non-orientable surface
unorientable surface

Some non-orientable surfaces have mapping class groups with simple presentations.

The same concept works equally well for non-orientable surfaces.

For a non-orientable surface, a hole is equivalent to two cross-caps.

The article on the fundamental polygon describes the higher non-orientable surfaces.

Vector fields can not be integrated on non-orientable surfaces.

We also remark that the closed genus three non-orientable surface N has:

The closed surface so produced is the real projective plane, yet another non-orientable surface.

It is a non-orientable surface.

Mathematicians call this a non-orientable surface.

By gluing together projective planes successively we get non-orientable surfaces of higher demigenus.

The simplest non-orientable surface on which the Petersen graph can be embedded without crossings is the projective plane.

That these two cases exhaust all the possibilities for a compact non-orientable surface was shown by Henri Poincaré.

In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces.

The unit normal bundle of a non-orientable surface is a circle bundle that is not a principal bundle.

He first defined the Möbius strip in 1861 (rediscovered four years later by Möbius), as an example of a non-orientable surface.

Dehn twists can also be defined on a non-orientable surface S, provided one starts with a 2-sided simple closed curve c on S.

Lickorish also obtained an analogous result for non-orientable surfaces, which require not only Dehn twists, but also "Y-homeomorphisms."

His graduate student François Apéry later discovered (in 1986) another parametrization of Boy's surface, which conforms to the general method for parametrizing non-orientable surfaces.

D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc.

It follows that all non-orientable surfaces, except the real projective plane, are aspherical as well, as they can be covered by an orientable surface genus 1 or higher.

In particular, it is equal to 2 2g for a closed oriented surface with genus g and to 2 k for a non-orientable surface with k crosscaps.

The non-orientable genus of a graph is the minimal integer n such that the graph can be embedded in a non-orientable surface of (non-orientable) genus n.

For Dr. Gerhard Ringel, head of the Math department at UCSC, Risan worked on embeddings on orientable and non-orientable surfaces.

The 5-regular Clebsch graph can be embedded as a regular map in the orientable manifold of genus 5, forming pentagonal faces; and in the non-orientable surface of genus 6, forming tetragonal faces.

This is because 1) two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible, and 2) the Möbius strip is the only surface that is topologically a subspace of every non-orientable surface.