Place a Rubik’s Cube on a flat surface, with one face facing towards you. Note the colours on each side. Now spin it around, toss it from hand to hand, and put it down again, again with one face towards you. Chances are the colours on the faces have changed.
In one sense, the Rubik’s Cube is oriented the same way both times. In both cases it is sitting on a face rather than balancing on an edge or a vertex. In both cases there is a face facing directly towards you. If you had peeled the stickers off your Rubik’s Cube beforehand, then you wouldn’t be able to tell the difference between the two orientations.
It is only the different colours on the faces that make it possible to tell the difference between these two orientations. Why? Because the faces have been swapped around, but wherever there was a face before, there is still a face. The edges have been swapped around too, but where an edge was before, there is still an edge. The vertices have been exchanged too, but where a vertex was before, there is still a vertex.
When two orientations of a polyhedron put faces, edges and vertices all in the same places, we say that those two orientations are isometric, and we call them isometries. These three Rubik’s Cubes are oriented isometric to each other:
If we now balanced our cube on one of its vertices, that orientation would not be isometric to the others. It was sitting on a face before, so it clearly had a bottom face and a top face. Now there are no bottom and top faces: there are bottom and top vertices, and all the faces are oblique.
Balancing it on an edge would not be an isometry either, and nor would sitting it on a face but with an edge facing forwards. None of these Cubes are oriented isometric to each other:
How many isometries does the cube have? Or, to put it another way, how many ways can a Rubik’s Cube be placed face down with a face towards you?
We don’t need to answer this by trial and error, we can work it out. Firstly, choose a face to be the top face. The cube has six faces to choose from, so there are six different ways to choose the top face. Having chosen the top face, the bottom face is determined too. But the cube can still be rotated to bring any of the four side faces to the front. Therefore the number of isometries is 6 × 4 = 24.
When it comes to counting isometries, there is nothing special about our chosen orientation. We might have decided instead to balance our cube on a vertex, with one incident edge facing forward. There are eight vertices, and each one has three incident edges, so the number of isometries is 8 × 3 = 24. Alternatively, we might have decided to balance our cube on an edge, with an incident face facing forwards. There are 12 edges, and each has two incident faces, so the number of isometries is 12 × 2 = 24. You’ll always get the same number, because 24 isometries is a property of the cube itself, not any particular orientation of the cube.
(Strictly speaking, we should say that the cube has 24 proper or rotational isometries. If we allowed reflection as well as rotation, we would unlock a further 24 isometries for a total of 48. But twisty puzzles do not support a reflection operation, so we restrict ourselves to proper isometries at all times.)
Applications to twisty puzzles
Isometries are important to understand twisty puzzles, because they explain why puzzles like the Rubik’s Cube can be thoroughly scrambled, yet they always maintain their cube shape. The key to understanding this is to think of the core of the Rubik’s Cube as a mechanism for moving part of a cube to another isometry, while leaving the rest of the cube behind. After making such a move, those two parts of the cube have different orientations, but because the orientations are isometric, together they still form a cube.
For example, have a look at these three Rubik’s Cubes:
The first two are solved, but oriented differently. Most importantly, these two orientations are isometric.
Now look at the third cube. The bottom two rows match the bottom two rows of the first cube, but the top row matches the top row of the second cube. It looks like we’ve started with the first cube, but turned the top layer to orient just that layer with the second cube. We now have a cube that is partly in one orientation and partly in another. But the two orientations are isometric, so when you put them together you still have a cube.
Now contrast this with Fisher’s Cube. Like the Rubik’s Cube, Fisher’s Cube puts a Rubik’s Cube core inside a cube shell, but the shell and core are oriented differently to each other. We still think of the core as a mechanism for reorienting part of the cube while leaving the rest of the cube behind. But in some cases the new orientation will not an isometry of the old. So when you join together those two pieces, the result is not a cube at all.
The first two images below show solved Fisher’s Cubes in different orientations. Note that they are not isometric: one is sitting on a face, the other is balancing on an edge. The third Fisher’s Cube has two layers oriented like the first cube, and one layer oriented like the second. Since the two orientations are not isometric, the third Fisher’s Cube is not a cube at all.
We have now arrived at an explanation for why some puzzles retain their shape — they scramble without jumbling — while others jumble readily. A cube scrambles without jumbling if every turn of an axis takes a layer from isometry to isometry. It jumbles if some turns take a layer from one orientation to another and these orientations are not isometric.
Ultimately, whether of not a turn takes a layer to an isometry or not is determined by how the turning axes of the puzzle align with the axes of symmetry of the shell.a href=”http://cubemeister.com/about-polyhedra/isometry/fishers-cube-orientation-turn/” rel=”attachment wp-att-529″