Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had largely been completed. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography Hahn (2002). In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. In dimensions other than 3, they are sometimes called Bieberbach groups. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchanged. In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The first m indicates the mirror plane perpendicular to the c-axis (a), the second m indicates the mirror planes parallel to the c-axis (b), and the c indicates the glide planes (b) and (c). The space group of hexagonal H 2O ice is P6 3/ mmc.
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