# FTM Antipodes of the Edge Group

I have done my own independent breadth-first search of the edge group using the face-turn metric. I used symmetry/antisymmetry equivalence classes to reduce the number of elements in the search space. I confirm the "Unique mod M+inv" values for this group/metric that Rokicki reported in 2004.

I reduced the "coordinate space" for the search to 5022205*2048=10285475840 elements by using symmetry/antisymmetry equivalence classes of the edge permutation group. (This gives a much more compact overall coordinate space than using an edge orientation sym-coordinate, at a cost of more time required to calculate representative elements. This allowed me to keep track of reached equivalence classes with a ~1.3 GB bitvector in RAM and 5022205 KB disk files to keep track of distances.)

Below is a summary of the 24 symmetry/antisymmetry equivalence classes representing the antipodes at distance 14 from Start. I include a move sequence to generate one element of the equivalence class, along with the number of elements in that equivalence class, and the order of each element (how many repetitions of the sequence gets all edge pieces back to their starting positions, including orientation).

Representative move sequence elements order U D L F D R F B' L' F' B D B L 1 2 U D L F2 D2 L F' B' D B2 D2 L' R B 3 2 U D F' L' D B U' D2 L2 R' B L U F2 12 4 U D2 F R' U' R' F2 B2 R' U' R' B' U' F2 6 2 U D L' F2 U2 L' F B U L' R B2 D2 F 3 2 U D' F U2 F2 L R' U L2 D2 F' B R' B2 12 4 U D L2 F' L' U2 B2 D2 L' U2 D' B' R' F' 24 4 U D F L R D' R2 U' F' B R U2 F2 L 12 4 U D L F' B' R' U' F' U2 R2 F2 D R2 F' 12 4 U D F2 B' L2 B2 D' L' B' L B2 U' L R' 12 4 U D R' F2 D' F D' B' U2 R F2 R2 F' L 24 4 U D F' L U' R' B2 L' B R2 D' L F B2 6 2 U D2 L' F2 U' R B' L' D F2 R U2 D' L2 6 2 U D L F U F' B D F2 B R' U' D' L' 6 2 U D R2 F' L' D2 F' R U L' B D2 R F' 6 2 U D L' U F R B' D' L F2 U2 R2 B D2 24 6 U D L' U F' D' L2 F' R' U L' F2 U2 D 24 4 U D' L2 F L' U R B2 U F L' B U2 L' 12 4 U D F D2 L' F2 U F' R' B' R2 D' L D' 6 4 U D' L D2 B R F' U' B2 L' D2 R F L2 1 2 U D L' F2 U' D R2 B' L R D' R2 B2 U 6 2 U D L' U2 B2 L' U D B D2 B2 L R' U 6 2 U D' F' L R D2 B2 U F2 B L' F D2 R 16 6 U D L U' D B2 L2 F L' R' U L2 F2 D 8 6