change the state of the puzzle
>
> *Redefining the S move
> *
> As some of you might have noticed Sy is quite different from Sx and Sz (f=
rom rep(UD) on the physical puzzle).. Sy is a rotation while Sx and Sz are =
non-elementary macro-moves. To better capture the rotations I'll shortly in=
troduce the G move. The rep(UD) Sx, rep(UD) Sz and similar moves from other=
reps are defined in the same way as previously for the physical cube. All =
other physical S moves (multiples of Sy from rep(UD) and similar from other=
reps) and all virtual S moves are removed from the notation (as the new G =
move covers these). The next paragraph contains the new definition of the p=
hysical S moves (as a reminder or for those new to the notation).
>
> The S moves are only defined for the physical 2^4 puzzle. The 'S' stands =
for "stacking". The S move is performed by first splitting the cube into tw=
o 4x2x1 blocks and then, without rotating the 4x2x1 blocks, putting them ba=
ck together in the opposite order. From rep(UD), Sx and Sz are possible. Th=
e axis specifies the normal to the plane in which the cube is split (i.e th=
e axis orthogonal to the splitting-plane). Thus, from rep(UD), Sx would tak=
e the right half of the puzzle and put it to the left whilst Sz would take =
the front half of the puzzle and put it at the back. Only axes orthogonal t=
o the longest edge of the puzzle are allowed, i.e the cube must be split in=
to two 4x2x1 halves.
>
> *Introducing the G move
> *
> All G moves are rotations. The 'G' (the best name I was able to come up w=
ith) stands for=C2=A0 "girabit" (which is Latin and translates to "rotate")=
and a handwritten G looks a bit like a circular arrow (indicating rotation=
). The G rotations (as opposed to the O rotations which orient the puzzle i=
n 3D-space) are rotations around a non-projected plane (remember that a 4D =
rotation is not a rotation around an axis but a plane). Gy =3D I(CUED) is a=
90 degree rotation which takes the stickers on C to U, the stickers on U t=
o E, the stickers on E to D, the stickers on D to C, leave the stickers on =
R on R and leave the stickers on L on L. The axis (y in the previous exampl=
e) specifies in which direction the stickers on C=C2=A0 should move (being =
a rotation around a non-projected plane, C is always involved). So, Gx woul=
d move the stickers on C in the x-direction, taking them to R. As with othe=
r moves, Gz2 (for instance) is simply just Gz Gz. In MC4D, Gy can be perfor=
med by ctrl+right-clicking=20
> on the cell in the y-direction (U) and similarly for the other moves.
>
> On the physical puzzle, how the G move is performed depends on which G mo=
ve it is and which rep the puzzle is currently in. The rule: a physical G m=
ove should (always) correspond to the same virtual G move. Let me walk you =
through how the G moves are performed from rep(UD):
> Gx =3D Uz Dz'
> Gx' =3D Uz' Dz
> Gx2 =3D Uz2 Dz2
>
> Gz =3D Ux' Dx
> Gz' =3D Ux Dx'
> Gz2 =3D Ux2 Dz2
>
> Gy: take the top 2x2 cap and put it at the bottom
> Gy': take the bottom 2x2 cap and put it at the top
> Gy2: take the top 2x2x2 half and put it at the bottom
>
> *Benefits of the revision*
> The benefits of the revision are simple:
> 1) all (physically possible) elementary twists and all rotations are desc=
ribed in the exact same way for the physical and virtual puzzles (previousl=
y rep(UD) Sx_physical !=3D Sx_virtual)
> 2) G moves can be used instead of e.g Uz Dz', which more clearly shows th=
at this is a rotation
> 3) O and G describe all possible rotations (although some physical O and =
G moves have the side-effect of changing the rep of the puzzle)
>
> Best regards,
> Joel
>
>
> 2018-01-07 22:27 GMT+01:00 Joel Karlsson o:joelkarlsson97@gmail.com>>:
>
> A follow-up post:
>
> *Short comments
> *
> I made a little mistake in my first post. I was inconsistent with how=
I describe Sy (from rep(UD)) for the physical and virtual puzzle. Sy shoul=
d take the stickers on U to C (for both the physical and the virtual cube),=
meaning that you should take the bottom cap and put it on top for the phys=
ical cube.
>
> To be clear with what rep to start from, when writing down a sequence=
, I simply put it at the beginning of the sequence. For instance rep(UD) Uy=
2 Rx2 Uy2 Rx2 (perhaps not the most useful sequence).
> *
> *
> Emil, I believe that you are correct, what I refer to as faces (which=
are indeed 3-cubes for a 4-cube) is also quite commonly called cells (I'll=
use this notion throughout the post).
>
> Melinda, honestly, I pretty much came up with E before I found a word=
that started with E so "edge" was a bit contrived. "End" might be a better=
name, I'll start to use it right away. Marc had a comment about naming the=
C and E faces "outer" and "inner" or O and I but as a mathematician, I fee=
l that 'I' is reserved for the identity (and I already use O for orienting =
the whole puzzle). I'll introduce you to a notation describing rotations us=
ing 'I' later in this post (since a rotation doesn't change the state of th=
e puzzle it's the identity permutation).
>
> *Elementary twists and rotations*
> Elementary twists from rep(UD) (with an elementary twist, I mean a tw=
ist that is a rotation of the 8 pieces that all have a sticker in a specifi=
c cell):
> =C2=A0=C2=A0 -=C2=A0=C2=A0 U, D: no restrictions here, all rotations =
of the U and D cells are elementary
> =C2=A0=C2=A0 -=C2=A0=C2=A0 F, B: only Fz2 and Bz2 physically possible=
(or at least, easy to perform) and these are elementary
