Symmetrical Twist Assignment, a chimera

A corner cubie may be moved to any of the eight corner cubicles in three different ways; untwisted, with clockwise twist or with counterclockwise twist. The standard convention is to assign the twist with reference the orientation of the cubie's U/D facelet vis-a-vis the cubicle's U/D face. If the cubie's U/D facelet is on the cubicle's U/D face the cubie is untwisted. If the cubie's U/D facelet is rotated 120° clockwise from the cubicle's U/D face the cubie has clockwise twist and vice versa. The disturbing thing about this convention is that it is unsymmetrical. Under this definition the 12 q-turns have different effects on the twist of the cubies turned. Turns of the U and D faces have no effect on the twist of the cubies while a q-turn of any of the other four faces twist two cubies clockwise and two cubies counterclockwise. Since all the faces are symmetry equivalent it has always seemed to me that there ought to be a way of defining corner cubie orientation which preserves this equivalence.

I looked through the cube lovers archives and found a post by Jerry Bryan back in 1996 which mitigates against my supposition. Jerry related the twist to the conjugacy class of the permutations giving the three orientations for a cubie ⇒ cubicle transformation. I look on these transformations as rotations about the symmetry axes of the cube, that is, as elements of the cubic rotational point group. For example the the UFR cubie may be moved to the URB cubicle by; 1. a 90° ccw rotation about the U/D (y axis), 2. a 90° cw rotation about the R/L (x axis) and 3. a 180° rotation about the two fold axis going through the UR and DL edge cubies (the axis bisecting the angle made by the x and y axes). To evaluate Jerry's argument, I printed out the rotations giving no twist ( 0 ), clockwise twist ( 1 ), and counterclockwise twist ( 2 ) for all the cubie to cubicle transformations using Schoenflies notation:

UFR ⇒ UFR
    0  x, y, z  E  
    1  y, z, x  C23xyz
    2  z, x, y  C3xyz 
UFR ⇒ UBL
    0 -x, y,-z  C2y 
    1 -y, z,-x  C23xy'z'
    2 -z, x,-y  C3x'y'z 
UFR ⇒ DFL
    0 -x,-y, z  C2z 
    1 -y,-z, x  C23x'yz'
    2 -z,-x, y  C3xy'z' 
UFR ⇒ DBR
    0  x,-y,-z  C2x 
    1  y,-z,-x  C23x'y'z
    2  z,-x,-y  C3x'yz' 
UFR ⇒ URB
    0  z, y,-x  C4y 
    1  x, z,-y  C34x
    2  y, x,-z  C2xy 
UFR ⇒ ULF
    0 -z, y, x  C34y
    1 -x, z, y  C2yz 
    2 -y, x, z  C4z 
UFR ⇒ DRF
    0  z,-y, x  C2xz 
    1  x,-z, y  C4x 
    2  y,-x, z  C34z
UFR ⇒ DLB
    0 -z,-y,-x  C2xz' 
    1 -x,-z,-y  C2yz' 
    2 -y,-x,-z  C2xy' 

***

UBL ⇒ UFR
    0 -x, y,-z  C2y 
    1  y,-z,-x  C23x'y'z
    2 -z,-x, y  C3xy'z' 
UBL ⇒ UBL
    0  x, y, z  E  
    1 -y,-z, x  C23x'yz'
    2  z,-x,-y  C3x'yz' 
UBL ⇒ DFL
    0  x,-y,-z  C2x 
    1 -y, z,-x  C23xy'z'
    2  z, x, y  C3xyz 
UBL ⇒ DBR
    0 -x,-y, z  C2z 
    1  y, z, x  C23xyz
    2 -z, x,-y  C3x'y'z 
UBL ⇒ URB
    0 -z, y, x  C34y
    1 -x,-z,-y  C2yz' 
    2  y,-x, z  C34z
UBL ⇒ ULF
    0  z, y,-x  C4y 
    1  x,-z, y  C4x 
    2 -y,-x,-z  C2xy' 
UBL ⇒ DRF
    0 -z,-y,-x  C2xz' 
    1 -x, z, y  C2yz 
    2  y, x,-z  C2xy 
UBL ⇒ DLB
    0  z,-y, x  C2xz 
    1  x, z,-y  C34x
    2 -y, x, z  C4z 

***

DFL ⇒ UFR
    0 -x,-y, z  C2z 
    1 -y, z,-x  C23xy'z'
    2  z,-x,-y  C3x'yz' 
DFL ⇒ UBL
    0  x,-y,-z  C2x 
    1  y, z, x  C23xyz
    2 -z,-x, y  C3xy'z' 
DFL ⇒ DFL
    0  x, y, z  E  
    1  y,-z,-x  C23x'y'z
    2 -z, x,-y  C3x'y'z 
DFL ⇒ DBR
    0 -x, y,-z  C2y 
    1 -y,-z, x  C23x'yz'
    2  z, x, y  C3xyz 
DFL ⇒ URB
    0  z,-y, x  C2xz 
    1 -x, z, y  C2yz 
    2 -y,-x,-z  C2xy' 
DFL ⇒ ULF
    0 -z,-y,-x  C2xz' 
    1  x, z,-y  C34x
    2  y,-x, z  C34z
DFL ⇒ DRF
    0  z, y,-x  C4y 
    1 -x,-z,-y  C2yz' 
    2 -y, x, z  C4z 
DFL ⇒ DLB
    0 -z, y, x  C34y
    1  x,-z, y  C4x 
    2  y, x,-z  C2xy 

