Geometry is considered as an indigenous branch of mathematics that evolved in the Greeks Hellenistic period, as a practical way for dealing with lengths, areas, and volumes. By the 3rd century BC, Euclid gave an axiomatic frame to study geometry, namely the Euclidean Geometry. Euclid’s work set a standard for many centuries to follow. René Descartes and Pierre de Fermat gave a new perspective of studying geometry on a solid analytic footing by the early 17th century. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object.

Mathematically speaking, a surface is a connected two-dimensional manifold.

The surface of our earth is thus a two-dimensional manifold, meaning that it locally looks like a Euclidean plane. That is why our linear dynamic laws used in daily life are based on the planar geometry, in spite of the fact that the earth is spherical in shape. In simple words, we say that a 3-dimensional sphere looks like a plane closed to a point.

Surfaces are generally divided into two big classes, namely, “orientable surfaces” and “non-orientable surfaces”. This characterization is based on their topological behavior. Roughly speaking, topological study of a surface is concerned with the properties that are preserved under all possible deformations, if the surface is a rubber piece (that can be stretched, crumpled and twisted but not teared or glued). An interesting fact about the surfaces is that they can be constructed from polygons. For instance, a torus can be constructed by two transformations on a rectangle. First, we bend the rectangle along one direction, joining opposite sides, creating a cylinder. Then, we bend the cylinder so the ends are joined. The resulting figure is a torus.

A torus is an example of an orientable surface, mathematically denoted as S1. All orientable closed surfaces are topologically equivalent to Sk, here k represents the count of handles on the sphere. An illustration of adding handles to a sphere is given below.

An important discovery in the nineteenth century was that non-orientable surfaces exist. August Ferdinand Möbius (1790–1868) discovered a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary. A Möbius strip can be obtained by joining the opposite sides of the rectangle in the direction of A given below.

A non-orientable surface is a self-intersecting surface obtained by adding cross caps to the sphere.

Adding a cross cap to the sphere involves the cutting off of a disc from the surface of the sphere, attaching a Möbius strip at the position of the disc, and matching the boundary of the Möbius strip with the boundary of the disc. A sphere with k cross caps is marked as Nk. Every closed non-orientable surface is topologically equivalent to Nk. A Klein bottle is a non-orientable surface N2.   