Date: Thu, 19 Jan 95 18:41:47 -0500
From: michael reid <mreid@ptc.com >
~~~ ~~~ Subject: symmetric maneuvers

mark writes

```p = R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3    (12 q)

Then  p + (p * Sm) = Superflip
```

This is Mike's process slightly patched, with the last two (commuting)
cube turns swapped in position.

i'm surprised this hasn't been pointed out previously. however, i would
write the above as

```(R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3  C_X)^2
```

where i use C_X for central reflection. this fits in with mark's
idea of "cyclic decomposition".

i've noticed that a number of minimal (or presumed to be minimal)
maneuvers for pretty patterns have some symmetry. here i'll use
commutator notation:

```[ A , B ]    refers to   A  B  A'  B'
```

also i'll use bandelow's notation for rotations of the whole cube:

```C_U , C_RF , C_URF ,
```

denote rotation about a face-face axis, edge-edge axis, corner-corner
axis, respectively.

now some patterns:

```anaconda:            B1 R1 D3 R3 F1 B3 D1 F3 U1 D3 L1 F1 L3 U3
= [ B1 R1 D3 R3 F1 B3 D1 , C_UB ]

python:              D2 F3 U3 L1 F3 B1 D3 B1 U1 D3 R3 F1 U1 B2
= [ D2 F3 U3 L1 F3 B1 D3 , C_UF ]

6 x's (order 3):     R2 L3 D1 F2 R3 D3 F1 B3 U1 D3 F1 L1 D2 F3 R1 L2
= [ R2 L3 D1 F2 R3 D3 F1 B3 , C_UB ]

my favorite example is
four twisted peaks:  U3 D1 B1 R3 F1 R1 B3 L3 F3 B1 L1 F1 R3 B3 R1 F3 U3 D1
= [ U3 D1 B1 R3 F1 R1 B3 L3 F3 , C_R2 ]
```

i'd hoped to find maneuvers for "cube within a cube" and "cube within a
cube within a cube", but no such luck.

mike