Thread: "My duoprism method"

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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=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 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, 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) =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.

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 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.
>=20
>

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