mapping of cube by itself From exam textbook I am given to solve following problem: question is like  this
  in space how many lines  are such by which if turn  cube by $180^\circ$ it will map itself?   I was thinking   about this  problem many times I though  it should be axis of symmetry  for which answer would be  $4$ but in answers there is not 4  so I did not find solution of it  yet please help me to make it clear for me
 A: Ross correctly enumerated the possible lines.
You should take into account that when you rotate the cube about a body diagonal, you have to rotate an integer multiple of 120 degrees in order to get the cube to map back to itself. So for the purposes of this question the body diagonals don't count. 9 is the correct answer.
Perhaps the easiest way to convince you of this is that there are 3 edges meeting at each corner. Rotation about a 3D-diagonal permutes these 3 edges cyclically, and is therefore of order 3 as a symmetry. Yet another way of seeing this is that if we view the cube as a subset $[0,1]\times[0,1]\times[0,1]\subset\mathbf{R}^3$, then the linear mapping $(x,y,z)\mapsto (y,z,x)$ keeps the opposite corners $(0,0,0)$ and $(1,1,1)$ as fixed, obviously maps the cube back to itself, and as this mapping is orientation preserving (in $SO(3)$), it must be a rotation about this diagonal. As it is of order 3, the angle of rotation must be 120 degrees.
A: What 4 did you get?  Clearly the axis has to pass through the center of the cube.  There are three through the face centers, six through the centers of edges, and four body diagonals.  You can see a figure here
