Message #3945

From: Joel Karlsson <>
Subject: Re: [MC4D] Re: Notation
Date: Wed, 10 Jan 2018 20:18:51 +0100

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,

2018-01-10 0:37 GMT+01:00 Melinda Green
[4D_Cubing] <>:

> 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
> On 1/9/2018 1:02 PM, Joel Karlsson [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 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,
> 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 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.