In my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there?
The teacher gives me the definition:
Definition: If we rotate a 3-dimension object around the line d for 180 degrees and it result in an exactly same shape in an exactly same position, line d is a axis of symmetry of that object.
I have found 9 axis of symmetry, 3 of which pass through the centers of 2 opposite faces, the other 6 pass through the midpoints of the 2 opposite edges. But my teacher told that 9 is wrong and said that the correct answer is not what we're going to expect.
So, what is the correct answer? And how can we prove it?
An axis of symmetry can only passes through
(1) mid-points of two opposite edges. (As a cube has 12 edges, there are $12\div2=6$ axes of this type.)
(2) two opposite vertices. (As a cube has 8 vertices, there are $8\div2=4$ axes of this type.)
(3) the centres of two opposite faces. (As a cube has 6 faces, there are $6\div2=3$ axes of this type.)
So it has $13$ axes of symmetry.
Note: The number of symmetry is equal to
$$\frac{E}{2}+\frac{V}{2}+\frac{F}{2}=\frac{E}{2}+\frac{E+2}{2}=E+1$$