H-symmetric positions

These are the positions that are invariant under all symmetries producing an even permutation of long diagonals of the cube. There are 24 such positions; 4 have more symmetry, namely M-symmetry. The other 20 positions pair up into 10 patterns, as shown below. I have even calculated all minimal maneuvers, using my optimal cube solver.

Superfliptwist U R F' B U' D' F U' D F L F' L' U R D F U R L (20q*, 20f*) Herbert Kociemba [1, 3] was the first to find a 20f maneuver; (it's different from this one).

Supertwist U R' B D B U L D B' D2 R U' F L F R D L F' L2 (22q*, 20q) B' L2 U D R2 B' D2 F2 D' R2 F B L2 D' B2 U2 (16f*, 24q) Dik Winter [4] was the first to find a 16f maneuver; it's different from this one.

6 H's D2 L2 B2 U2 D2 B2 R2 D2 (16q*, 8f*) Up to H-symmetry, this is the only maneuver that is minimal in face turns. (It is equivalent to its inverse.) James Saxe [2] was the first to find this maneuver.

Superflip composed with 6 H's U F' L' F' B U R F' B U' B' U D' R2 L' B U' (18q*, 17f*) Up to symmetry and inversion, this is the only maneuver that is minimal in either quarter turns or face turns.

Superfliptwist composed with Pons Asinorum U F B D R L U' F2 B2 R L D' F B D R L D F' B' (22q*, 20f*) Up to symmetry and inversion, this is the only maneuver that is minimal in quarter turns. It has the remarkable property that it is invariant under a 180 degree rotation about the U-D axis! Dik Winter [5] was the first to find a 20f maneuver (different from this one).

Supertwist composed with Pons Asinorum F L D F U' B2 R F R' F' R F L2 U' R D B R (20q*, 18f) B2 L U2 F' B' U2 R' F2 L2 F' U2 R' L' U2 B R2 (16f*, 24q) The 20q maneuver is the unique minimal maneuver in quarter turns, up to symmetry. (It is equivalent to its inverse.) The 16f maneuver is the unique minimal maneuver in face turns, up to symmetry and inversion.

	Superfliptwist composed with 6 H's (type 1) U F B U' R L U F B R2 L2 D' F B U' R L D' R' L' (22q*, 20f*) This is the unique maneuver, up to symmetry, that is minimal in quarter turns. It has the remarkable property that it is invariant under a 180 degree rotation about the U-D axis!
	Superfliptwist composed with 6 H's (type 2) Inverse of type 1

	Supertwist composed with 6 H's (type 1) U2 L U B D L U B' R' L' F' D R U F D L' U2 (20q*, 18f) L' B2 U' D' B2 R' U2 L2 U B2 R L F2 U R2 U2 (16f*, 24q)
	Supertwist composed with 6 H's (type 2) Inverse of type 1

References

[1] Herbert Kociemba, Close to God's Algorithm, Cubism For Fun 28 (April 1992) pp. 10-13.
[2] James Saxe, 16 qtw algs for CC and H patterns, cube-lovers e-mail, December 16, 1980.
[3] Dik T. Winter, Are we approaching God's algorithm?, cube-lovers e-mail, May 4, 1992.
[4] Dik T. Winter, Kociemba's algorithm, cube-lovers e-mail, May 17, 1992.
[5] Dik T. Winter, Re: Diameter of cube group?, cube-lovers e-mail, August 25, 1993.

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Updated October 28, 2006.