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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

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up vote 4 down vote accepted

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.

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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

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no i have read in internet that cube has 4 of axis of symmetry so it means that there is 7 such line? – dato datuashvili Jul 8 '11 at 13:58
but answer is 9 maybe two in base? – dato datuashvili Jul 8 '11 at 14:00
@user3196: added some info-missed some. I don't understand the last question. – Ross Millikan Jul 8 '11 at 14:05
i think in text book here is some mistakes no there is not info-missed just in answers there is said that in space exist 9 such line no more i think it is very brief information to say how much line exist which satisfy condition given in problem – dato datuashvili Jul 8 '11 at 14:21
9 is the correct answer; see Jyriki's answer. Use a die (from a pair of dice, to convince yourself. – amWhy Jul 8 '11 at 16:33

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