Möbius strip in the context of "Recycling symbol"

In mathematics, the Klein bottle (/ˈklaɪn/) is an example of a surface with no distinct inside or outside. In other words, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, it is an example of a non-orientable surface, a two-dimensional manifold on which one cannot define a consistent direction perpendicular to the surface (normal vector) that varies continuously over the whole shape.

The Klein bottle is related to other non-orientable surfaces like the Möbius strip, which also has only one side but does have a boundary. In contrast, the Klein bottle is boundaryless, like a sphere or torus, though it cannot be embedded in ordinary three-dimensional space without intersecting itself.

In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". It generalizes the concept of curve orientation, which for a plane simple closed curve is defined based on whether the curve interior is to the left or to the right of the curve. A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space.

Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. A generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

A roller coaster is a type of amusement ride employing a form of elevated railroad track that carries passengers on a train through tight turns, steep slopes, and other elements. Roller coasters are usually designed to produce a thrilling experience, though some roller coasters aim to provide a more gentle experience. Trains consist of open cars connected in a single line, and tracks are typically built and designed as a complete circuit in which trains depart from and return to the same loading station. One notable exception is Racer at Kennywood, West Mifflin, PA, which has the train return to the opposite loading track due to its unique Möbius design. The rides are typically found in amusement parks around the world but can also be located in shopping malls and zoos. The Roller Coaster DataBase estimates there are approximately 6,000 extant roller coasters as of October 2025.

The earliest progenitors to the modern roller coasters were the "Russian Mountains", which first appeared in the 17th century. LaMarcus Adna Thompson obtained one of the first known patents for a track-based roller coaster design in 1885, based on the Switchback Railway which opened a year earlier at Coney Island. Today, most roller coasters are built out of steel, which can allow for more intense forces and inversions (where the rider is upside down).