Space group

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A space group is the set of all symmetry operations that describe the repeating pattern of a structure in space. In simple terms, it’s all the ways you can move or flip a crystal so that it still looks the same.

In three dimensions, there are 230 distinct space groups (219 if you treat mirror-image versions as the same). Space groups come from combining translational symmetry (how the pattern repeats in space) with point-symmetry operations (like rotations, reflections and inversions) and extra symmetries that tie translation to rotation, called screw axes and glide planes.

What makes up a space group
- Translations: the repeating unit of the pattern, called a Bravais lattice. There are 14 Bravais lattices organized into 7 lattice systems.
- Point-group symmetry: rotations, reflections, and improper rotations (rotoinversions) that leave at least one point fixed.
- Glide planes: a reflection in a plane followed by a translation parallel to that plane.
- Screw axes: a rotation about an axis followed by a translation along that axis.

A space group can be thought of as a combination of a lattice (translations) and a point group (the rotational and reflection symmetries). When you factor out the lattice translations, you’re left with a finite point-group symmetry.

Chirality and types
- 65 Sohncke groups have no mirror planes, inversions, or related symmetries, so they can produce chiral crystals (crystals that are not identical to their mirror image).
- A space group that includes mirrors or related elements generally yields achiral crystals.

Other related ideas
- In crystallography there are magnetic space groups (also called Shubnikov groups) that include time-reversal symmetry. In 3D, there are 1,651 magnetic space groups.
- In two dimensions, the analogous symmetry classifications are the 17 wallpaper groups.
- There are many ways to name and classify space groups, and different naming systems can describe the same group.

Why space groups matter
They provide a compact way to describe all the possible symmetries of a crystal. Knowing a crystal’s space group helps predict its physical properties, such as how it diffracts X-rays, how it vibrates, and how its atoms are arranged.