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Pattern Sub-Groups

Submitted by B MacKenzie on Sat, 01/12/2008 - 22:10.

I've been fooling around with symmetric Rubik cube patterns: "Cross" patterns, involving corners only; "Check" patterns, involving edges only; and "Dots" patterns, involving edges and corners.

By rotating the eight corners a quarter turn about one of the cube axes a cube state with a cross on four faces is produced. A second such state may be produced by rotating about a different cube axes: 

    R U B R U R D R U' D B' U' B' D' B' R' D' B'
R B D R B R F R B' F D' B' D' F' D' R' F' D'

By recursively forming binary products from these two generators a group of 24 cube states is produced composed of all the different pure cross patterns plus the identity element. This Cross group is composed of five conjugate classes:

  1. The identity cube
  2. Six patterns via rotating the corners a quarter turn about the four fold cube axes
  3. Three by rotating a half turn about the four fold cube axes
  4. Eight by rotating a third of a turn about the cube's three fold axes
  5. Six by rotating a half turn about the cube's two fold axes.

These patterns are well known, the three fold is Plummer's cross, the two fold is Christman's cross, etc. What is interesting is that the cross cube states form a sub-group within the Rubik group. What is more that group is isomorphic with the cubic pure rotation point group, Schoenflies group O.

Generators for the check patterns are formed by rotation the edges a third turn about two of the cubes three fold axes:

    U U D' F L L U' F' R' L F' B L D F F L' U D D
U U D' B R R U' B' L' R B' F R D B B R' U D D

The group formed with these two generators has twelve elements in three classes (actually four classes, the three fold patterns divide into clockwise and anti-clockwise in this group.)

  1. The Identity cube
  2. Eight patterns by rotating the edges a third of a turn about the three fold axes
  3. Three patterns by rotating a half turn about the four fold axes

This group is isomorphic with the tetrahedral pure rotation point group, Schoenflies group T. These Check group patterns composed with the corresponding elements from the Cross group form the Dots pattern group. The Dots group again is isomorphic with the T point group. The Check group may be expanded by adding a third generator, a cube with check patterns on two opposite faces:

    F F U U D D F B' U U D D F' B'

This group has 24 elements and is isomorphic with the Th point group. The Th point group is not a pure rotation group. It includes reflection elements. This makes the added elements difficult to describe. For example, the group includes Pons Asinorum which is homologous to the inversion symmetry operation. The generator above is homologous to a reflection across a mirror plane symmetry element.

I don't know if this analysis has been done before. I suspect that it has. But it tickled me when I saw the connection between the pattern sub-groups within the Rubik group and point group symmetry. I thought I would share it.

If you combine the generators for the Cross and Check sub-groups you generate a sub-group comprised of patterns with combinations of Checks, Crosses and Dots on the various faces. There are 576 elements in the group. When you eliminate elements which differ only in the orientation the cube is in when a particular pattern is applied you get 46 unique patterns. That is to say the group reduced by rotational symmetry has 46 elements. Representative elements from the 46 symmetry classes are listed below. The minimal solutions were found using Herbert Kociemba's Optimal Cube Solver.

    Forming group from 5 generators

Group Order: 576 elements in 40 conj. classes
symmetry reduced to 46 elements.

Class 1: 1 element, 1 sym class, 1 unique

	E

Class 2: 1 element, 1 sym class, 1 unique

	U U D D R R L L F F B B  (12q*) "Check group, Pons Asinorum"

Class 3: 3 elements, 1 sym class, 1 unique

	F F U U D D F B' U U D D F' B'  (14q*) "Check Group, generator"

Class 4: 3 elements, 1 sym class, 1 unique

	R R U U D D F F B B R R  (12q*)

Class 5: 3 elements, 1 sym class, 1 unique

	U U F F U U R L' F F U U F F R' L  (16q*) "Cross Group"

Class 6: 3 elements, 1 sym class, 1 unique

	U U F F U U R R L L U U B B U U  (16q*) "Check Group"

Class 7: 4 elements, 1 sym class, 1 unique

	U R B D B U' F' D' B' D' R' L F B R U' L' D' B' L  (20q*) "Check Group"

Class 8: 4 elements, 1 sym class, 1 unique

	U U D' F L L U' F' R' L F' B L D F F L' U D D  (20q*) "Check Group"

Class 9: 4 elements, 1 sym class, 0 unique

Class 10: 4 elements, 1 sym class, 0 unique

Class 11: 6 elements, 1 sym class, 1 unique

	U F F U U D R L' U' D R U U L' U' D R L'  (18q*)

Class 12: 6 elements, 1 sym class, 1 unique

	R U D L L U D F' B' R R F' B' L'  (14q*)

