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2x2x2 Cube Antipodes

Submitted by B MacKenzie on Wed, 12/12/2012 - 13:34.

I have written a GUI NxNxN cube program to which I just added a 2x2x2 cube auto solve function. To test the performance of the solution algorithm I wanted try it on the 14 q-turn antipodes. So I did the depth-wise expansion of the group, found the 276 antipodes and reduced them with M† symmetry. In the context of the fixed DBL cubie 2x2x2 model, that is the <R U F> group model, M† symmetry classes are formed by ( c * m' * q * m ) where q is a group element, m ranges over the cubic symmetry group and c is the whole cube rotation needed to place the conjugate back in the group.

For those who might be interested, here are the results of the symmetry reduction and turn sequences for representative elements of the antipode classes. 

Class size:  4 Count:  1
Class size:  6 Count: 36
Class size:  8 Count:  1
Class size: 12 Count:  4
Class count: 42
 
 R R U R' U F U' F F R F' R F F
 R R U' R' U F U' R F R' F U' R F'
 R R U R' F R' F R F' U F F U' F
 R R U F R' U U F U' F' R U' R R
 R R U R U' F U' R R U' R' U' F' R
 R R U R F R' U R R U' F U' F' U'
 R R U' R' F' U R' U' R F' U' R' U R
 R R U R F' U U R F' R' F R' U U
 R R U R' U F U' R F R' U' R U R
 R R F R F' R U' R F' U U F R R
 R R U R' U' F' U U R F' U' F F U
 R R U R' U' F' U U F' U F R' F U
 R R F F R F' U R U U F' R U' R'
 R R F R F R' F U' R F' R F' R' F'
 R R U R' U' R R U F' R U F R' F
 R R U R U' F R' F' R F' U U F F
 R R U R R F U' R' U' R F' U F F
 R R U R R U F U' R F' U F U' F
 R R U R R F U' F' U' R F' R F F
 R R U R' U' R U' F' U U R U' R F
 R R U U F' R F R U F F R' F' U
 R R U F U' R F F R' U' R F' R U'
 R R F U R U' F U' R U' F U' R' U'
 R R U R' U F' R U F U' R' F' U F'
 R R F F R U' R U R R F' R U' F'
 R R U R U F' U F' U' F U' F R F
 R R U R' F U' R U F R' U' F' U F'
 R R F R U R' F U' F R' F R' F' R'
 R R U R' F' R R F R' F U R U' R
 R R U F R' U F F U' F' U F' R U'
 R R U U R U' R' F R R U R' U F'
 R R U U R F' U' F R R U F' R U'
 R R U' F R U F' U U F' U' R U' R
 R R U R' U U F U U F' R U' F U'
 R R U R' U' F' R U' F F R' F R' F
 R R U R' F' R' U R R U R F' U F'
 R R U R F' R F' R R U F' U' F' U'
 R R U R R U R F' U F' U F U' R
 R R U R' U' R F U' R' F' R U' R U'
 R R U' R' U R F' U R F' R' F R' U
 R R U F R' U' F U' R' F U R' U R'
 R R U F' R R F' R' F F U R' F' U
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2x2x2 Cube Antipodes

Submitted by Pangaia on Sun, 10/01/2017 - 07:57.
I am native german and maybe i am choosing the wrong words to explain what i mean, so please do not care about it.
There is a second thread about the antipodes of the pocket cube, but i think here it is easier for me to start.
I wrote myself a programm which do the calculations of the Pocket Cube. Its output is besides the count of the positions a text file where all local maxmia positions are listed with a sequence to reach it.
I picked out all 14 qtm sequences printed them out. Then i started to moved my cube to the first position. I rotated the cube to an other view, analysed the 'new' postion and marked it on the list as an equal positon.
So i could reduce the 276 antipodes to 16 positions. 14 of them can be seen as a pair. Seeing the positon in a mirror you will have an other positon. First my list. A letter to 'name' them; the sequence; in brackets the amount of positions you get when you turn the scrambled cube