> =C2=A0=C2=A0 -=C2=A0=C2=A0 R, L: only Rx2 and Lx2 (see F, B above)
> =C2=A0=C2=A0 -=C2=A0=C2=A0 C: only multiples of Cy (Cy, Cy' and Cy2) =
as well as Cx2 and Cz2 (although the last two might be a bit hard to perfor=
m)
> =C2=A0=C2=A0 -=C2=A0=C2=A0 E: only multiples of Ey (the Ex2 and Ez2 w=
ould indeed be elementary but is hard to perform)
> Note that this covers all the known elementary twists that are possib=
le on the physical 2^4 (at least as far as I know). A 2x2x2 block can be or=
iented in 24 different ways and there are precisely 23 U moves (from rep(UD=
)) in my notation; Ux, Ux', Ux2 and similar for the other axes makes 9, the=
re is one Uxyz or similar for every corner so that's 8 more, there are 6 di=
fferent Uxy twists (note that Uxy=3DUx'y') and 9+8+6 =3D 23. The 24th one i=
s the identity (leaving the block as it is).
>
> The single move rotations described by my notation are (a rotation is=
a move or sequence of moves that leaves the state of the puzzle unchanged)=
:
> =C2=A0=C2=A0 -=C2=A0=C2=A0 O: all O moves (regardless of rep)
> =C2=A0=C2=A0 -=C2=A0=C2=A0 S: multiples of Sy (Sy, Sy' and Sy2) (from=
rep(UD) or rep(Cy))
> Note that the Sy and Sy' changes the representation from rep(UD) to r=
ep(Cy) or the other way around.
>
> Note that (after the correction above regarding Sy from rep(UD)) for =
all elementary twists and single move (pure) rotations P in my notation, it=
is true that: physical(P) =3D virtual(P).
>
> *The non-elementary S moves*
> To get the whole set of legal states we need to introduce a non-eleme=
ntary move that can be used to compose rotations. I've chosen the last two =
S moves for this: Sx and Sz (from rep(UD)). Following is a relation between=
these S moves for the physical puzzle and elementary moves in MC4D. To avo=
id confusion I will, in the following section use Sx_p and Sz_p for the S m=
oves on the physical puzzle and simply Sx and Sy for the virtual S moves (c=
trl-clicking in MC4D, these are pure rotations).
> rep(UD) Sx_p Sy' =3D Oy2 Ox Rx2 Fz2 Rx2 Uy2 Uz' Dz' Fz2 Rx2 Uy2
> I included the Sy' on the left-hand side (could have put Sy at the en=
d of the right-hand side instead) to not change the rep of the physical puz=
zle.
>
> I'll attach an MC4D macro file with this sequence. For reference, I c=
hose xyz on C. Apologies if I'm breaking any convention in how to chose ref=
erence stickers for the macro.
>
> *The I notation*
> This is an addition to my notation. 'I' can be used to describe seque=
nces that don't change the state of the puzzle, i.e rotations. The physical=
puzzle has two attributes apart from the 2^4 puzzle's state: the rep and w=
hich colour being in which cell (in the solved state). A general rotation c=
an thus be described with how it changes the rep and how it permutes the ce=
lls. The rep is quite easy; if the puzzle is in rep(UD) before the rotation=
and rep(RL) after,=C2=A0 we can use rep(UD) I(rep(RL)) to describe this. S=
o I(rep(RL)) is a rotation which takes you from wherever you are to rep(RL)=
. However, if the rep isn't changed we can leave this part out.
>
> A permutation of the faces can be broken down into cycles and a cycle=
is quite easy to write down. For example, FRU is the cycle which takes the=
stickers on F to R, the stickers on R to U and the stickers on U to F. Ano=
ther example is RL which takes R to L and L to R. There are some constraint=
s for these cycles to be possible. They need to have a kind of symmetry; if=
R->L then L->R and if R->U then L->D and so on (to keep opposite colours o=
pposite). Thus, it's enough to specify the cycles including R, U, F and C. =
Moreover, all cells not moving can be left out, i.e R->R don't have to be s=
pecified. Let's now look at how we can use this.
>
> One easy (but not so useful) example is:
> rep(UD) I(rep(RL), ULDR)=C2=A0 =3D rep(UD) Oz
> So the I is a rotation which takes you from rep(UD) to rep(RL) and cy=
cles ULDR (thus leaving F, B, C and E where they are) and this is precisely=
what Oz does.
>
> Another example:
> rep(UD) I(rep(Cy), UCDE) =3D rep(UD) Sy
>
> Now on to a useful example.
> The equality which relates Sx_p to elementary twists and rotations (a=
bove)=C2=A0 can be rearranged to get a sequence (for the physical puzzle) w=
hich describes a rotation:
> rep(UD) I(UF RL) =3D rep(UD) Sx_p Sy' Uy2 Rx2 Fz2 Uz Dz Uy2 Rx2 Fz2 R=
x2
> This is a rotation (starting and ending in rep(UD)) which consists of=
three 2-cycles: UF, DB and RL. The DB is not written explicitly since it f=
ollows implicitly from UF (the opposite colour cell of U is D and the oppos=
ite of F is B). Using my notation, this is the shortest sequence I have fou=
nd which preserves rep(UD) while permuting U and D with other cells (which =
is exactly what is needed in addition to the elementary moves of the physic=
al puzzle to get all states of a 2^4 cube)
>
> Note that this notation can be used for the virtual puzzle as well. H=
owever, it's not very useful there since the representation is symmetrical =
and all rotations can easily be written down with O and S moves.
>
> Best regards,
> Joel
>
> PS. I'll make sure to post on the "Canonical moves" subject as soon a=
s I get the time.
>
> 2018-01-05 18:26 GMT+01:00 emil.indjev@gmail.com gmail.com> [4D_Cubing] <4D_Cubing@yahoogroups.com oups.com>>:
>
> Great post, though I don't see how to notate a "whole cube reorie=
ntation". BTW you keep calling them faces, but the are whole cubes and are =
called cells.
>
>
>
>
>
>=20


