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In mathematics, a Möbius strip, Möbius band, or Möbius loop [a] is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE .
The Möbius strip is a nontrivial bundle over the circle. Perhaps the simplest example of a nontrivial bundle E {\displaystyle E} is the Möbius strip . It has the circle that runs lengthwise along the center of the strip as a base B {\displaystyle B} and a line segment for the fiber F {\displaystyle F} , so the Möbius strip is a bundle of the ...
In the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz Friedrich Tietze, who showed in 1910 that the Möbius strip can be subdivided into six regions that all touch each other – three along the boundary of the strip and three along its center line – and therefore that graphs that are embedded ...
The homology of a topological space X is a set of topological invariants of X represented by its homology groups (), (), (), … where the homology group () describes, informally, the number of holes in X with a k-dimensional boundary.
The Möbius strip can be constructed by a non-trivial gluing of two trivial bundles on open subsets U and V of the circle S 1.When glued trivially (with g UV =1) one obtains the trivial bundle, but with the non-trivial gluing of g UV =1 on one overlap and g UV =-1 on the second overlap, one obtains the non-trivial bundle E, the Möbius strip.
A torus is an orientable surface The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. The Roman surface is non-orientable.. In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". [1]
Meet baby MilliAnna, the fourth generation of women in her family to be born with a wild streak of white hair. Baby takes after 3 generations of women to be born with streak of white hair Skip to ...
An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map h takes a homotopy class of maps from X to K(G, i) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor. [1]