A: U R U' R F' U R' U F' U F R' U2 (12)
B: R2 U R U' R F' U R' U2 R' U2 (24)
C: U' R' U' R U' F R' U R' F U F' U2 (24)
D: F2 U2 F U' F R F' R U' R' U2 (12)
E: U R' F R' U' R' U' F2 R' U R' U2 (12)
F: F2 U' R' U' R F U' F' U F R' U2 (24)
G: R' F R' U R' U F' R' F U2 R' U2 (24)
H: R' F U F' R F2 R' U R' U' R U2 (4)
I: U R' U F' R' F U' R F' R' U' R' U2 (8)
J: U F' R F' R F2 R' U R U R' U2 (8)
K: R' F' R U' R' F' R F' R U' F R' U2 (24)
L: R U R F' U F2 U' R F' U' R' U2 (4)
M: U' F R' U R' F2 R F2 U' R U2 (24)
N: R U F' U R' F R F' U' R' U R' U2 (24)
O: U R' F U' R' F R2 F' R' U R' U2 (24)
P: F U' R' F' R F2 R' U' R' U R' U2 (24)

the pairs are: B-C / D-E / G-K / H-L / I-J / M-P / N-O
H-L are the pattern "6 flags" and can be faster done with this sequence: U F2 U F2 U2 R2 U R2 U


My knowledge of math is not so good, that i know what M+ symetry means (so maybe my comment is wrong), but on the list of the 42 up there - from my view - the first, third and fourth sequence are equal:
I am using the colors of my cube - some other cubes my vary:
U: white / F: red / R: blue
B: purple / D: yellow / L: green
If you are doing the first sequence holding the cube with the colors i listed and then turn the scrambled cube, so that these colors are on the DBL position : B: yellow / D: purple / L: blue and then do the third sequence backwards, you get a solved cube. The same will happen with the fourth sequence backwards if the DBL cube is: B: green / D: red / L: white
(on my list it is positon F)
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2x2x2 antipodes

Submitted by B MacKenzie on Wed, 10/11/2017 - 18:34.
You are right. The above post is in error. I revisited this question in a later post:

2x2x2 Cube

In the later post I found I could reduce the 276 antipodes to 8.
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I have checked your list of 8

Submitted by Pangaia on Thu, 02/08/2018 - 06:53.
I have checked your list of 8 antipodes and compared them with my list of 16 and the difference is that i am having the pair M-P which is not on your list of eight:

M: U' F R' U R' F2 R F2 U' R U2 (24)
P: F U' R' F' R F2 R' U' R' U R' U2 (24)

Is this an error of my calculation?
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symmetry reduction

Submitted by B MacKenzie on Fri, 02/09/2018 - 17:11.
If you reduce the antipodes using just the 24 rotational symmetries of a cube (O symmetry) you get the 16 equivalence classes you found. If you add in the 24 mirrored symmetries (Oh symmetry) they compress down to 10 equivalence classes. Then if you include the inverses (Oh+ symmetry) you get just 8 equivalence classes. Here is a table listing representative elements of the equivalence classes. The third column is an identifier string for the Oh+ class and the forth column is a number to identify each class.

Generator Sequence Oh+ Eq Class Identifier Class
O Symmetry


1 R U R' U R R F U' R' U' R R U' F (12) BUR BRD DFL ULF FRU UBL DLB FDR 1
2 R R U R R U F U' R F' U F U' F (12) DRF DBR UBL LDF URB LFU DLB UFR 2
3 R R U F F U' F R' U F' U' R R F (12) DRF DBR UBL LDF URB LFU DLB UFR 2
4 R R F U' R' U F' R F' U F' R F U (24) FUL RBU DBR LDF FRU RFD DLB LUB 3
5 R R U U R U' F U R R F' U F' U' (24) FUL RBU DBR LDF FRU RFD DLB LUB 3
6 R R U R' F U F' U U R F' U R R (24) RUF BRD FDR RBU LUB LFU DLB DFL 4
7 R R U' F U' R' U' F U' R F' U F' R (24) UFR LDF BRD UBL URB RFD DLB FUL 5
8 R R U R' U U R' F' R R U R' F' R (24) UFR LDF BRD UBL URB RFD DLB FUL 5
9 R R U R' U F U U R U F U' F' U (24) UFR LDF BRD UBL URB RFD DLB FUL 5
10 R R U R' F' R' U R' F' R U' F R' U' (24) UFR LDF BRD UBL URB RFD DLB FUL 5
11 R R U R F R' U R R U' F U' F' U' ( 4) UFR ULF DBR DRF URB UBL DLB DFL 6
12 R R U' R' U' F R' F F R U' F R U ( 4) UFR ULF DBR DRF URB UBL DLB DFL 6
13 R R U R R U F U' R F' U R U' R ( 8) UFR ULF RDB DRF URB LUB DLB LDF 7
14 R R U R' U R U F' U F' U F F R' ( 8) UFR ULF RDB DRF URB LUB DLB LDF 7
15 R R U R' U R F U' R' F' R U' F R' (24) ULF BUR DBR FLD FRU DRF DLB LUB 8
16 R R F' U F' R F' R U' R' U' R U' R' (24) ULF BUR DBR FLD FRU DRF DLB LUB 8




