The octahedral point group has 13 axes of symmetry. Three of them are the familiar face-turning axes of the cube.

The set of turning axes provided by the mechanism in the core is a fundamental factor in determining the nature of a twisty puzzle: the way it turns, the level of difficulty, and so on.

Of primary importance is the point group that the puzzle is based upon. Is the internal mechanism tetrahedral? Cubic? Dodecahedral? Each point group has its own isometries and its own axes of rotational symmetry. These axes fall naturally into groups called conjugacy classes. That sounds a little scary but this is actually pretty straightforward:

Conjugacy classes

Consider the axes of rotation of the cube. One axis passes through the centres of two opposite faces. This axis has four-fold symmetry, which means when you rotate around that axis you hit an isometry every 90°, so there are four isometries per revolution.

There is another axis of rotation that passes through the centres of a different pair of opposite faces, and this axis, too, has four-fold symmetry. In fact, since the cube has three pairs of opposite faces, there are a total of three axes of rotation that pass through opposite face centres, and all three of them have the same four-fold symmetry. These three axes of symmetry are fundamentally similar to each other; they share many properties. In mathematical terminology they are conjugate to each other. So we group these three axes together, and simply refer to them as the face-turning axes.

In the same way, the cube has four axes of rotation that pass through opposite vertices. Each such axis has three-fold symmetry. These four axes are conjugate to each other too.

So too are the six two-fold axes of symmetry that pass through opposite edge centres.

So introducing the concept of conjugacy classes makes things easier for us, not harder. Instead of having to think in terms of thirteen different axes of rotational symmetry, we only have to think about three conjugacy classes: the face-turning axes, the vertex-turning axes, and the edge-turning axes.

While we’re talking about cubes, note that there is no cube puzzle that supports turning around every axis of symmetry. The Rubik’s Cube supports only the three face-turning axes. The Dino Cube supports only the four vertex-turning axes. The Helicopter Cube supports only the six edge-turning axes. And so on.

More to the point, note that these puzzles don’t just support a random selection of axes: every one of them supports a single conjugacy class! Actually there is nothing to prevent a puzzle from supporting more than one conjugacy class of axes — indeed there are puzzles that do — but you won’t find a puzzle core that only half-supports a conjugacy class.

Generating sets

What have we lost by not supporting every possible axis of symmetry? That depends. Recall that for every pair of isometries, there is an axis of rotational symmetry that allows one to rotate directly from one isometry to another. But being able to go directly from one state to another is not all that desirable for a puzzle. If we can only get from one state to another indirectly, by making more than one turn, then our puzzle is probably that little bit harder to solve. We’ve lost nothing, and arguably we’ve gained.

A set of axes that lets you get from any isometry to any other, perhaps indirectly, is called a generating set. Ideally, you want your set of axes to be a generating set.

Consider the cube. The axes of symmetry that pass through opposite face centres are a generating set for the isometries of the cube: it is possible to reorient a cube from any orientation to any isometric orientation, using face turns alone.

What does this mean for a face-turning cube puzzle such as the Rubik’s Cube? It means that for the Rubik’s Cube it will always be possible to move any edge piece into any edge position in any orientation, and any corner piece into any corner position in any orientation, and so forth (though not necessarily all at once).

Mosaic Cube, scrambled. Note the green-red piece is correctly positioned and oriented relative to the adjacent corner piece. No combination of moves can put it in this position upside down.

What if our set of axes is not a generating set? That does make a big different, but it isn’t the end of the world. For example, take the vertex-turning axes of the cube. These are not a generating set for cube isometries. Starting at a given orientation, some isometric orientations cannot be reached at all using only vertex rotations. For example, it is impossible to swap neighbouring faces using vertex rotations alone. That means that in any vertex-turning cube puzzle — the Dino Cube, the Mosaic Cube, the Rex Cube — it is impossible to place an edge piece in its correct position without also orienting it correctly!