# Message #3940

From: Joel Karlsson <joelkarlsson97@gmail.com>

Subject: Re: [MC4D] Re: Notation

Date: Sun, 07 Jan 2018 22:27:44 +0100

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 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 cubes and are called

> cells.

>

>