space group
/ spays groop /
Think about everything you can do to a patterned wallpaper that leaves it looking exactly the same: you can shift it sideways by one full motif, you can rotate it, you can mirror it, and you can do clever combinations of shifting-and-mirroring at once. List every such move that works, and you have written down the wallpaper's complete symmetry. For a three-dimensional crystal, that complete list is the space group.
A space group is the full catalogue of all the symmetry operations that map a crystal onto itself, including not only the rotations and reflections of the point group but also the translations of the lattice and the hybrid moves that mix the two — screw axes (rotate while sliding) and glide planes (mirror while sliding). Combining the lattice translations with the allowed point-group symmetries, mathematicians proved there are exactly 230 distinct space groups in three dimensions. Every periodic crystal in the universe belongs to one of these 230.
Space groups matter because they are the complete identity card of a crystal structure: tell someone the space group plus the cell dimensions and the atom positions, and they can reproduce the structure exactly. This is the language in which the worldwide databases of solved crystal structures are written. The subtle point worth flagging is that the space group includes those mixed screw and glide operations, which is why there are 230 of them rather than just the 32 point groups times the 14 lattices — the genuinely combined symmetries add many extra possibilities.
When researchers solve a protein's structure by X-ray crystallography, the very first thing they report is its space group. That single label tells other scientists how the molecules are packed and related throughout the crystal.
A solved structure's space group is the first thing crystallographers report.
The 230 figure is for three dimensions. The two-dimensional analogues — the patterns you can make on flat wallpaper — number just 17, the so-called wallpaper groups. The count depends entirely on how many dimensions you allow.