Message #3942

From: Joel Karlsson <>
Subject: Re: [MC4D] Re: Notation
Date: Tue, 09 Jan 2018 22:02:06 +0100

A third post, revising the notation:

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 "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 = 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 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 = Uz Dz’
Gx’ = Uz’ Dz
Gx2 = Uz2 Dz2

Gz = Ux’ Dx
Gz’ = Ux Dx’
Gz2 = 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 != 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,

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 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).
> 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):
> - U, D: no restrictions here, all rotations of the U and D cells are
> 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 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=Ux’y’) and 9+8+6 = 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) = 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’ = 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, 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 this.
> One easy (but not so useful) example is:
> rep(UD) I(rep(RL), ULDR) = 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.
> Another example:
> rep(UD) I(rep(Cy), UCDE) = 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) = 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 [4D_Cubing] <
>> 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.