From mreid@ptc.com Thu Jan 19 18:30:05 1995
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Date: Thu, 19 Jan 1995 18:41:47 -0500
From: mreid@ptc.com (michael reid)
Message-Id: <9501192341.AA09110@ducie.ptc.com>
To: cube-lovers@ai.mit.edu
Subject: symmetric maneuvers
Content-Length: 1504
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