In fact, as we shall see below, we only need to start with 2 permutations to generate all possible rotations. This has 24 possible rotations, we can generate these by starting with the identity element and the 90° rotations about the x, y and z axis, then by combining these in different sequences we can generate all 24 permutations. realise that there are 8 vertices and so we can place it on each vertex in a given position in 3 possible ways giving 8*3 = 24 possible orientations.realise that there are 6 faces and so we can place it on each face in turn in 4 possible ways giving 6*4 = 24 possible orientations.
If we realise that there is a one to one correspondence between rotations and possible orientations of the cube, then all we have to do is count the possible orientations, for instance we could: There is a quicker way to discover that there are 24 rotations.
#Camtasia 9 cube rotate plus#
This gives 9 + 8 + 6 = 23 possible rotations of the cube, plus the identity element (leave it where it is giving 24 possible rotations in total. There are 24 made up of 1 identity element, 9 rotations about opposite faces, 8 rotations about opposite vertices and 6 rotations about opposite lines.
We now need to work out all the permutations of these rotations. So a 90° rotation about each of the x, y or z axis could be defined as follows (as a cycle There are a number of ways to analyise this, one way is to number the vertices and track the effect of each rotation on these vertices. These other platonic solids are described on this page. We could have chosen other platonic solids, other then a cube, such as a dodecahedron. multiples of 90° rotations about x, y or z)Īlthough we will keep track of the rotations possibly by markings on Take as an example the rotation of a cube in ways that does notĬhange its shape (i.e.