2D Chirality

Chirality in dimension two

If we were to live in a flat world, e.g. inside a plane, the notion of chirality would still exist. Furthermore, its definition would almost be unchanged. We would simply carry out reflections through straight lines instead of planar mirrors, as in the following example:

I am triangle (a) I am a reflection line I am triangle (b)
Triangle (a) Reflection line Triangle (b)

Taking triangle (a) and carrying out the reflection through the line, we get triangle (b). The triangle has been somehow inverted in the process, just like a left hand is converted into a right hand by mirror reflection. As a result, triangle (a) and triangle (b) cannot be superimposed by any displacement in the plane. Just try and check. Remember that moving triangles out of the plane is forbidden as we agreed to live, hence to perform any movement, inside the plane.

Drop me a mail if you succeed...
Trying to superimpose (a) and (b)

Conclusion: triangle (a) is chiral as a two-dimensional (2D) object. Let us say that it is 2D chiral. Triangle (b) is also 2D chiral. Next, consider the following equilateral triangle:

I am triangle (c) So what? Guess who I am?
Triangle (c) Reflection line Triangle (d)

Now, surprise, surprise, we can superimpose (c) and (d)! Rotate (d) clockwise by 90° and move it over (c). This works fine:

(c) and (d) can be superimposed

Conclusion: (c) and (d) are one and the same object. So triangle (c) is 2D achiral.

Chirality can be similarly defined in every conceivable dimension; As far as principles go, chirality is not related to dimension three in any way. Since describing chirality is simpler in the 2D case, the tutorial on the gauge description of chirality is focused on the 2D case.