Oh Symmetry


1 R U R' U R R F U' R' U' R R U' F (12) BUR BRD DFL ULF FRU UBL DLB FDR 1
2 R R U F F U' F R' U F' U' R R F (24) DRF DBR UBL LDF URB LFU DLB UFR 2
3 R R U U R U' F U R R F' U F' U' (24) FUL RBU DBR LDF FRU RFD DLB LUB 3
4 R R F U' R' U F' R F' U F' R F U (24) FUL RBU DBR LDF FRU RFD DLB LUB 3
5 R R U R' F U F' U U R F' U R R (24) RUF BRD FDR RBU LUB LFU DLB DFL 4
6 R R U R' U U R' F' R R U R' F' R (48) UFR LDF BRD UBL URB RFD DLB FUL 5
7 R R U R' U F U U R U F U' F' U (48) UFR LDF BRD UBL URB RFD DLB FUL 5
8 R R U' R' U' F R' F F R U' F R U ( 8) UFR ULF DBR DRF URB UBL DLB DFL 6
9 R R U R' U R U F' U F' U F F R' (16) UFR ULF RDB DRF URB LUB DLB LDF 7
10 R R F' U F' R F' R U' R' U' R U' R' (48) ULF BUR DBR FLD FRU DRF DLB LUB 8




Oh+ Symmetry


1 R U R' U R R F U' R' U' R R U' F (12) BUR BRD DFL ULF FRU UBL DLB FDR 1
2 R R U F F U' F R' U F' U' R R F (24) DRF DBR UBL LDF URB LFU DLB UFR 2
3 R R F U' R' U F' R F' U F' R F U (48) FUL RBU DBR LDF FRU RFD DLB LUB 3
4 R R U R' F U F' U U R F' U R R (24) RUF BRD FDR RBU LUB LFU DLB DFL 4
5 R R U R' U U R' F' R R U R' F' R (96) UFR LDF BRD UBL URB RFD DLB FUL 5
6 R R U' R' U' F R' F F R U' F R U ( 8) UFR ULF DBR DRF URB UBL DLB DFL 6
7 R R U R' U R U F' U F' U F F R' (16) UFR ULF RDB DRF URB LUB DLB LDF 7
8 R R F' U F' R F' R U' R' U' R U' R' (48) ULF BUR DBR FLD FRU DRF DLB LUB 8