***

DBR ⇒ UFR
    0  x,-y,-z  C2x 
    1 -y,-z, x  C23x'yz'
    2 -z, x,-y  C3x'y'z 
DBR ⇒ UBL
    0 -x,-y, z  C2z 
    1  y,-z,-x  C23x'y'z
    2  z, x, y  C3xyz 
DBR ⇒ DFL
    0 -x, y,-z  C2y 
    1  y, z, x  C23xyz
    2  z,-x,-y  C3x'yz' 
DBR ⇒ DBR
    0  x, y, z  E  
    1 -y, z,-x  C23xy'z'
    2 -z,-x, y  C3xy'z' 
DBR ⇒ URB
    0 -z,-y,-x  C2xz' 
    1  x,-z, y  C4x 
    2 -y, x, z  C4z 
DBR ⇒ ULF
    0  z,-y, x  C2xz 
    1 -x,-z,-y  C2yz' 
    2  y, x,-z  C2xy 
DBR ⇒ DRF
    0 -z, y, x  C34y
    1  x, z,-y  C34x
    2 -y,-x,-z  C2xy' 
DBR ⇒ DLB
    0  z, y,-x  C4y 
    1 -x, z, y  C2yz 
    2  y,-x, z  C34z

***

URB ⇒ UFR
    0 -z, y, x  C34y
    1  y, x,-z  C2xy 
    2  x,-z, y  C4x 
URB ⇒ UBL
    0  z, y,-x  C4y 
    1 -y, x, z  C4z 
    2 -x,-z,-y  C2yz' 
URB ⇒ DFL
    0  z,-y, x  C2xz 
    1 -y,-x,-z  C2xy' 
    2 -x, z, y  C2yz 
URB ⇒ DBR
    0 -z,-y,-x  C2xz' 
    1  y,-x, z  C34z
    2  x, z,-y  C34x
URB ⇒ URB
    0  x, y, z  E  
    1 -z, x,-y  C3x'y'z 
    2  y,-z,-x  C23x'y'z
URB ⇒ ULF
    0 -x, y,-z  C2y 
    1  z, x, y  C3xyz 
    2 -y,-z, x  C23x'yz'
URB ⇒ DRF
    0  x,-y,-z  C2x 
    1 -z,-x, y  C3xy'z' 
    2  y, z, x  C23xyz
URB ⇒ DLB
    0 -x,-y, z  C2z 
    1  z,-x,-y  C3x'yz' 
    2 -y, z,-x  C23xy'z'

***

ULF ⇒ UFR
    0  z, y,-x  C4y 
    1  y,-x, z  C34z
    2 -x, z, y  C2yz 
ULF ⇒ UBL
    0 -z, y, x  C34y
    1 -y,-x,-z  C2xy' 
    2  x, z,-y  C34x
ULF ⇒ DFL
    0 -z,-y,-x  C2xz' 
    1 -y, x, z  C4z 
    2  x,-z, y  C4x 
ULF ⇒ DBR
    0  z,-y, x  C2xz 
    1  y, x,-z  C2xy 
    2 -x,-z,-y  C2yz' 
ULF ⇒ URB
    0 -x, y,-z  C2y 
    1  z,-x,-y  C3x'yz' 
    2  y, z, x  C23xyz
ULF ⇒ ULF
    0  x, y, z  E  
    1 -z,-x, y  C3xy'z' 
    2 -y, z,-x  C23xy'z'
ULF ⇒ DRF
    0 -x,-y, z  C2z 
    1  z, x, y  C3xyz 
    2  y,-z,-x  C23x'y'z
ULF ⇒ DLB
    0  x,-y,-z  C2x 
    1 -z, x,-y  C3x'y'z 
    2 -y,-z, x  C23x'yz'

***

DRF ⇒ UFR
    0  z,-y, x  C2xz 
    1 -y, x, z  C4z 
    2  x, z,-y  C34x
DRF ⇒ UBL
    0 -z,-y,-x  C2xz' 
    1  y, x,-z  C2xy 
    2 -x, z, y  C2yz 
DRF ⇒ DFL
    0 -z, y, x  C34y
    1  y,-x, z  C34z
    2 -x,-z,-y  C2yz' 
DRF ⇒ DBR
    0  z, y,-x  C4y 
    1 -y,-x,-z  C2xy' 
    2  x,-z, y  C4x 
DRF ⇒ URB
    0  x,-y,-z  C2x 
    1  z, x, y  C3xyz 
    2 -y, z,-x  C23xy'z'
DRF ⇒ ULF
    0 -x,-y, z  C2z 
    1 -z, x,-y  C3x'y'z 
    2  y, z, x  C23xyz
DRF ⇒ DRF
    0  x, y, z  E  
    1  z,-x,-y  C3x'yz' 
    2 -y,-z, x  C23x'yz'
DRF ⇒ DLB
    0 -x, y,-z  C2y 
    1 -z,-x, y  C3xy'z' 
    2  y,-z,-x  C23x'y'z

