open Möbius The space of lines in the hyperbolic plane can be parameterized by unordered pairs 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 These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the affine transformations, and the symmetries of hyperbolic lines include the The affine transformations and Möbius transformations both form Lie groups, topological spaces having a compatible algebraic structure describing the composition of 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 solvmanifolds, 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 a compact solvmanifold 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 cosets 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 consists of all symmetries that take the axis to itself. Each line corresponds to a coset, the set of symmetries that map to the Therefore, the 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 Applications Beyond the already-discussed applications of Möbius strips to the design of mechanical belts that wear evenly on their entire surface, and of the Plücker conoid to the design of gears, other applications of Möbius strips include: Graphene ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism Möbius aromaticity, a property of organic chemicals whose molecular structure forms a cycle, with molecular orbitals aligned along the cycle in the pattern of a Möbius strip The Möbius resistor, a strip of conductive material covering the single side of a dielectric Möbius strip, in a way that cancels its own self-inductance Resonators with a compact design and a resonant frequency that