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">


Hello Joel,



Thank you for another interesting post. This time I understand most
of it. :-)



I like the idea of 'G' for the whole-puzzle reorientations, which is
a mouthful. I would suggest calling it "gyro" rather than "girabit".
Gyro is the Greek root for circle, which is appropriate, and people
can associate it with gyroscopes. I mainly think of the move as a
"big" rotation but we do need a good name and notation for it, and I
think 'G' will do nicely.



The only other thing I'll mention is that I thought we had decided
to call the two overall puzzle forms "projections" instead of
"representations", right? So abbreviated "proj" instead of "rep".
Please correct me if I'm wrong. I forget everything. Maybe an even
better term will appear once we really begin to understand exactly
what these two forms mean and how best to think about them.



Everything else seems fine as far as I understand it. I'd love to
see a video of your solution as well as one or more showing these
sequences you are describing and notating. Without some sort of
pictorial form, it's difficult to know when I'm fully understanding
your text.



Best,

-Melinda

cite=3D"mid:CAEohJcH87EAhuDUbm3C6=3D4=3Dn8tdhO9uL0DLu5C_PzWKdT8N6Tw@mail.gm=
ail.com">

A third post, revising the notation:

Vocabulary:

elementary twist: a move rotating the set of
pieces which all have a sticker in a
specific cell (the only twists possible in
MC4D)

rotation: a move that doesn't change the
state of the puzzle

Redefining the S move

As some of you might have noticed Sy is
quite different from Sx and Sz (from rep(UD)
on the physical puzzle).. Sy is a rotation
while Sx and Sz are non-elementary
macro-moves. To better capture the rotations
I'll shortly introduce the G move. The rep(UD)
Sx, rep(UD) Sz and similar moves from other
reps are defined in the same way as previously
for the physical cube. All other physical S
moves (multiples of Sy from rep(UD) and
similar from other reps) and all virtual S
moves are removed from the notation (as the
new G move covers these). The next paragraph
contains the new definition of the physical S
moves (as a reminder or for those new to the
notation).

The S moves are only defined for the
physical 2^4 puzzle. The 'S' stands for
"stacking". The S move is performed by first
splitting the cube into two 4x2x1 blocks and
then, without rotating the 4x2x1 blocks,
putting them back together in the opposite
order. From rep(UD), Sx and Sz are possible.
The axis specifies the normal to the plane in
which the cube is split (i.e the axis
orthogonal to the splitting-plane). Thus, from
rep(UD), Sx would take the right half of the
puzzle and put it to the left whilst Sz would
take the front half of the puzzle and put it
at the back. Only axes orthogonal to the
longest edge of the puzzle are allowed, i.e
the cube must be split into two 4x2x1 halves.

