Proof that cube has 24 rotational symmetries I was doing a combinatorics problem which states this definition of symmetry: for a subset $S$ of $\mathbb{R}^3$ a symmetry is "rigid motion" $f:\mathbb{R^3}\rightarrow \mathbb{R^3}$ such that any $x\in S$ has its image $f(x)$ also in S. (To provide some fodder for the problem, the author dumbs it down for me that a symmetry is some operation which leaves $S$ in the same place although the individual points may be rearranged) 
From this working definition it is asked to find the number of rotational symmetries of the cube. So far I have seen two different arguments and I still dont know because drawing on paper is getting confusing. Also my solution was incorrect and I dont know where I undercounted.
Ignore my method.Drawing 4 axis perpendicular with the faces, each with 4 rotational symmetries $(0^\circ,90^\circ,180^\circ,270^\circ)$ and I get $4\times 4 =16$ symmetries. Plus the four body diagonals$=20$. This is wrong and the answer is $24$.
The book gives the short solution: "move any vertex to any other and rotate the edges leading to it in three ways". I don't get it. There are 8 vertices, and 3 ways of what?
I google and get another argument from wikipedia: "The group of orientation-preserving symmetries is S4, or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite sides of the octahedron." Indeed $4!=24$ but I cannot convince myself that aall permutations of the pair of opposite edges is all the symmetries there can be (I clearly undercounted when I got 16 and still have no insight why, so this does not help me as I dont know for sure there cant be more)
My imagination and visualization skills are very weak. Is there a good online software that allow me to visualize and play around with a labelled cube?
 A: Counting the possible orientations of the cube we know that there are 24 rotational symmetries, by considering faces, edges or corners and the their respective number of orientations, giving 6 * 4 = 12 * 2 = 8 * 3 = 24.
It seems you also want to explicitly know the rotations, instead of just counting the rotational symmetries.  The rotations are:


*

*1 identity rotation that does nothing.

*9 rotations around axes through the middle of one of 3 pairs of opposing faces, with rotations of 90°, 180° and 270°.

*6 rotations around axes through the middle of one of 6 pairs of opposing edges, with a rotation of 180°.

*8 rotations around axes through one of 4 pairs of opposing corners, with rotations of 120° and 240°.
A: The threefold rotations about a corner/long diagonal can be quite hard to visualise. The count which is most intuitive to me is that, with the cube on the table in front of me I can place any one of the six faces on the table, then choose any one of four faces to be facing me.
However, it is useful to work on the harder to visualise perspectives in a simple example such as this where the answer is clear, because then intuition is sharpened for more complex situations.
Another way of looking at this, which is more complex, is to analyse what happens to the two tetrahedra which are formed by picking out alternate vertices of the cube, or to the set of long diagonals of the cube.
A: "Drawing 4 axis perpendicular with the faces, each with 4 rotational symmetries (0∘,90∘,180∘,270∘) and I get 4×4=16 symmetries"
The cube has 6 faces, so 6 axis, and gives $6\cdot 4 = 24$ as needed. (Sorry, this seemed like something only worth a comment, but I don't see where to comment... It could be I don't have enough rep to do so yet? Either that or I'm blind :) )
Edit: If you want to visualize objects, and view different symmetries, it may be helpful to give each vertex a name (so for the cube, a,b,c,d,e,f,g,h could be your vertices). This is going to be easier than looking at a single object and imagining rotating it.
A: The point of the comment in the book is this: given a cube, pick one corner and call it $A$. You can certainly rotate the cube so that $A$ either stays put or moves to any chosen other corner, of which there are 7. So you have 8 choices for where you want $A$ to be.
Once you've done this, the other corners are obviously not free to move as they wish. Let $B$ be some corner adjacent to $A$, meaning it shares an edge of the cube with $A$. If A moves to $A'$, $B$ must move to some corner adjacent to $A'$ - otherwise the edge $AB$ gets distorted. How many corners are adjacent to $A'$? That's where the 3 comes from.
Finally you'll need to convince yourself that once $A$ and $B$ have been placed, all other corners are "forced" - their final location is already fixed. See if you can do that just using the relationships of adjacency, sharing a face etc. 
A: A cube has 12 edges, so it has 24 oriented edges (each edge can be oriented in exactly two ways) It is pretty obvious, if you have a cube to play with, that the group $G$ of rotations acts simply transitively on these orientied edges. Therefore $G$ has 24 elements.
Exactly the same argument counts the number of rotational symmetries of each regular polyhedron, of course.
A: I was just considering this problem for my own work, and I think I found a unique solution.  There are eight symmetries of a square: four rotations and four reflections.  If you consider a cube to be a sqauare that has been expanded in a third dimension, then you can see that a cube has 24 symmetries - eight for each dimension (this does not consider turning the cube inside out).
A: The point-group symmetry of a cube is m-3m, which has 48 symmetry operations. These include the mirror reflections. If we insist on rigid-body operations only, with no change of "handedness" (inversions), then the mirror operations must be removed  and the point-group symmetry is reduced to 432 – four-fold rotation axes about each face (parallel to the edges), 3-fold rotation axes parallel to each body-diagonal and 2-fold rotation axes parallel each face-diagonal; all axes pass through the cube center. This gives 4x3x2 = 24 orientations. The 432 group is a subgroup of m-3m. The latter includes the additional 24 orientations that arise after each orientation of the rigid cube is reflected.
