How to do a Moebius strip.
1. Start with a long rectangle (ABCD) made of paper.
2. Give the rectangle a half twist.
This surface is called a Möbius Strip or Möbius Band, named after August Ferdinand Möbius, a nineteenth century German mathematician and astronomer, who was a pioneer in the field of topology. Möbius, along with his better known contemporaries, Riemann, Lobachevsky and Bolyai, created a non-Euclidean revolution in geometry. Möbius strips have found a number of surprising applications that exploit a remarkable property they possess: one-sidedness. Joining A to C and B to D (no half twist) would produce a simple belt-shaped loop with two sides and two edges — impossible to travel from one side to the other without crossing an edge. But, as a result of the half twist, the Möbius Strip has only one side and one edge!
Giant Möbius Strips have been used as conveyor belts (to make them last longer, since “each side” gets the same amount of wear) and as continuous-loop recording tapes (to double the playing time). In the 1960’s Sandia Laboratories used Möbius Strips in the design of versatile electronic resistors. Free-style skiers have christened one of their acrobatic stunts the Möbius Flip.