page_title: Möbius strip
section_title: Applications
hierarchical_section_title: Möbius strip / Applications
caption_attribution_description: A demonstration of the mathematical basis of crochet: a scarf made as a moebius strip. Acrylic with novelty yarn. Original design.
context_page_description: In mathematics, a Möbius strip, band, or loop, also spelled Mobius or Moebius, is a surface with only one side and only one boundary curve. The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858, though similar structures can be seen in Roman mosaics c. 200–250 AD.
  An example of a Möbius strip can be created by taking a paper strip, giving one end a half-twist, and then joining the ends to form a loop; its boundary is a simple closed curve which can be traced by single unknotted string. Any topological space homeomorphic to this example is also called a Möbius strip, allowing for a very wide variety of geometric realizations as surfaces with a definite size and shape. For example, any rectangle can be glued left-edge to right-edge with a reversal of orientation. Some, but not all, of these can be smoothly modeled as surfaces in Euclidean space.
context_section_description: There have been several technical applications for the Möbius strip.  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).  Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, as they let the ribbon be twice as wide as the print head while using both halves evenly.
  A Möbius resistor is an electronic circuit element that cancels its own inductive reactance. Nikola Tesla patented similar technology in 1894: "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.
  The Möbius strip is the configuration space of two unordered points on a circle.  Consequently, in music theory, the space of all two-note chords, known as dyads, takes the shape of a Möbius strip; this and generalizations to more points is a significant application of orbifolds to music theory.
  In physics/electro-technology as:
  A compact resonator with a resonance frequency that is half that of identically constructed linear coils
  An inductionless resistor
  Superconductors with high transition temperature
  Möbius resonator
  In chemistry/nano-technology as:
  Molecular knots with special characteristics (Knotane [2], Chirality)
  Molecular engines
  Graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism
  A special type of aromaticity: Möbius aromaticity
  Charged particles caught in the magnetic field of the Earth that can move on a Möbius band
  The cyclotide (cyclic protein) kalata B1, active substance of the plant Oldenlandia affinis, contains Möbius topology for the peptide backbone.
caption: Mathematical art: a scarf designed as a Möbius strip
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