All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells that are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types. Here are all the names that differ in short and full notation. The remaining names are "p"1, "p"2, "p"3, "p"3"m"1, "p"31"m", "p"4, and "p"6. Orbifold notation for wallpaper groups, advocated by John Horton Conway (Conway, 1992) (Conway 2008), is based not on crystallography, but on topology. We fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols. Consider the group denoted in crystallographic notation by cmm; in Conway's notation, this will be 2*22. The 2 before the * says we have a 2-fold rotation centre with no mirror through it. The * itself says we have a mirror. The first 2 after the * says we have a 2-fold rotation centre on a mirror. The final 2 says we have an independent second 2-fold rotation centre on a mirror, one that is not a duplicate of the first one under symmetries. The group denoted by pgg will be 22×. We have two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold. Coxeter's bracket notation is also included, based on reflectional Coxeter groups, and modified with plus superscripts accounting for rotations, improper rotations and translations. An orbifold can be viewed as a polygon with face, edges, and vertices which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a spherical symmetry group or Hyperbolic symmetry group. The type of space the polygons tile can be found by calculating the Euler characteristic, "χ" = "V" − "E" + "F", where "V" is the number of corners (vertices), "E" is the number of edges and "F" is the number of faces.