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Graphs and Surfaces
Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005
The sphere, torus, and cylinder can all be considered as two-sided surfaces -they have an inside and outside. One way to define this is to imagine a small clockwise-oriented circle drawn in the surface. If we reflect this circle in the surface, we obtain a circle of the opposite orientation. On the sphere, torus, and cylinder it is not possible to walk along the surface, taking the oriented circle with us, until it coincides with its opposite orientation (we are not allowed to walk over the boundary of the cylinder or Möbius band). On the Möbius band, it is possible to make these two circles coincide. We therefore say that the Möbius band is a one-sided surface. A two-sided surface is said to be orientable. We can assign an orientation to the surface, by partitioning the set of oriented circles defined at every point of the surface. One orientation is called the inside, and the other the outside. A one-sided surface is non-orientable, as the set of oriented circles does not have this partition into two subsets. For surfaces like the sphere and torus which can be constructed in euclidean 3-space, we could alternatively use a normal vector to the surface, and its reflexion in the surface in place of an oriented circle and its reflexion.
Mathematical Preliminaries
Published in William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain, of Physical Systems, 2020
William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain
To ensure that λ, n, and ν will be uniquely defined, only orientable surfaces and simple curves are considered herein. A surface is orientable if the positive normal direction, indicated by the vector n in figure 2.2, can be defined in a unique and continuous way around the surface [Kreyszig, 1962]. Physically an orientable surface such as S in figure 2.2 has two distinguishable sides which permit selection of a positive normal direction. A well known example of a non-orientable surface is the Möbius strip (A Möbius strip may be constructed by taking a long flat piece of rectangular paper with sides indicated as “a” and “b”, rotating one of the narrow ends 180°, and joining the narrow ends of the paper to form a loop by overlapping the ends such that side “a” is the only exposed side at the point of joining.). For this surface, the positive normal direction reverses when the normal vector is displaced continuously along the surface from a starting point, around the loop, back to the starting point. Of course, when cut, the Möbius strip becomes an orientable surface. So, for any orientable surface, the normal vector to the surface (i.e. n) uniquely defines that surface. Note that one degree of arbitrariness exists in that two choices exist for the positive normal direction. Provided one is consistent with manipulations, this degree of freedom is unimportant. Here, for a closed surface, n will be selected as positive outward. The vectors ν and λ are not uniquely defined for a surface, though they are constrained to be orthogonal to n and to each other.
Fundamental Theorems of Multivariable Calculus
Published in John Srdjan Petrovic, Advanced Calculus, 2020
In order to make this meaningful, our surface M must have 2 sides. We say that such a surface is orientable. Geometrically, this means that if the normal vector n is moved along a closed curve in M, upon returning to the same point it will not have the opposite direction. A typical example of a non-orientable surface is the Mobius strip (Figure 15.11). Most surfaces that we encounter are orientable.
Design in the service of topology and the classification theorem
Published in International Journal of Mathematical Education in Science and Technology, 2023
We know the Möbius strip has a boundary or end, since it can be constructed from paper of a certain size. We will say that a surface does not have an end if any point on the surface is not the ‘last point’ in the sense of boundary points (e.g. a sphere, torus, or Klein bottle). We also know that an ant can walk continuously and cover the whole surface of the strip – this property is called non-orientability, i.e. a surface with one side. A Möbius strip or Klein bottle are non-orientable, while a sphere or torus are orientable, i.e. surfaces with two sides (for more details, see Braselton et al., 2002; Cohen & Gul, 2020). These properties can help us define and identify surfaces that are homeomorphic to the Möbius strip.