Class 13: 6 elements, 1 sym class, 1 unique

	F U U F F R L' D D B B D D R' L F  (16q*) "Cross Group"

Class 14: 6 elements, 1 sym class, 1 unique

	R U B R U R D R U' D B' U' B' D' B' R' D' B'  (18q*) "Cross Group, generator"

Class 15: 8 elements, 1 sym class, 1 unique

	U U D B' L L U B U D' F B' U' B' R R U F' B B  (20q*) "Cross Group"

Class 16: 8 elements, 1 sym class, 1 unique

	U U D D R F D D L' F' R L' F B' L F U U L' B'  (20q*)

Class 17: 9 elements, 2 sym classes, 2 unique

	U U F F B B U U R L F F B B R L  (16q*)

	U U F B' R R L L F B' U U  (12q*) "Dots Group"

Class 18: 9 elements, 2 sym classes, 2 unique

	U U D D F B' U U D D F' B  (12q*)

	R R U D' R R L L U' D L L  (12q*) 

Class 19: 12 elements, 1 sym class, 1 unique

	U U D D F' L' U U B L U' D R' L U' F' L L U B  (20q*)

Class 20: 12 elements, 1 sym class, 1 unique

	U U B L' B' L' B' L U U F' B R D F' U' L' F' R U  (20q*)

Class 21: 12 elements, 1 sym class, 0 unique

Class 22: 12 elements, 1 sym class, 0 unique

Class 23: 18 elements, 2 sym classes, 2 unique

	R F F B B R R U D F F B B U D R  (16q*)

	U R F D B U L' F' R' B' D' L' U D D F' B R  (18q*)

Class 24: 18 elements, 2 sym classes, 2 unique

	U U R U B L F R D' B' U' F' L' D' R' F' B D'  (18q*)

	U U D F F U U R R L L U U B B U  (16q*) 

Class 25: 18 elements, 2 sym classes, 2 unique

	U F F U U D F' B U D' F L L B' U' D F' B  (18q*)

	R U D R R U D F B L L F B L'  (14q*)

Class 26: 18 elements, 2 sym classes, 2 unique

	R U D R R U U R L B B R L U D' R'  (16q*)

	U D' R L' U R R U D D R' L U' D R B B R'  (18q*)

Class 27: 24 elements, 2 sym classes, 2 unique

	U' D F U F L D R U B' L' F' R' D' B' D'  (16q*)

	U R R U U F B R' L' B B R R U' D' F  (16q*)

Class 28: 24 elements, 2 sym classes, 2 unique

	U U F U' D R' L' U D' B D D B R L' D D F' B L  (20q*)

	R F F R L' D D R L U' D' F' B R' L' U  (16q*)

Class 29: 24 elements, 2 sym classes, 2 unique

	R U U B B U U F' B' R L U' D' F' B U'  (16q*)

	U D' R B B L' F' B R L' F' B B D D B' U D'  (18q*)

Class 30: 24 elements, 2 sym classes, 2 unique

	U D' R U U R L L F' B R L' B L L B' U' D  (18q*)

	U' R L' U D F' B' R L U U L L U U B  (16q*)

Class 31: 24 elements, 1 sym class, 1 unique

	U R' B' D F B' R' U F' B U' L F B' D' F L U' F' B  (20q*)

Class 32: 24 elements, 1 sym class, 1 unique

	U L' F B U D' B' R' F B' R F U' D F' B' R D R' L'  (20q*)

Class 33: 24 elements, 2 sym classes, 0 unique

Class 34: 24 elements, 2 sym classes, 0 unique

Class 35: 24 elements, 2 sym classes, 0 unique

Class 36: 24 elements, 2 sym classes, 0 unique

Class 37: 32 elements, 4 sym classes, 4 unique

	U U F R' F' R' F' R U D R L' B R D' L' B' D' F L  (20q*)

	U U R L F F U' D L L F B U U R L'  (16q*)

	U' R' B B U R F B' R L' B' D' R R B U' D D F B'  (20q*)

	U F B' R' L U D' B'  (8q*) "Dots Group"

Class 38: 32 elements, 4 sym classes, 4 unique

	U R' B' D' F' U R L U' D B' R' B' L' F' R B R U B  (20q*)

	U U D D R' L U D' F' B R' L  (12q*)

	U' R' L' B' D' R' L U' R' F D F B' U L B U F U' D  (20q*)

	R R F' B R L' U' D F' B D D  (12q*)

Class 39: 32 elements, 4 sym classes, 0 unique

Class 40: 32 elements, 4 sym classes, 0 unique
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Revised Table

Submitted by B MacKenzie on Mon, 01/14/2008 - 19:13.