Introducing the G move

All G moves are rotations. The 'G' (the best name
I was able to come up with) stands for=C2=A0 "girabit=
"
(which is Latin and translates to "rotate") and a
handwritten G looks a bit like a circular arrow
(indicating rotation). The G rotations (as opposed
to the O rotations which orient the puzzle in
3D-space) are rotations around a non-projected
plane (remember that a 4D rotation is not a
rotation around an axis but a plane). Gy =3D I(CUED)
is a 90 degree rotation which takes the stickers
on C to U, the stickers on U to E, the stickers on
E to D, the stickers on D to C, leave the stickers
on R on R and leave the stickers on L on L. The
axis (y in the previous example) specifies in
which direction the stickers on C=C2=A0 should move
(being a rotation around a non-projected plane, C
is always involved). So, Gx would move the
stickers on C in the x-direction, taking them to
R. As with other moves, Gz2 (for instance) is
simply just Gz Gz. In MC4D, Gy can be performed by
ctrl+right-clicking on the cell in the y-direction
(U) and similarly for the other moves.

On the physical puzzle, how the G move is
performed depends on which G move it is and which
rep the puzzle is currently in. The rule: a
physical G move should (always) correspond to the
same virtual G move. Let me walk you through how
the G moves are performed from rep(UD):

Gx =3D Uz Dz'

Gx' =3D Uz' Dz

Gx2 =3D Uz2 Dz2

Gz =3D Ux' Dx

Gz' =3D Ux Dx'

Gz2 =3D Ux2 Dz2

Gy: take the top 2x2 cap and put it at the
bottom

Gy': take the bottom 2x2 cap and put it at the
top

Gy2: take the top 2x2x2 half and put it at the
bottom

Benefits of the revision

The benefits of the revision are simple:

1) all (physically possible) elementary twists and all
rotations are described in the exact same way for the
physical and virtual puzzles (previously rep(UD)
Sx_physical !=3D Sx_virtual)

2) G moves can be used instead of e.g Uz Dz', which more
clearly shows that this is a rotation

3) O and G describe all possible rotations (although some
physical O and G moves have the side-effect of changing the
rep of the puzzle)

Best regards,

Joel

2018-01-07 22:27 GMT+01:00 Joel
Karlsson < href=3D"mailto:joelkarlsson97@gmail.com" target=3D"_blank"
moz-do-not-send=3D"true">joelkarlsson97@gmail.com>an>:

.8ex;border-left:1px #ccc solid;padding-left:1ex">

A follow-up post:

Short comments

I made a little mistake in my first post. I was
inconsistent with how I describe Sy (from rep(UD))
for the physical and virtual puzzle. Sy should
take the stickers on U to C (for both the physical
and the virtual cube), meaning that you should
take the bottom cap and put it on top for the
physical cube.

To be clear with what rep to start
from, when writing down a sequence, I simply put
it at the beginning of the sequence. For instance
rep(UD) Uy2 Rx2 Uy2 Rx2 (perhaps not the most
useful sequence).=C2=A0

Emil, I believe that you are correct, what I refer
to as faces (which are indeed 3-cubes for a 4-cube)
is also quite commonly called cells (I'll use this
notion throughout the post).

Melinda, honestly, I pretty much came up with E
before I found a word that started with E so "edge"
was a bit contrived. "End" might be a better name,
I'll start to use it right away. Marc had a comment
about naming the C and E faces "outer" and "inner"
or O and I but as a mathematician, I feel that 'I'
is reserved for the identity (and I already use O
for orienting the whole puzzle). I'll introduce you
to a notation describing rotations using 'I' later
in this post (since a rotation doesn't change the
state of the puzzle it's the identity permutation).

Elementary twists and rotations

Elementary twists from rep(UD) (with an elementary
twist, I mean a twist that is a rotation of the 8
pieces that all have a sticker in a specific cell):

=C2=A0=C2=A0 -=C2=A0=C2=A0 U, D: no restrictions her=
e, all rotations
of the U and D cells are elementary

=C2=A0=C2=A0 -=C2=A0=C2=A0 F, B: only Fz2 and Bz2 ph=
ysically possible
(or at least, easy to perform) and these are
elementary

=C2=A0=C2=A0 -=C2=A0=C2=A0 R, L: only Rx2 and Lx2 (s=
ee F, B above)

=C2=A0=C2=A0 -=C2=A0=C2=A0 C: only multiples of Cy (=
Cy, Cy' and Cy2)
as well as Cx2 and Cz2 (although the last two might
be a bit hard to perform)

=C2=A0=C2=A0 -=C2=A0=C2=A0 E: only multiples of Ey (=
the Ex2 and Ez2
would indeed be elementary but is hard to perform)>

Note that this covers all the known elementary
twists that are possible on the physical 2^4 (at
least as far as I know). A 2x2x2 block can be
oriented in 24 different ways and there are
precisely 23 U moves (from rep(UD)) in my notation;
Ux, Ux', Ux2 and similar for the other axes makes 9,
there is one Uxyz or similar for every corner so
that's 8 more, there are 6 different Uxy twists
(note that Uxy=3DUx'y') and 9+8+6 =3D 23. The 24th one
is the identity (leaving the block as it is).

