Rotation reflection is a single geometric transformation that combines two fundamental movements: a rotation around an axis and a reflection across a plane that is perpendicular to that axis. It is also referred to by terms like rotoflection, rotary reflection, or an improper rotation in contexts like crystallography and molecular symmetry.
This combined operation leverages the actions described in the reference:
- Rotation: This is the turning of an object around a fixed point or line. The reference illustrates this: when the clock face on the left is rotated 90 counterclockwise, the result is the clock face on the right. This shows how a shape changes its orientation through rotation.
- Reflection: This involves creating a mirror image of an object. As the reference defines it, To reflect an object means to produce its mirror image with respect to a line, which is called the line of reflection. In the context of rotation reflection, this mirror image is typically produced across a plane perpendicular to the rotation axis (in three dimensions).
Essentially, a rotation reflection operation means you first rotate an object by a specific angle around an axis, and then reflect the result through a plane perpendicular to that same axis.
A Concrete Example: S₄ Symmetry
A classic example of rotation reflection symmetry (specifically known as S₄ symmetry) is found in the tetrahedral arrangement of atoms in a molecule like methane (CH₄).
Here's how the S₄ operation works on methane:
- Choose an Axis: Imagine an axis passing through the central carbon atom and extending outwards through the midpoint between two opposite hydrogen atoms.
- Perform Rotation: Rotate the molecule by 90 degrees around this axis.
- Perform Reflection: Reflect the rotated molecule through a plane that passes through the carbon atom and is perpendicular to the rotation axis you chose.
If the molecule possesses S₄ symmetry, performing these two steps consecutively will return the molecule to a state that is indistinguishable from its original configuration, even though individual atoms might have moved positions. The S₄ symmetry element represents this combined rotation and reflection transformation.
Other examples can be found in crystallography and the symmetry of certain composite objects, where a simple rotation or a simple reflection alone would not bring the object back to its original appearance, but the combination does.
