That is, every orbit of the group (the images of a given point under all group elements) forms a discrete set. All finite symmetry groups are discrete. Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversions and rotoinversions – i.e., the finite subgroups of O(n); (2) infinite lattice groups, which include only translations; and (3) infinite space groups containing elements of both previous types, and perhaps also extra transformations like screw displacements and glide reflections. There are also continuous symmetry groups (Lie groups), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is O(3), the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E(n) (the isometry group of Rn). Two geometric figures have the same symmetry type when their symmetry groups are conjugate subgroups of the Euclidean group: that is, when the subgroups H1, H2 are related by for some g in E(n). For example: two 3D figures have mirror symmetry, but with respect to different mirror planes. two 3D figures have 3-fold rotational symmetry, but with respect to different axes. two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction. In the following sections, we only consider isometry groups whose orbits are topologically closed, including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by a rational number; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail. One dimension The isometry groups in one dimension are: the trivial cyclic group C1 the groups of two elements generated by a reflection; they are isomorphic with C2 the infinite discrete groups generated by a translation; they are isomorphic with Z, the additive group of the integers the infinite discrete groups generated by a translation and a reflection; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D∞ (which is a semidirect product of Z and C2). the group generated by all translations (isomorphic with the additive group of the real numbers R); this group cannot be the symmetry group of a Euclidean figure, even endowed with a pattern: such a pattern would be homogeneous, hence could also be reflected. However, a constant one-dimensional