Your Classes


A: U R U' R F' U R' U F' U F R' U2 (12) BUR BRD DFL ULF FRU UBL DLB FDR 1
D: F2 U2 F U' F R F' R U' R' U2 (12) DRF DBR UBL LDF URB LFU DLB UFR 2
E: U R' F R' U' R' U' F2 R' U R' U2 (12) DRF DBR UBL LDF URB LFU DLB UFR 2
B: R2 U R U' R F' U R' U2 R' U2 (24) FUL RBU DBR LDF FRU RFD DLB LUB 3
C: U' R' U' R U' F R' U R' F U F' U2 (24) FUL RBU DBR LDF FRU RFD DLB LUB 3
F: F2 U' R' U' R F U' F' U F R' U2 (24) RUF BRD FDR RBU LUB LFU DLB DFL 4
M: U' F R' U R' F2 R F2 U' R U2 (24) UFR LDF BRD UBL URB RFD DLB FUL 5
N: R U F' U R' F R F' U' R' U R' U2 (24) UFR LDF BRD UBL URB RFD DLB FUL 5
O: U R' F U' R' F R2 F' R' U R' U2 (24) UFR LDF BRD UBL URB RFD DLB FUL 5
P: F U' R' F' R F2 R' U' R' U R' U2 (24) UFR LDF BRD UBL URB RFD DLB FUL 5
H: R' F U F' R F2 R' U R' U' R U2 (4) UFR ULF DBR DRF URB UBL DLB DFL 6
L: R U R F' U F2 U' R F' U' R' U2 (4) UFR ULF DBR DRF URB UBL DLB DFL 6
I: U R' U F' R' F U' R F' R' U' R' U2 (8) UFR ULF RDB DRF URB LUB DLB LDF 7
J: U F' R F' R F2 R' U R U R' U2 (8) UFR ULF RDB DRF URB LUB DLB LDF 7
G: R' F R' U R' U F' R' F U2 R' U2 (24) ULF BUR DBR FLD FRU DRF DLB LUB 8
K: R' F' R U' R' F' R F' R U' F R' U2 (24) ULF BUR DBR FLD FRU DRF DLB LUB 8
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Class 5 difference

Submitted by Pangaia on Fri, 02/09/2018 - 19:01.
Thank you for your detailed answer.
I have used your sequences of Class 5 (O symmetry) on my Pocket Cube and
Revenge (as Pocket) and compared them. 7/8 & 9/10 are different.

As example here are sequences 8 & 10:

abload.de/img/pc4eusm.jpg

Pocket Cube shows sequence 8
Revenge shows sequence 10
In the upper picture the Pocket cube shows on the top the Up side, left the Front and on the right the Right.
The Revenge shows on the top the Back side, left the Up side and on the right the Right side.
(to make the difference obvious)
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anti-symmetry

Submitted by B MacKenzie on Fri, 02/09/2018 - 20:02.
Oh symmetry classes 6 and 7 above are related by inversion. The inverses of class 6 form class 7 and the two classes merge into the Oh+ class 5 when using anti-symmetry. If you take the inverse of instance 6 and compare it to instance 7 you will find they are Oh conjugates and visually similar.

R' F R U' R2 F R U2 R U' R2

R2 U R' U F U2 R U F U' F' U

Addendum:

The second cube may be transformed into the first by conjugating with the diagonal mirror plane perpendicular to the UB-DF axis (bisects the right face and reflects the U face onto F): 
R2 U  R' U  F  U2 R  U  F  U' F' U

R2 F' R  F' U' F2 R' F' U' F  U  F'  y-z mirror conj
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I have posted this list befor

Submitted by B MacKenzie on Sat, 02/10/2018 - 01:06.
I have posted this list before but I'll post it again here. The list gives the re-mapping of the cube turns produced by conjugating with the 48 cubic symmetries. The m column gives the permutation of the x,y,z coordinate axes produced by the symmetry transform. For example symmetry element 2, a two fold rotation about the x or right axis, rotates the -y (down) axis onto the y axis and the -z (back) axis onto the z axis. The last column gives the Schoenflies symbol for the symmetry element.

