ABSTRACT
A point group is a collection of symmetry elements passing through a point, and therefore, necessarily does not include translations. Space groups, in contrast, include translations that are fractions of a repeat unit, for example, a 2 1 $ 2_1 $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315114910/a1f6aa37-c8b0-4273-bc79-d03791cd487d/content/inline-math4_1.tif"/> axis which involves a rotation of 180 ∘ $ ^\circ $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315114910/a1f6aa37-c8b0-4273-bc79-d03791cd487d/content/inline-math4_2.tif"/> followed by a translation of 1 2 $ \frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315114910/a1f6aa37-c8b0-4273-bc79-d03791cd487d/content/inline-math4_3.tif"/> of the repeat distance along the axis. Glide planes involve reflections followed by fractional translations. These translations are small, and hence do not manifest when the point group is determined from the external shape of well-formed crystals. However, they have consequences in structure determination and on other properties of crystals. There are 230 space groups which are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices.