The single move rotations described by my
notation are (a rotation is a move or sequence of
moves that leaves the state of the puzzle
unchanged):

=C2=A0=C2=A0 -=C2=A0=C2=A0 O: all O moves (regardles=
s of rep)

=C2=A0=C2=A0 -=C2=A0=C2=A0 S: multiples of Sy (Sy, S=
y' and Sy2) (from
rep(UD) or rep(Cy))

Note that the Sy and Sy' changes the
representation from rep(UD) to rep(Cy) or the other
way around.

Note that (after the correction above regarding
Sy from rep(UD)) for all elementary twists and
single move (pure) rotations P in my notation, it is
true that: physical(P) =3D virtual(P).

The non-elementary S moves

To get the whole set of legal states we need to
introduce a non-elementary move that can be used to
compose rotations. I've chosen the last two S moves
for this: Sx and Sz (from rep(UD)). Following is a
relation between these S moves for the physical
puzzle and elementary moves in MC4D. To avoid
confusion I will, in the following section use Sx_p
and Sz_p for the S moves on the physical puzzle and
simply Sx and Sy for the virtual S moves
(ctrl-clicking in MC4D, these are pure rotations).

rep(UD) Sx_p Sy' =3D Oy2 Ox Rx2 Fz2 Rx2 Uy2 Uz' Dz'
Fz2 Rx2 Uy2

I included the Sy' on the left-hand
side (could have put Sy at the end of the right-hand
side instead) to not change the rep of the physical
puzzle.

I'll attach an MC4D macro file with this
sequence. For reference, I chose xyz on C. Apologies
if I'm breaking any convention in how to chose
reference stickers for the macro.

The I notation

This is an addition to my notation.
'I' can be used to describe sequences that don't
change the state of the puzzle, i.e rotations. The
physical puzzle has two attributes apart from the
2^4 puzzle's state: the rep and which colour being
in which cell (in the solved state). A general
rotation can thus be described with how it changes
the rep and how it permutes the cells. The rep is
quite easy; if the puzzle is in rep(UD) before the
rotation and rep(RL) after,=C2=A0 we can use rep(UD)
I(rep(RL)) to describe this. So I(rep(RL)) is a
rotation which takes you from wherever you are to
rep(RL). However, if the rep isn't changed we can
leave this part out.=C2=A0

A permutation of the faces can be
broken down into cycles and a cycle is quite easy to
write down. For example, FRU is the cycle which
takes the stickers on F to R, the stickers on R to U
and the stickers on U to F. Another example is RL
which takes R to L and L to R. There are some
constraints for these cycles to be possible. They
need to have a kind of symmetry; if R->L then
L->R and if R->U then L->D and so on (to
keep opposite colours opposite). Thus, it's enough
to specify the cycles including R, U, F and C.
Moreover, all cells not moving can be left out, i.e
R->R don't have to be specified. Let's now look
at how we can use this.=C2=A0

One easy (but not so useful) example
is:

rep(UD) I(rep(RL), ULDR)=C2=A0 =3D rep(=
UD) Oz

So the I is a rotation which takes you
from rep(UD) to rep(RL) and cycles ULDR (thus
leaving F, B, C and E where they are) and this is
precisely what Oz does.=C2=A0

Another example:

rep(UD) I(rep(Cy), UCDE) =3D rep(UD) Sy=

Now on to a useful example.=C2=A0

The equality which relates Sx_p to
elementary twists and rotations (above)=C2=A0 can be
rearranged to get a sequence (for the physical
puzzle) which describes a rotation:

rep(UD) I(UF RL) =3D rep(UD) Sx_p Sy'
Uy2 Rx2 Fz2 Uz Dz Uy2 Rx2 Fz2 Rx2

This is a rotation (starting and ending in
rep(UD)) which consists of three 2-cycles: UF, DB
and RL. The DB is not written explicitly since it
follows implicitly from UF (the opposite colour cell
of U is D and the opposite of F is B). Using my
notation, this is the shortest sequence I have found
which preserves rep(UD) while permuting U and D with
other cells (which is exactly what is needed in
addition to the elementary moves of the physical
puzzle to get all states of a 2^4 cube)

Note that this notation can be used for the
virtual puzzle as well. However, it's not very
useful there since the representation is symmetrical
and all rotations can easily be written down with O
and S moves.

Best regards,

Joel

PS. I'll make sure to post on the "Canonical moves"
subject as soon as I get the time.

