Mirror Mirror

vixen · April 25th, 2006, 9:01 am

It is well known that if a plane figure has a line of symmetry then the mirror image of the figure can be rotated/translated in such a way that it can be superposed onto the original figure.Show that this is not the case in 3D. In other words, show that there exists ( or give an example ) shapes in 3 dimensions that have no plane of symmetry and yet the mirror image can be superposed onto the original shape.

sevvost · April 25th, 2006, 11:22 am

Snub cube?

kws · April 25th, 2006, 2:56 pm

Something with C_i symmetry.

vixen · April 25th, 2006, 3:11 pm

I don't know about the snub cube. I will investigate further, though I have this sneaking suspicion that it won't be superposable.Not too clear kws. What do you mean by C_i symmetry?

alexandreC · April 25th, 2006, 3:51 pm

alexandreC · April 25th, 2006, 3:54 pm

kws · April 25th, 2006, 4:23 pm

A system where the only symmetry element besides E is a point of inversion. I could draw hydrazine to have C_i symmetry, for example.

sevvost · April 25th, 2006, 9:03 pm

I think I now recall that there was an example of an achiral system lacking symmetry planes that consisted of a finite number of points (8?) Will that be acceptable?

kws · April 25th, 2006, 9:36 pm

I think anything that has an improper rotation axis (S_n operation -- can be transformed onto itself by a 2*pi/n radian rotation followed by reflection through a plane) but no plane of symmetry would work here. Another example is tetramethylcyclooctatetraene, which has S_4 symmetry. The C_i point group is equivalent to S_2.