Currently having problems understanding 3D point groups. I am reading the wkipedia article: http://en.wikipedia.org/wiki/Point_groups_in_three_dimensions.

Under the Chapter "Group structure" it says:

SO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed.

As I understand the direct isometries are rotations and translations because it preserves the orientation. Reflection though is an indirect isometry which changes the orientation. What I dont understand is that a 180° rotation is the same as

Question

Currently having problems understanding 3D point groups. I am reading the wkipedia article: http://en.wikipedia.org/wiki/Point_groups_in_three_dimensions.

Under the Chapter "Group structure" it says:

SO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed.

As I understand the direct isometries are rotations and translations because it preserves the orientation. Reflection though is an indirect isometry which changes the orientation. What I dont understand is that a 180° rotation is the same as

anonymous · Answer

a reflection?

anonymous · Answer

It deleted my last part.

The last sentence should have been:
What I dont understand is that a 180° rotation is the same as point reflection, but isnt that contradictory to the idea of direct isometries?
Thank you

anonymous · Answer

I'm still unfamiliar with Groups (or anything in Abstract Algebra), but I always thought that all rotations can be thought of as composites of reflections.