2018-01-05 18:26 GMT+01:00
target=3D"_blank" moz-do-not-send=3D"true">emil.ind=
jev@gmail.com
[4D_Cubing] < href=3D"mailto:4D_Cubing@yahoogroups.com"
target=3D"_blank" moz-do-not-send=3D"true">4D_Cub=
ing@yahoogroups.com>:

0px 0px 0.8ex;border-left:1px solid
rgb(204,204,204);padding-left:1ex">

=C2=A0
id=3D"m_6988725919983488718gmail-m_-4527176556827551992m_167837619620329765=
2m_-5122883874215409483ygrp-mlmsg">
id=3D"m_6988725919983488718gmail-m_-4527176556827551992m_167837619620329765=
2m_-5122883874215409483ygrp-msg">
id=3D"m_6988725919983488718gmail-m_-4527176556827551992m_167837619620329765=
2m_-5122883874215409483ygrp-text">

Great post, though I don't see how to
notate a "whole cube reorientation".
BTW you keep calling them faces, but
the are whole cubes and are called
cells.

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Hi Melinda,

That's great! I might attempt writing the notation down more carefully on a
wiki-page (suggestions?) but this will have to wait since it's still
developing quite quickly.

In MC4D, with ctrl-click rotating "by cubie", I believe that different
projections can be reached by ctrl-clicking on different pieces. However, I
wouldn't call the different physical representations "different
projections" mainly because I don't think that the transformations which
take the 4D-puzzle to the physical 3D-representations are projections. I'll
elaborate this point in a future post (I have a small project that I think
all of you will like). Whether the community has agreed to call them
"projections" or not is more than I know.

Best regards,
Joel

2018-01-10 0:37 GMT+01:00 Melinda Green melinda@superliminal.com
[4D_Cubing] <4D_Cubing@yahoogroups.com>:

