Editing Möbius strip (section)

===Spaces of lines===
The family of lines in the plane can be given the structure of a smooth space, with each line represented as a point in this space. The resulting space of lines is [[diffeomorphic|topologically equivalent]] to the open Möbius {{nowrap|strip.{{r|parker}}}} One way to see this is to extend the Euclidean plane to the [[real projective plane]] by adding one more line, the [[line at infinity]]. By [[projective duality]] the space of lines in the projective plane is equivalent to its space of points, the projective plane itself. Removing the line at infinity, to produce the space of Euclidean lines, punctures this space of projective {{nowrap|lines.{{r|bickel}}}} Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the open Möbius {{nowrap|strip.{{r|seifert-threlfall}}}} The space of lines in the [[hyperbolic plane]] can be parameterized by [[unordered pair]]s of distinct points on a circle, the pairs of points at infinity of each line. This space, again, has the topology of an open Möbius {{nowrap|strip.{{r|mangahas}}}}

These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the [[affine transformation]]s, and the symmetries of hyperbolic lines include the {{nowrap|[[Möbius transformation]]s.{{r|ramirez-seade}}}} The affine transformations and Möbius transformations both form {{nowrap|6-dimensional}} [[Lie group]]s, topological spaces having a compatible [[Symmetry group|algebraic structure]] describing the composition of {{nowrap|symmetries.{{r|fomenko-kunii|isham}}}} Because every line in the plane is symmetric to every other line, the open Möbius strip is a [[homogeneous space]], a space with symmetries that take every point to every other point. Homogeneous spaces of Lie groups are called [[solvmanifold]]s, and the Möbius strip can be used as a [[counterexample]], showing that not every solvmanifold is a [[nilmanifold]], and that not every solvmanifold can be factored into a [[Direct product of groups|direct product]] of a [[compact space|compact]] solvmanifold {{nowrap|with <math>\mathbb{R}^n</math>.}} These symmetries also provide another way to construct the Möbius strip itself, as a ''group model'' of these Lie groups. A group model consists of a Lie group and a [[stabilizer subgroup]] of its action; contracting the [[coset]]s of the subgroup to points produces a space with the same topology as the underlying homogenous space. In the case of the symmetries of Euclidean lines, the stabilizer of the {{nowrap|<math>x</math>-axis}} consists of all symmetries that take the axis to itself. Each line <math>\ell</math> corresponds to a coset, the set of symmetries that map <math>\ell</math> to the {{nowrap|<math>x</math>-axis.}} Therefore, the [[Quotient space (topology)|quotient space]], a space that has one point per coset and inherits its topology from the space of symmetries, is the same as the space of lines, and is again an open Möbius {{nowrap|strip.{{r|gor-oni-vin}}}}