I messed around with my program today and cleaned up the output a bit. I fudged the conjugate classes so that the symmetry classes for elements incorporating C3 and S6 operations on the edge pattern aren't split between two conjugate classes. There really are forty conjugate classes rather than thirty. I added the order of the symmetry class and Schoenflies notation for the Cross, Check and Dots point group homologs. The revised table is below. 

    Full Pattern Group

Group order: 576
Conjugate Classes: 30
Symmetry Classes: 46
Conjugate Class 1; elements: 1; symmetry classes: 1
	Symmetry Class 1; elements: 1:
	  Representative Element:  E

Conjugate Class 2; elements: 1; symmetry classes: 1
	Symmetry Class 1; elements: 1:
	  Representative Element: U U D D R R L L F F B B  (12q*)  Check Group, i, Pons Assinorum

Conjugate Class 3; elements: 3; symmetry classes: 1
	Symmetry Class 1; elements: 3:
	  Representative Element:  U U R R L L U D' R R L L U' D'  (14q*)  Check Group, sigma h

Conjugate Class 4; elements: 3; symmetry classes: 1
	Symmetry Class 1; elements: 3:
	  Representative Element:  U U R R L L F F B B U U  (12q*) 

Conjugate Class 5; elements: 3; symmetry classes: 1
	Symmetry Class 1; elements: 3:
	  Representative Element:  U U R R F B' U U R R U U F' B R R  (16q*)  Cross Group, C2y 

Conjugate Class 6; elements: 3; symmetry classes: 1
	Symmetry Class 1; elements: 3:
	  Representative Element:  F F R R F F D D U U F F L L F F  (16q*) Check Group, C2y  

Conjugate Class 7; elements: 6; symmetry classes: 1
	Symmetry Class 1; elements: 6:
	  Representative Element:  B D R B D B U B D' U R' D' R' U' R' B' U' R'  (18q*)  Cross Group, C4y 

Conjugate Class 8; elements: 6; symmetry classes: 1
	Symmetry Class 1; elements: 6:
	  Representative Element:  F' U D B B U D R' L' F F R' L' B  (14q*)

Conjugate Class 9; elements: 6; symmetry classes: 1
	Symmetry Class 1; elements: 6:
	  Representative Element:  U D' R' L U L L D' R' L U D' R' L L B B L'  (18q*) 

Conjugate Class 10; elements: 6; symmetry classes: 1
	Symmetry Class 1; elements: 6:
	  Representative Element:  U U D F B R L F F B B R L F B D  (16q*)  Cross Group, C2xz

Conjugate Class 11; elements: 8; symmetry classes: 1
	Symmetry Class 1; elements: 8:
	  Representative Element:  D' D' U' R B' B' D' R' B F' R' L B U R' R' B' D U' U'  (20q*)  Check Group, C3xyz 

Conjugate Class 12; elements: 8; symmetry classes: 1
	Symmetry Class 1; elements: 8:
	  Representative Element:  U U D' B R R U' B' U' D B F' U B L L U' B B F  (20q*)  Cross Group, C3xyz 

Conjugate Class 13; elements: 8; symmetry classes: 1
	Symmetry Class 1; elements: 8:
	  Representative Element:  U U D D F' L' U U B L F' B R L' B' L' D D B R  (20q*) 

Conjugate Class 14; elements: 8; symmetry classes: 1
	Symmetry Class 1; elements: 8:
	  Representative Element:  B' D' R' F' R' B L F R F D U' L' R' D' B U F R U'  (20q*)  Check Group, S6xyz 

Conjugate Class 15; elements: 9; symmetry classes: 2
	Symmetry Class 1; elements: 3:
	  Representative Element:  F F D U' R R L L D U' F F  (12q*)  Dots Group, C2y 

	Symmetry Class 2; elements: 6:
	  Representative Element:  U U F F U D F F B B U D F F U U  (16q*) 

Conjugate Class 16; elements: 9; symmetry classes: 2
	Symmetry Class 1; elements: 3:
	  Representative Element:  U D' R R L L U' D R R L L  (12q*) 

	Symmetry Class 2; elements: 6:
	  Representative Element:  F F U D' F F B B U' D B B  (12q*) 

Conjugate Class 17; elements: 18; symmetry classes: 2
	Symmetry Class 1; elements: 12:
	  Representative Element:  R R L F F R L F F B B R L F F R  (16q*) 

	Symmetry Class 2; elements: 6:
	  Representative Element:  U U R U B L F R D' B' U' F' L' D' R' F' B D'  (18q*) 