>
>
> Hello Joel,
>
> Thank you for another interesting post. This time I understand most of it=
.
> :-)
>
> I like the idea of 'G' for the whole-puzzle reorientations, which is a
> mouthful. I would suggest calling it "gyro" rather than "girabit". Gyro i=
s
> the Greek root for circle, which is appropriate, and people can associate
> it with gyroscopes. I mainly think of the move as a "big" rotation but we
> do need a good name and notation for it, and I think 'G' will do nicely.
>
> The only other thing I'll mention is that I thought we had decided to cal=
l
> the two overall puzzle forms "projections" instead of "representations",
> right? So abbreviated "proj" instead of "rep". Please correct me if I'm
> wrong. I forget everything. Maybe an even better term will appear once we
> really begin to understand exactly what these two forms mean and how best
> to think about them.
>
> Everything else seems fine as far as I understand it. I'd love to see a
> video of your solution as well as one or more showing these sequences you
> are describing and notating. Without some sort of pictorial form, it's
> difficult to know when I'm fully understanding your text.
>
> Best,
> -Melinda
>
> On 1/9/2018 1:02 PM, Joel Karlsson joelkarlsson97@gmail.com [4D_Cubing]
> wrote:
>
> A third post, revising the notation:
>
> Vocabulary:
> elementary twist: a move rotating the set of pieces which all have a
> sticker in a specific cell (the only twists possible in MC4D)
> rotation: a move that doesn't change the state of the puzzle
>
>
> *Redefining the S move *
> As some of you might have noticed Sy is quite different from Sx and Sz
> (from rep(UD) on the physical puzzle).. Sy is a rotation while Sx and Sz
> are non-elementary macro-moves. To better capture the rotations I'll
> shortly introduce the G move. The rep(UD) Sx, rep(UD) Sz and similar move=
s
> from other reps are defined in the same way as previously for the physica=
l
> cube. All other physical S moves (multiples of Sy from rep(UD) and simila=
r
> from other reps) and all virtual S moves are removed from the notation (a=
s
> the new G move covers these). The next paragraph contains the new
> definition of the physical S moves (as a reminder or for those new to the
> notation).
>
> The S moves are only defined for the physical 2^4 puzzle. The 'S' stands
> for "stacking". The S move is performed by first splitting the cube into
> two 4x2x1 blocks and then, without rotating the 4x2x1 blocks, putting the=
m
> back together in the opposite order. From rep(UD), Sx and Sz are possible=
.
> The axis specifies the normal to the plane in which the cube is split (i.=
e
> the axis orthogonal to the splitting-plane). Thus, from rep(UD), Sx would
> take the right half of the puzzle and put it to the left whilst Sz would
> take the front half of the puzzle and put it at the back. Only axes
> orthogonal to the longest edge of the puzzle are allowed, i.e the cube mu=
st
> be split into two 4x2x1 halves.
>
>
> *Introducing the G move *
> All G moves are rotations. The 'G' (the best name I was able to come up
> with) stands for "girabit" (which is Latin and translates to "rotate") a=
nd
> a handwritten G looks a bit like a circular arrow (indicating rotation).
> The G rotations (as opposed to the O rotations which orient the puzzle in
> 3D-space) are rotations around a non-projected plane (remember that a 4D
> rotation is not a rotation around an axis but a plane). Gy =3D I(CUED) is=
a
> 90 degree rotation which takes the stickers on C to U, the stickers on U =
to
> E, the stickers on E to D, the stickers on D to C, leave the stickers on =
R
> on R and leave the stickers on L on L. The axis (y in the previous exampl=
e)
> specifies in which direction the stickers on C should move (being a
> rotation around a non-projected plane, C is always involved). So, Gx woul=
d
> move the stickers on C in the x-direction, taking them to R. As with othe=
r
> moves, Gz2 (for instance) is simply just Gz Gz. In MC4D, Gy can be
> performed by ctrl+right-clicking on the cell in the y-direction (U) and
> similarly for the other moves.
>
> On the physical puzzle, how the G move is performed depends on which G
> move it is and which rep the puzzle is currently in. The rule: a physical=
G
> move should (always) correspond to the same virtual G move. Let me walk y=
ou
> through how the G moves are performed from rep(UD):
> Gx =3D Uz Dz'
> Gx' =3D Uz' Dz
> Gx2 =3D Uz2 Dz2
>
> Gz =3D Ux' Dx
> Gz' =3D Ux Dx'
> Gz2 =3D Ux2 Dz2
>
> Gy: take the top 2x2 cap and put it at the bottom
> Gy': take the bottom 2x2 cap and put it at the top
> Gy2: take the top 2x2x2 half and put it at the bottom
>
> *Benefits of the revision*
> The benefits of the revision are simple:
> 1) all (physically possible) elementary twists and all rotations are
> described in the exact same way for the physical and virtual puzzles
> (previously rep(UD) Sx_physical !=3D Sx_virtual)
> 2) G moves can be used instead of e.g Uz Dz', which more clearly shows
> that this is a rotation
> 3) O and G describe all possible rotations (although some physical O and =
G
> moves have the side-effect of changing the rep of the puzzle)
>
> Best regards,
> Joel
>
>
> 2018-01-07 22:27 GMT+01:00 Joel Karlsson :
>
>> A follow-up post:
>>
>>
>> *Short comments *
>> I made a little mistake in my first post. I was inconsistent with how I
>> describe Sy (from rep(UD)) for the physical and virtual puzzle. Sy shoul=
d
>> take the stickers on U to C (for both the physical and the virtual cube)=
,
>> meaning that you should take the bottom cap and put it on top for the
>> physical cube.
>>
>> To be clear with what rep to start from, when writing down a sequence, I
>> simply put it at the beginning of the sequence. For instance rep(UD) Uy2
>> Rx2 Uy2 Rx2 (perhaps not the most useful sequence).
>>
>> Emil, I believe that you are correct, what I refer to as faces (which ar=
e
>> indeed 3-cubes for a 4-cube) is also quite commonly called cells (I'll u=
se
>> this notion throughout the post).
>>
>> Melinda, honestly, I pretty much came up with E before I found a word
>> that started with E so "edge" was a bit contrived. "End" might be a bett=
er
>> name, I'll start to use it right away. Marc had a comment about naming t=
he
