symmetries of shape?

Question

if you have a 3d cube. How many symmetries take vertex 1 to vertex 2? is there six. if there is how do you explain them

what happens if vertex 1 and 2 are on the same face, but also next to each other??

Answer 1

If the vertices lie on the same face (diagonally opposite), then there are 6 2-fold planes of symmetry, 2 that cover each parallel pairs of faces and there are 3 such pairs. These are equivalent to 6 axes of 2-fold rotational symmetry through the midpoints of parallel edges (These are symbolised in crystallography as 6A2)

If the vertices lie diagonally opposite each other in the cube then there are 4 axes of 3-fold rotational symmetry as there are 4 diagonals (8 vertices .. 4 opposite pairs) (These are symbolised in crystallography as 4A3)

There are also 3 axes of 4-fold symmetry (yep 3A4) which go through the midpoints of the opposite faces.

In crystal structures, the cube is the most symmetrical possible.

Answer 2

It might be easier if you viewed it as 2 cubes: the original, and it's translation.

There are 3 edges from each vertex. They are considered to be symmetrical. Which means there are 3 edges from both vertex 1 and vertex 2.

Call the 3 points connected to 1, 1a, 1b, 1c
and the 3 points connected to 2, 2a, 2b 2c.

The different ways of mapping 1a, 1b, 1c are
(2a, 2b, 2c) Same,
(2b, 2c, 2a) rotate the cube along it's diagonal by 120 deg
(2c, 2a, 2b) rotate the cube along it's diagonal by 120 deg
(2a, 2c, 2b) reflect the cube along (2, 2a)
(2c, 2b, 2a) reflect the cube along (2, 2b)
(2b, 2a, 2c) reflect the cube along (2, 2c)

Answer 3

Try this link, good luck!

http://homepages.cwi.nl/~dik/english/mathematics/poly/intro.html