I've read the responses to the question `"Lack of inversion symmetry" in crystal?' but I'm still unsure about the meaning of inversion symmetry. Which of the following two dimensional shapes are inversion symmetric?

(a) $\longleftrightarrow$

(b) $\leftharpoonup\rightharpoondown$

(c) $\leftharpoonup\rightharpoonup$

(d) $\nleftarrow\nrightarrow$

  • $\begingroup$ Space inversion is also known as the parity transformation, and inversion-symmetric systems are sometimes called parity-symmetric or parity-conserving systems. $\endgroup$ – rob Apr 9 '18 at 18:23
  • 1
    $\begingroup$ More precisely, space inversion and parity differ by a rotation in the plane relative which the parity transformation is performed. $\endgroup$ – mavzolej Apr 17 at 4:41

I believe it means that the transformation $\textbf{r}\rightarrow -\textbf{r}$ produces the same figure. That is, if you see a point at $(x,y,z)$ you should also see one at $(-x,-y,-z)$.

In this case, if you place the origin at the center of each line, (a),(b), and (d) seem to be inversion symmetric.


From a geometrical point of view, inversion symmetry is equal to point symmetry, i.e. invariance under a point reflection.

Shapes (a), (b) and (d) are invariant under point reflection if the point of inversion is located directly between the two arrows. Hence, they are inversion symmetric. Shape (c) is not point/inversion symmetric.


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