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God's Algorithm to 11f, Summarized by Symmetry Class

Submitted by Jerry Bryan on Mon, 02/26/2007 - 12:21.
God's Algorithm to 11f has been posted before, but it has not been summarized by symmetry class. The symmetry classes in the table below follow Dan Hoey's taxonomy. 
           
|x| Symmetry  Size           Patterns        Positions
     Class     of
                xM

  0      M       1               1               1
      Total                      1               1
      
  1      CR     12               1              12
         Q       6               1               6
      Total                      2              18
      
  2      I      48               4             192
         HW     24               1              24
         CR     12               1              12
         R       6               1               6
         CX      6               1               6
         X       3               1               3
      Total                      9             243
      
  3      I      48              61            2928
         HW     24               4              96
         ES     24               8             192
         HQ     12               2              24
      Total                     75            3240
      
  4      I      48             873           41904
         HV     24               1              24
         HW     24              17             408
         ES     24              34             816
         HS     12               2              24
         EX     12               4              48
         HX      6               2              12
         X       3               1               3
      Total                    934           43239
      
  5      I      48           11886          570528
         HV     24               4              96
         HW     24              48            1152
         ES     24             123            2952
         HS     12               2              24
         CR     12               2              24
         HQ     12               8              96
         EX     12               2              24
         Q       6               2              12
      Total                  12077          574908
      
  6      I      48          158347         7600656
         HV     24              22             528
         AV     24               3              72
         CV     24               2              48
         HW     24             176            4224
         ES     24             507           12168
         V      12               1              12
         W      12               3              36
         HS     12               5              60
         CR     12               8              96
         HQ     12               4              48
         CS     12               4              48
         EX     12              27             324
         R       6               3              18
         HX      6              11              66
         CX      6               4              24
         X       3               3               9
         M       1               1               1
      Total                 159131         7618438
      
  7      I      48         2098601       100732848
         HV     24             102            2448
         AV     24               9             216
         CV     24              33             792
         HW     24             555           13320
         ES     24            2182           52368
         V      12               1              12
         W      12               1              12
         HS     12               7              84
         AS     12               1              12
         CR     12               7              84
         HQ     12              33             396
         AP     12               7              84
         CS     12               6              72
         EX     12              18             216
         S       6               2              12
         Q       6               1               6
         R       6               1               6
         P       6               3              18
         CX      6               5              30
      Total                2101575       100803036
      
  8      I      48        27752374      1332113952
         HV     24             440           10560
         AV     24              20             480
         CV     24              78            1872
         HW     24            1662           39888
         ES     24            7210          173040
         V      12               7              84
         W      12               3              36
         HS     12              42             504
         AS     12               1              12
         CR     12               7              84
         HQ     12              17             204
         AP     12              16             192
         CS     12              22             264
         EX     12             152            1824
         HT      8               1               8
         S       6               4              24
         Q       6               3              18
         R       6               3              18
         P       6               8              48
         HX      6              20             120
         AX      6               1               6
         CX      6               5              30
         X       3               6              18
         H       2               1               2
      Total               27762103      1332343288
      
  9      I      48       366575597     17595628656
         HV     24            1846           44304
         AV     24             117            2808
         CV     24             365            8760
         HW     24            4977          119448
         ES     24           28017          672408
         ET     16               3              48
         V      12              10             120
         W      12              18             216
         HS     12              33             396
         AS     12               1              12
         CR     12               7              84
         HQ     12              34             408
         AP     12              25             300
         CS     12              10             120
         EX     12             132            1584
         HT      8               3              24
         S       6               1               6
         Q       6               1               6
         P       6              13              78
         CX      6               1               6
         X       3               1               3
      Total              366611212     17596479795
      
 10      I      48      4838439044    232245074112
         HV     24            8036          192864
         AV     24             419           10056
         CV     24            1861           44664
         HW     24           14062          337488
         ES     24           99641         2391384
         ET     16               6              96
         V      12              33             396
         W      12              34             408
         HS     12             115            1380
         AS     12              22             264
         CR     12              42             504
         HQ     12             111            1332
         AP     12              48             576
         CS     12             111            1332
         EX     12             515            6180
         HT      8               2              16
         S       6              11              66
         Q       6               6              36
         P       6              10              60
         HX      6               2              12
         AX      6              11              66
         CX      6               3              18
         X       3               2               6
      Total             4838564147    232248063316
      
 11      I      48     63818314101   3063279076848
         HV     24           32617          782808
         AV     24            1439           34536
         CV     24            6713          161112
         HW     24           40016          960384
         ES     24          323655         7767720
         ET     16               5              80
         V      12              72             864
         W      12              58             696
         HS     12             423            5076
         AS     12              92            1104
         CR     12             177            2124
         HQ     12             280            3360
         AP     12             110            1320
         CS     12             230            2760
         EX     12             641            7692
         HT      8               2              16
         CT      8               1               8
         S       6              28             168
         Q       6              10              60
         R       6              10              60
         P       6              22             132
         AX      6               9              54
         CX      6               5              30
      Total            63818720716   3063288809012
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Its nice to see that the numb

Submitted by silviu on Tue, 02/27/2007 - 09:15.
Its nice to see that the number of symmetric cubes corresponds to the ones posted by Herbert on my symmetric cube article. Did you do the whole calculation again or did you have the number of symmetric cubes stored from before?

Regards,

Silviu
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From Before

Submitted by Jerry Bryan on Tue, 02/27/2007 - 14:03.
This was all from before. I've had all the symmetry data out to 10f and 12q since about 1997. Extending to 11f and 13q was a little more recent.

It's just that it made no sense to post the data until I had some reasonable way to describe the symmetry classes. That reasonable way arrived because Dan Hoey gave me permission to post his taxonomy and because I found Herbert Kociemba's description of his use of Shoenflies symbols in Cube Explorer.

By the way, I agree that Shoenflies symbols are the best way to go in the future, even though no system is perfect. However, some of the names in Dan Hoey's taxonomy do make mneumonic sense. That's because he tried to relate names for certain symmetry classes to the corresponding names for certain pretty cube patterns. For example, certain cubes with H's for faces may have H symmetry and certain cubes with X's for for faces may have X symmetry. I do not remember all the details of how this was worked out. But there are too many different symmetry classes for all of them to make sense. Many of the names in the taxonomy are clearly arbitrary.

Another aspect of the taxonomy that makes sense is that many of the two letter names in the taxonomy are based on sort of "combining" two of the one letter names. Symmetry can be calculated separately for the corners and the edges. Just as we can calculate Symm(x) for the entire cube, we can calculate Symm(xc) for the corners and Symm(xe) for the edges. Then, it's the case that Symm(x) = Symm(xc) intersect Symm(xe).

It's my experience that the eye can often see the symmetries of the corners and the symmetries of the edges separately, and therefore that the eye often perceives a position as having more symmetry than a straightforward calculation of Symm(x) would suggest. For example, if Symm(xc)=A1 and Symm(xe)=X1, then we have that Symm(x)=AX1. But I think the eye perceives both the A1 symmetry and the X1 symmetry, not just the lesser AX1 symmetry.
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