***

DLB ⇒ UFR
    0 -z,-y,-x  C2xz' 
    1 -y,-x,-z  C2xy' 
    2 -x,-z,-y  C2yz' 
DLB ⇒ UBL
    0  z,-y, x  C2xz 
    1  y,-x, z  C34z
    2  x,-z, y  C4x 
DLB ⇒ DFL
    0  z, y,-x  C4y 
    1  y, x,-z  C2xy 
    2  x, z,-y  C34x
DLB ⇒ DBR
    0 -z, y, x  C34y
    1 -y, x, z  C4z 
    2 -x, z, y  C2yz 
DLB ⇒ URB
    0 -x,-y, z  C2z 
    1 -z,-x, y  C3xy'z' 
    2 -y,-z, x  C23x'yz'
DLB ⇒ ULF
    0  x,-y,-z  C2x 
    1  z,-x,-y  C3x'yz' 
    2  y,-z,-x  C23x'y'z
DLB ⇒ DRF
    0 -x, y,-z  C2y 
    1 -z, x,-y  C3x'y'z 
    2 -y, z,-x  C23xy'z'
DLB ⇒ DLB
    0  x, y, z  E  
    1  z, x, y  C3xyz 
    2  y, z, x  C23xyz

( A note about the above notation. The x , y , z axes map the the R , U , F faces. This comes from the common way the coordinate axes are drawn on the page. The x axis points to the right on the page, the y axis points to the top of the page and the z axis is perpendicular to the page pointing toward you. The subscripts designate the positive axis direction, xy'z denotes the 1,-1, 1 axis vector, xz' denotes the 1, 0,-1 axis vector, etc.

I define the sense of a rotation using the right hand rule: placing the thumb of the right hand pointing in the positive axis direction, a positive rotation is the direction the fingers move in making a fist, i.e. counterclockwise. With this convention C_3x'y'z is a 120° counterclockwise rotation about the -1,-1, 1 vector (the three fold axis emerging from the LDF cubicle.)

The conventional twist can be related to the D4 cubic group subgroup having y as the principal axis and its two cosets.

D4 Symmetry: E   C2x  C2y  C2z  C4y  C34y C2xz  C2xz' 
Coset One: C3xyz  C3x'yz'  C3x'y'z  C3xy'z'  C2xy  C4z  C34z C2xy' 
Coset Two: C23xyz C23x'y'z C23xy'z' C23x'yz' C34x C2yz  C4x  C2yz' 

D4 rotations give the untwisted orientation, coset one gives twist in one sense and coset two gives twist in the opposite sense.

For fans of permutation notation an element of the C3 permutation group may be extracted from the axis mapping. First flatten the mapping by removing any minus signs. The corner cubies may be grouped into two tetrahedral sets: T1. UFR UBL DFL DBR and T2. URB ULF DRF DLB. For transforms within each set the flattened axis mapping is an element of C3. For example the transforms for UFR ⇒ UBL:

   
 0 -x, y,-z ⇒  x, y, z ⇒ 1,2,3
 1 -y, z,-x ⇒  y, z, x ⇒ 2,3,1
 2 -z, x,-y ⇒  z, x, y ⇒ 3,1,2

For transforms between the two sets the flattened mapping is not an element of C3 and must be transformed into one. Using the y axis as the reference axis swap the x and z mappings. For the UFR ⇒ URB transforms:

0  z, y,-x ⇒ z, y, x ⇒ x, y, z ⇒ 1,2,3
1  x, z,-y ⇒ x, z, y ⇒ y, z, x ⇒ 2,3,1
2  y, x,-z ⇒ y, x, z ⇒ z, x, y ⇒ 3,1,2

After these digressions lets get back to Jerry's analysis. Looking at the transformations within each tetrahedral domain, T1 and T2, the assignment of twist is actually independent of any reference direction. It doesn't matter which face one uses as reference; U/D , R/L , F/B, the twist assigned is the same. The transformations within each domain are the D2 rotations and its cosets within the T rotational group. The D2 subgroup is symmetrical within the cubic rotational group which makes it independent of any reference axis. It looks the same regardless.

The mischief comes in when cubies are moved from one tetrahedral domain to the other. Jerry observed that with one exception the three transforms for each cubie to cubical rearrangement were not of the same conjugate class and that this might be the basis of a symmetrical definition of twist. The exception is when a cubie is moved to its antipodal position, e.g. UFR ⇒ DLB:

UFR ⇒ DLB
    0 -z,-y,-x  C2xz' 
    1 -x,-z,-y  C2yz' 
    2 -y,-x,-z  C2xy' 

These three transforms are symmetry equivalent. They are members of the same conjugate class. The dilemma is that in any scheme of assigning twist one of the above must be designated as untwisted and doing so breaks the symmetry and defines a reference axis. Jerry concluded, and I agree, that this points up that a symmetrical definition of twist is a chimera. Any consistent assignment of twist must break the cubic symmetry of the cube.