Notation

The following standard notation comes from [MC], so I hear, and all the books except [SS] use it. Imagine you're holding a cube so that one face points upward and one face points toward you. In that situation we call the faces left (L) and right (R), front (F) and back (B), and up (U) and down (D). We then use those letters to represent clockwise quarter turns of the corresponding faces. We also think of the letters as group elements under multiplication, so that for example L² represents a half turn of the left face, L³ = L⁻¹ represents a counterclockwise quarter turn, and LR represents L followed by R. In that way any sequence of turns can be written down as a sequence of letters.

That notation doesn't include any way to describe reorientation of the whole cube. We could add one, but it's not necessary—anything that can be done can be done without reorientation—and in fact not even all that useful. If you're reorienting in the middle of a sequence, that's usually just to get a better grip on the cube, and you can figure out how to do that yourself after running through the sequence a few times. And, if you're reorienting between sequences, that's usually a process that requires looking at the cube and making decisions. If we added a way to describe that, we'd be able to write down specifications for cube-based Turing machines!

There is one useful thing that's missing from that notation, though. If you do LR⁻¹ and then reorient the cube in the opposite direction, the net effect is to turn a center slice. For such slice moves I'll use the notation invented by the authors of [WW]: we can turn the tire-like slice away (α) or back (β), the steering-wheel-like slice left (gauche, γ) or right (droite, δ), and the equator-like slice east (ε) or west (ω). We don't really need all six symbols, since for example β = α⁻¹, but it's a nice convention, and the extra symbols make it more memorable.

A few points:

• Don't underestimate the power of having good names, symbols, and notation! I put a lot of effort into that when I'm writing and programming.
• We can use slice moves to write down reorientations, but it's awkward. For example, RL⁻¹α is a reorientation.
• Technically, even three symbols is more than we need. Here's ε in terms of α and γ.

  ε = LR⁻¹α⁻¹ BF⁻¹γ RL⁻¹α UD⁻¹

• Along the same lines, here's a nice result by David Benson that was noted in [WW].

  D = RL⁻¹ F²B² RL⁻¹ U RL⁻¹ F²B² RL⁻¹

Finally, when I show pictures of cubes, I'll show all six faces, like so. (Note that not all cubes have the same color scheme.)

To fit the two halves together, just imagine that the blue and white faces are connected by a small hinge so that you can swing the right half around behind the left. Sorry it looks like an optical illusion!

The orientation doesn't always matter, but the standard orientation for pictures like these is to let the area that's red be the up face and the area that's yellow be the front face. Just as an example, here's the result of UR.