Möbius’ strip (or Möbius’ band) is named after August Ferdinand Möbius (1790-1868). This is probably the most famous of all one-sided surface. To create a cylinder, take a strip of paper, bent it to join the two ends. For Möbius’ strip, the procedure is rather the same, except, before joining the ends, turn one of them half a turn, i.e. 180°, and then apply glue. Also, cutting Möbius’ strip also leads to unexpected results. 

  • Cut your band down the center line all the way around, what will happen…EX: Figure 7- 1 half turn, M = 3, the middle layer removed  & Figure 8: 2 half turns, M = 3, the middle layer removed.
  • Cut 1/3 in form edge, all the way around., what will happen….
  • What happens with two twists or more…

See more at: Möbius’ Strip on Cut-the-knot.org & http://jdh.hamkins.org

Topology is the mathematical study of the spatial properties that are preserved through the deformation, twisting and stretching of objects. Topological architectures are common in nature and can be seen, for example, in DNA molecules that condense and relax during cellular events 1. Synthetic topological nanostructures, such as catenanes and rotaxanes, have been engineered using supramolecular chemistry, but the fabrication of complex and reconfigurable structures remains challenging 2. Here, we show that DNA origami 3 can be used to assemble a Möbius strip, a topological ribbon-like structure that has only one side 4, 5, 6. In addition, we show that the DNA Möbius strip can be reconfigured through strand displacement 7 to create topological objects such as supercoiled ring and catenane structures. This DNA fold-and-cut strategy, analogous to Japanese kirigami 8, may be used to create and reconfigure programmable topological structures that are unprecedented in molecular engineering. (Source)

Image: Folding and cutting DNA into reconfigurable topological nanostructures by Dongran Han,  Suchetan Pal, Yan Liu and Hao Yan - Nontrivial topological states on a Möbius band W. Beugeling, A. Quelle, and C. Morais Smith - Mobius band (gifs) on Wikipedia - Mobius’ Strip on Cut-the-knot.org