Anyway, if one has a generator turn sequence for a cube, one may form symmetry conjugates by re-mapping the turns as listed here. 
               Turn Mapping                  m     Schoenflies Symbol
 1  R  R' U  U' F  F' L  L' D  D' B  B'   x, y, z  E  
 2  R  R' D  D' B  B' L  L' U  U' F  F'   x,-y,-z  C2x 
 3  L  L' U  U' B  B' R  R' D  D' F  F'  -x, y,-z  C2y 
 4  L  L' D  D' F  F' R  R' U  U' B  B'  -x,-y, z  C2z 
 5  B  B' R  R' D  D' F  F' L  L' U  U'  -z, x,-y  C3x'y'z 
 6  U  U' B  B' L  L' D  D' F  F' R  R'   y,-z,-x  C23x'y'z
 7  F  F' L  L' D  D' B  B' R  R' U  U'   z,-x,-y  C3x'yz' 
 8  D  D' B  B' R  R' U  U' F  F' L  L'  -y,-z, x  C23x'yz'
 9  B  B' L  L' U  U' F  F' R  R' D  D'  -z,-x, y  C3xy'z' 
10  D  D' F  F' L  L' U  U' B  B' R  R'  -y, z,-x  C23xy'z'
11  F  F' R  R' U  U' B  B' L  L' D  D'   z, x, y  C3xyz 
12  U  U' F  F' R  R' D  D' B  B' L  L'   y, z, x  C23xyz
13  R  R' B  B' U  U' L  L' F  F' D  D'   x,-z, y  C4x 
14  R  R' F  F' D  D' L  L' B  B' U  U'   x, z,-y  C34x
15  F  F' U  U' L  L' B  B' D  D' R  R'   z, y,-x  C4y 
16  B  B' U  U' R  R' F  F' D  D' L  L'  -z, y, x  C34y
17  D  D' R  R' F  F' U  U' L  L' B  B'  -y, x, z  C4z 
18  U  U' L  L' F  F' D  D' R  R' B  B'   y,-x, z  C34z
19  U  U' R  R' B  B' D  D' L  L' F  F'   y, x,-z  C2xy 
20  D  D' L  L' B  B' U  U' R  R' F  F'  -y,-x,-z  C2xy' 
21  F  F' D  D' R  R' B  B' U  U' L  L'   z,-y, x  C2xz 
22  B  B' D  D' L  L' F  F' U  U' R  R'  -z,-y,-x  C2xz' 
23  L  L' F  F' U  U' R  R' B  B' D  D'  -x, z, y  C2yz 
24  L  L' B  B' D  D' R  R' F  F' U  U'  -x,-z,-y  C2yz' 
25  U' U  R' R  F' F  D' D  L' L  B' B    y, x, z  σd_xy 
26  D' D  L' L  F' F  U' U  R' R  B' B   -y,-x, z  σd_xy' 
27  F' F  U' U  R' R  B' B  D' D  L' L    z, y, x  σd_xz 
28  B' B  U' U  L' L  F' F  D' D  R' R   -z, y,-x  σd_xz' 
29  R' R  F' F  U' U  L' L  B' B  D' D    x, z, y  σd_yz 
30  R' R  B' B  D' D  L' L  F' F  U' U    x,-z,-y  σd_yz' 
31  L' L  B' B  U' U  R' R  F' F  D' D   -x,-z, y  S4x 
32  L' L  F' F  D' D  R' R  B' B  U' U   -x, z,-y  S34x
33  F' F  D' D  L' L  B' B  U' U  R' R    z,-y,-x  S4y 
34  B' B  D' D  R' R  F' F  U' U  L' L   -z,-y, x  S34y
35  D' D  R' R  B' B  U' U  L' L  F' F   -y, x,-z  S4z 
36  U' U  L' L  B' B  D' D  R' R  F' F    y,-x,-z  S34z
37  L' L  D' D  B' B  R' R  U' U  F' F   -x,-y,-z  i  
38  L' L  U' U  F' F  R' R  D' D  B' B   -x, y, z  σh_x 
39  R' R  D' D  F' F  L' L  U' U  B' B    x,-y, z  σh_y 
40  R' R  U' U  B' B  L' L  D' D  F' F    x, y,-z  σh_z 
41  D' D  F' F  R' R  U' U  B' B  L' L   -y, z, x  S6x'y'z 
42  F' F  L' L  U' U  B' B  R' R  D' D    z,-x, y  S56x'y'z
43  U' U  F' F  L' L  D' D  B' B  R' R    y, z,-x  S6x'yz' 
44  B' B  R' R  U' U  F' F  L' L  D' D   -z, x, y  S56x'yz'
45  U' U  B' B  R' R  D' D  F' F  L' L    y,-z, x  S6xy'z' 
46  F' F  R' R  D' D  B' B  L' L  U' U    z, x,-y  S56xy'z'
47  D' D  B' B  L' L  U' U  F' F  R' R   -y,-z,-x  S6xyz 
48  B' B  L' L  D' D  F' F  R' R  U' U   -z,-x,-y  S56xyz
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