The Mobius Strip

The mobius strip is a structure that is a one sided band form by combining  ends of a rectangular strip after first giving one of the ends a half twist. One strip of the end  will be twisted 180°.  August Ferdinand Mobius  was the mathematician and professor that made the discovery of this shape late in his career. Johann Benedict Listing would also make contributions to the articulation of its topological properties. Both  mathematicians produced their findings around 1858. Although it is not realized the mobius strip has many applications to science, art, engineering, and mathematical theory. The one sided structure has been used to present geometric models of the universe, molecular structure, and architectural design. Magicians use the mobius strip for their tricks. The mobius strip can be commonly seen as a the recycling symbol on bins. The mobius strip did not gain wide recognition till after the death of the mathematicians who explored it attributes. The mobius strip can sometimes be described as a twisted cylinder. The one sided surface is non orientable. The mobius strip can not however, be classified as a true surface, rather a  surface with a boundary. This seems like a bizarre concept to many observers. A surface without a boundary describes a topological space that is revealed by edges and vertices of a set of triangles.

The mobius strip has to be expressed  in a form of equation. Parametric equations are ways in which certain elements of topology can be explained in Euclidean space. This is where classical geometry and topology differ. Topology focuses on the arrangement of shapes rather than strict emphasis on angle, distance, and measurement. Euclidean geometry has its limitations when describing particular shapes. The mobius strip would not have been discovered under a method using classical geometry. Topology breaks down shapes into nodes and connections. This approach allows shapes like the torus, mobius strip. and klein bottle to be comprehended. Topology was developed from graph theory in Leonard Euler's work The Seven Bridges of Konigsberg . This mathematical puzzle asked if it was possible to cross all seven bridges in the Baltic sea port with one route. Euler was able to demonstrate that this was not possible due to the number of connections. It was not a matter of the actual distance or orientation. It had to do with the connections or rather vertices. There was no solution to this mathematical problem, because a person would have to either cross a bridge twice or avoid one all together. This was written by Euler in 1736, but he had no idea how mathematics would advance in the 19th century. Euler laid the foundation for August Ferdinand Mobius  and Johann Benedict Listing. Topology seeks to see the nodes and connections in shapes. There is to this branch of mathematics a similarity  in shapes. Shapes remain the same no matter how much they are distorted. 

The mobius strip and the topology it represents branches off into other questions. Homotopy and homology. Homology examines the the structure in relation to algebraic sequence to topological structures. Homotopy puts emphasis on the study of the information related to the spaces of the shape. It also takes into account how functions behave and why they cause certain formations. Both homotopy and homology are needed to explain the attributes of the mobius strip. Topology also explores how geometric functions can be entangled or deformed. Multidimensional surfaces are known as manifolds and classical geometry would not have enough tools to describe these shapes. Topology can be applied to technological use. A pivotal use is for cooperative swarmbots. These diminutive robots manipulate topological spaces to monitor the environment. This allows for discovery of areas to place cell phone masts to enable reliable signals. Geographical information systems utilize topology in relation to maps through domains and boundaries. It may be possible that the mobius strip was stumbled upon prior to the 19th century. There is no recorded evidence that mathematicians came across such as structure. The mobius strip was the first one-sided surface studied by scientists. The shape is one of the most recognizable features of topology known to the general public.   

References 

Picker, Clifford. Math Book . New York: Sterling, 2009. 

Jackson, Tom. Mathematics An Illustrated History of Numbers. New York: 

             Shelter Harbor Press, 2012. 

  Weisstein, Eric. “Möbius Strip.” Wolfram MathWorld, Wolfram MathWorld , mathworld.wolfram.com/MoebiusStrip.html.

Britannica, The Editors of Encyclopaedia. “Möbius Strip.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 19 Apr. 2017, www.britannica.com/science/Mobius-strip.