Why don't crystals have 5 fold symmetry?


A crystal structure is made of atoms. A crystal lattice is made of points. A crystal system is a set of axes. In other words, the structure is an ordered array of atoms, ions or molecules. A unit cell is a crystal structure’s building block that also describes the whole crystal structure and symmetry, as well as the atom locations and principal axes. Lattice constants are the length, edges of principal axes, and angle between unit cells.

  • Crystals do appear to have 5-fold symmetry but these symmetries are not possible.
  • Crystals can only exist in the 2, 3, 4 or 6-fold rotational axis.
  • The external shape of a crystal is based on a geometric arrangement of atoms, which explains why crystals only have 2, 3, 4, and 6 folds. In fact, when we try to combine objects with 5-fold and 8-fold apparent symmetry, we can’t combine to fill the space completely.
  • Therefore, crystals cannot have 5, 7, 8, and other higher-fold rotational axes.

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