Möbius Strip

In the eighteenth century, Euler observed that for polyhedra the number of vertices minus the number of edges plus the number of faces equals two. But this relations does not states for all polyhedra, as for example, for a polyhedron with a hole. The astronomer and mathematician August Ferdinand Möbius (1790-1868) studied the geometrical theory of polyhedra and identified surfaces in terms of flat polygonal joined pieces. The Möbius strip, a continuous surface named after him, has only one side and one edge. Starting in any point in its surface, one can reach every point on the strip without even crossing an edge. The Möbius strip is a mathematical construction that shows an evolution from a two-dimensional space into a three-dimensional one, by merging inner and outer spaces it creates a single continuously curved surface. The Möbius strip is also a non oriented surface. In order to be an oriented surface it should, for any point on the surface, have normal vectors with opposite directions.

The torus has a parametric equation:
1. x = (R + L*Cos(Alpha)) * Cos(Theta)
2. y = (R + L*Cos(Alpha)) * Sin(Theta)
3. z = L*Sin(Alpha)
Alpha and Theta ranging from 0 to 360 degrees

The strip of Möbius has a very similar parametric equation:
1. x = (R + L*Cos(Alpha/2)) * Cos(Alpha)
2. y = (R + L*Cos(Alpha/2)) * Sin(Alpha)
3. z = L*Sin(Alpha/2)
Alpha ranging from 0 to 360 degrees, L ranging from -Lmax to +Lmax

The Möbius strip has provided inspiration both for sculptures and for graphical art. Maurits C. Escher is one of the artists who was especially fond of it and based several of his lithographs on this mathematical object. It is also a recurrent feature in science fiction stories, such as Arthur C. Clarke's The Wall of Darkness. Science fiction stories sometimes suggest that our universe might be some kind of generalised Möbius strip.
In the short story "A Subway Named Moebius", by A.J. Deutsch, the Boston subway authority builds a new line; the system becomes so tangled that it turns into a Möbius strip, and trains start to disappear.
There have been technical applications; giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time).