Skip to main content
Engineering LibreTexts

15.7: Symmetry

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    We have already met the concept symmetry in relation to crystal structures: the lattice generates the translational symmetry—the motif is repeated on every lattice point.

    Other types of symmetry exist, including:

    • rotation axes
    • mirror planes
    • centre of symmetry
    • inversion axes (combination of rotation and centre of symmetry operations)

    An n fold rotational symmetry operation rotates an object by 360°/n. Only n = 1, 2, 3, 4, and 6 are permitted in a periodic lattice

    Examples of n-fold rotational symmetry

    An object has mirror symmetry if reflection of the object in a plane brings it into coincidence with itself:

    Examples of mirror symmetry

    Some objects have special symmetry about an origin such that, for any point at position x, y, z, there is an exactly similar point at x, y, z. The origin is called a centre of symmetry ( “inversion centre”). Such an object is said to be centrosymmetric:

    An n-fold inversion axis is a combination of a rotation by 360/n followed by a centre of symmetry operation. An example of a 4-fold inversion axis is show in the following animation:

    15.7: Symmetry is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Dissemination of IT for the Promotion of Materials Science (DoITPoMS).

    • Was this article helpful?