Conjugate Class 18; elements: 18; symmetry classes: 2
	Symmetry Class 1; elements: 12:
	  Representative Element:  F U D B B U U F' B' L L F' B' U D' F'  (16q*) 

	Symmetry Class 2; elements: 6:
	  Representative Element:  U F F U' R L' U D' R' L L U U R' U' D R' L  (18q*) 

Conjugate Class 19; elements: 18; symmetry classes: 2
	Symmetry Class 1; elements: 12:
	  Representative Element: U D' R' L U L L U' R' L U' D R' L L F F R'  (18q*) 

	Symmetry Class 2; elements: 6:
	  Representative Element:  U' R L U U R L F B D D F B D  (14q*) 

Conjugate Class 20; elements: 18; symmetry classes: 2
	Symmetry Class 1; elements: 12:
	  Representative Element:  U R L' F B B D' B' R' U' L' D' F R B L U F  (18q*) 

	Symmetry Class 2; elements: 6:
	  Representative Element:  R F F B B R R U D F F B B U D R  (16q*) 

Conjugate Class 21; elements: 24; symmetry classes: 1
	Symmetry Class 1; elements: 24:
	  Representative Element:  U D' R B' U' L U' D B' R U D' R' F U' D L' D F R'  (20q*) 

Conjugate Class 22; elements: 24; symmetry classes: 1
	Symmetry Class 1; elements: 24:
	  Representative Element:  U L' D' R D F' R F R F L' B R' U' R R F' L' U D  (20q*) 

Conjugate Class 23; elements: 24; symmetry classes: 1
	Symmetry Class 1; elements: 24:
	  Representative Element:  U R F F U' R' U' D R L' U R B B U' R' F' B U' D  (20q*) 

Conjugate Class 24; elements: 24; symmetry classes: 1
	Symmetry Class 1; elements: 24:
	  Representative Element:  U D B' D' R L F B' L' D' R L' D R F' B R' L' U F'  (20q*) 

Conjugate Class 25; elements: 48; symmetry classes: 2
	Symmetry Class 1; elements: 24:
	  Representative Element: F R L' F' B' U' D' R' L' B B L L F F D'  (16q*) 

	Symmetry Class 2; elements: 24:
	  Representative Element: U D' R B B R L L F B' R' L B U U B' U' D  (18q*) 

Conjugate Class 26; elements: 48; symmetry classes: 2
	Symmetry Class 1; elements: 24:
	  Representative Element:  F U U F B U D R L' F B U D B B U'  (16q*) 

	Symmetry Class 2; elements: 24:
	  Representative Element:  U D' R' U' R' B' D' F' U' L B R F D L D  (16q*) 

Conjugate Class 27; elements: 48; symmetry classes: 2
	Symmetry Class 1; elements: 24:
	  Representative Element:  U R R U' F' B U D' F F B' U U B' R' L F' B  (18q*) 

	Symmetry Class 2; elements: 24:
	  Representative Element: F R L' F B U D R L B B R R F F D'  (16q*) 

Conjugate Class 28; elements: 48; symmetry classes: 2
	Symmetry Class 1; elements: 24:
	  Representative Element:  U U R L' F F B' R B D L U R F' D' B' U' L' F' L  (20q*) 

	Symmetry Class 2; elements: 24:
	  Representative Element:  F U U F B U D' R L U U R R F' B D  (16q*) 

Conjugate Class 29; elements: 64; symmetry classes: 4
	Symmetry Class 1; elements: 24:
	  Representative Element:  U U R L U D F F R R F' B' R' L' U U  (16q*) 

	Symmetry Class 2; elements: 8:
	  Representative Element:  U' B F' R' L U' D B  (8q*)  Dots Group, C3xyz

	Symmetry Class 3; elements: 8:
	  Representative Element:  U U L F' L' F' L' F U D F B' R F D' B' R' D' L B  (20q*) 

	Symmetry Class 4; elements: 24:
	  Representative Element:  U D' F R D D F' R' F' B R L' F R U U F' R' U D'  (20q*) 

Conjugate Class 30; elements: 64; symmetry classes: 4
	Symmetry Class 1; elements: 24:
	  Representative Element:  R R U D' R' L F B' U D' B B  (12q*) 

	Symmetry Class 2; elements: 8:
	  Representative Element:  U U D D R L' U D' F B' R L'  (12q*) 

	Symmetry Class 3; elements: 8:
	  Representative Element:  U R L B D R' L U L F' D' F' B U' R' B' U' F' U D'  (20q*) 

	Symmetry Class 4; elements: 24:
	  Representative Element:  U R F U F D' B' U' F' U' R' L F B R D' L' U' F' R  (20q*)
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