>> C and E faces "outer" and "inner" or O and I but as a mathematician, I f=
eel
>> that 'I' is reserved for the identity (and I already use O for orienting
>> the whole puzzle). I'll introduce you to a notation describing rotations
>> using 'I' later in this post (since a rotation doesn't change the state =
of
>> the puzzle it's the identity permutation).
>>
>> *Elementary twists and rotations*
>> Elementary twists from rep(UD) (with an elementary twist, I mean a twist
>> that is a rotation of the 8 pieces that all have a sticker in a specific
>> cell):
>> - U, D: no restrictions here, all rotations of the U and D cells ar=
e
>> elementary
>> - F, B: only Fz2 and Bz2 physically possible (or at least, easy to
>> perform) and these are elementary
>> - R, L: only Rx2 and Lx2 (see F, B above)
>> - C: only multiples of Cy (Cy, Cy' and Cy2) as well as Cx2 and Cz2
>> (although the last two might be a bit hard to perform)
>> - E: only multiples of Ey (the Ex2 and Ez2 would indeed be
>> elementary but is hard to perform)
>> Note that this covers all the known elementary twists that are possible
>> on the physical 2^4 (at least as far as I know). A 2x2x2 block can be
>> oriented in 24 different ways and there are precisely 23 U moves (from
>> rep(UD)) in my notation; Ux, Ux', Ux2 and similar for the other axes mak=
es
>> 9, there is one Uxyz or similar for every corner so that's 8 more, there
>> are 6 different Uxy twists (note that Uxy=3DUx'y') and 9+8+6 =3D 23. The=
24th
>> one is the identity (leaving the block as it is).
>>
>> The single move rotations described by my notation are (a rotation is a
>> move or sequence of moves that leaves the state of the puzzle unchanged)=
:
>> - O: all O moves (regardless of rep)
>> - S: multiples of Sy (Sy, Sy' and Sy2) (from rep(UD) or rep(Cy))
>> Note that the Sy and Sy' changes the representation from rep(UD) to
>> rep(Cy) or the other way around.
>>
>> Note that (after the correction above regarding Sy from rep(UD)) for all
>> elementary twists and single move (pure) rotations P in my notation, it =
is
>> true that: physical(P) =3D virtual(P).
>>
>> *The non-elementary S moves*
>> To get the whole set of legal states we need to introduce a
>> non-elementary move that can be used to compose rotations. I've chosen t=
he
>> last two S moves for this: Sx and Sz (from rep(UD)). Following is a
>> relation between these S moves for the physical puzzle and elementary mo=
ves
>> in MC4D. To avoid confusion I will, in the following section use Sx_p an=
d
>> Sz_p for the S moves on the physical puzzle and simply Sx and Sy for the
>> virtual S moves (ctrl-clicking in MC4D, these are pure rotations).
>> rep(UD) Sx_p Sy' =3D Oy2 Ox Rx2 Fz2 Rx2 Uy2 Uz' Dz' Fz2 Rx2 Uy2
>> I included the Sy' on the left-hand side (could have put Sy at the end o=
f
>> the right-hand side instead) to not change the rep of the physical puzzl=
e.
>>
>> I'll attach an MC4D macro file with this sequence. For reference, I chos=
e
>> xyz on C. Apologies if I'm breaking any convention in how to chose
>> reference stickers for the macro.
>>
>> *The I notation*
>> This is an addition to my notation. 'I' can be used to describe sequence=
s
>> that don't change the state of the puzzle, i.e rotations. The physical
>> puzzle has two attributes apart from the 2^4 puzzle's state: the rep and
>> which colour being in which cell (in the solved state). A general rotati=
on
>> can thus be described with how it changes the rep and how it permutes th=
e
>> cells. The rep is quite easy; if the puzzle is in rep(UD) before the
>> rotation and rep(RL) after, we can use rep(UD) I(rep(RL)) to describe
>> this. So I(rep(RL)) is a rotation which takes you from wherever you are =
to
>> rep(RL). However, if the rep isn't changed we can leave this part out.
>>
>> A permutation of the faces can be broken down into cycles and a cycle is
>> quite easy to write down. For example, FRU is the cycle which takes the
>> stickers on F to R, the stickers on R to U and the stickers on U to F.
>> Another example is RL which takes R to L and L to R. There are some
>> constraints for these cycles to be possible. They need to have a kind of
>> symmetry; if R->L then L->R and if R->U then L->D and so on (to keep
>> opposite colours opposite). Thus, it's enough to specify the cycles
>> including R, U, F and C. Moreover, all cells not moving can be left out,
>> i.e R->R don't have to be specified. Let's now look at how we can use th=
is.
>>
>> One easy (but not so useful) example is:
>> rep(UD) I(rep(RL), ULDR) =3D rep(UD) Oz
>> So the I is a rotation which takes you from rep(UD) to rep(RL) and cycle=
s
>> ULDR (thus leaving F, B, C and E where they are) and this is precisely w=
hat
>> Oz does.
>>
>> Another example:
>> rep(UD) I(rep(Cy), UCDE) =3D rep(UD) Sy
>>
>> Now on to a useful example.
>> The equality which relates Sx_p to elementary twists and rotations
>> (above) can be rearranged to get a sequence (for the physical puzzle)
>> which describes a rotation:
>> rep(UD) I(UF RL) =3D rep(UD) Sx_p Sy' Uy2 Rx2 Fz2 Uz Dz Uy2 Rx2 Fz2 Rx2
>> This is a rotation (starting and ending in rep(UD)) which consists of
>> three 2-cycles: UF, DB and RL. The DB is not written explicitly since it
>> follows implicitly from UF (the opposite colour cell of U is D and the
>> opposite of F is B). Using my notation, this is the shortest sequence I
>> have found which preserves rep(UD) while permuting U and D with other ce=
lls
>> (which is exactly what is needed in addition to the elementary moves of =
the
>> physical puzzle to get all states of a 2^4 cube)
>>
>> Note that this notation can be used for the virtual puzzle as well.
>> However, it's not very useful there since the representation is symmetri=
cal
>> and all rotations can easily be written down with O and S moves.
>>
>> Best regards,
>> Joel
>>
>> PS. I'll make sure to post on the "Canonical moves" subject as soon as I
>> get the time.
>>
>> 2018-01-05 18:26 GMT+01:00 emil.indjev@gmail.com [4D_Cubing] <
>> 4D_Cubing@yahoogroups.com>:
>>
>>>
>>>
>>> Great post, though I don't see how to notate a "whole cube
>>> reorientation". BTW you keep calling them faces, but the are whole cube=
s
>>> and are called cells.
>>>
>>
>>
>
>=20
>

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