Exotic spheres and curvature - American Mathematical Society

... manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps. It is far from obvious that there are manifolds which (up to diﬀeomorphism) admit more than one distinct smooth struc ...

... manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps. It is far from obvious that there are manifolds which (up to diﬀeomorphism) admit more than one distinct smooth struc ...

Pdf slides - Daniel Mathews

... Shortest distance between two points? I Between (x, y1 ) and (x, y2 ): a vertical line. I For other points: the hyperbolic line between them is not a straight Euclidean line. I Turns out it’s a circle intersecting the x-axis orthogonally. I So given point p and line l, many lines through p are paral ...

... Shortest distance between two points? I Between (x, y1 ) and (x, y2 ): a vertical line. I For other points: the hyperbolic line between them is not a straight Euclidean line. I Turns out it’s a circle intersecting the x-axis orthogonally. I So given point p and line l, many lines through p are paral ...

2_4_Postulates_Diagrams

... geometry that are so basic, they are assumed to be true without proof. – Sometimes called axioms. ...

... geometry that are so basic, they are assumed to be true without proof. – Sometimes called axioms. ...

Non-Euclidean Geometry, Topology, and Networks

... measures of the angles in any triangle is 180°. In Lobachevskian geometry, the sum of the measures of the angles in any triangle is less than 180°. Also, triangles of different sizes can never have equal angles, so similar triangles do not exist. The geometry of Euclid can be represented on a plane. ...

... measures of the angles in any triangle is 180°. In Lobachevskian geometry, the sum of the measures of the angles in any triangle is less than 180°. Also, triangles of different sizes can never have equal angles, so similar triangles do not exist. The geometry of Euclid can be represented on a plane. ...

Mathematics and Music Boxes

... vertically. This will make the pitch higher or lower, known to musicians as transposition. The two can be combined to play the music at both a different time and pitch, as seen in figure 2. ...

... vertically. This will make the pitch higher or lower, known to musicians as transposition. The two can be combined to play the music at both a different time and pitch, as seen in figure 2. ...

Learning with a twist

... One way to represent the Möbius strip as a subset of threedimensional Euclidean space is using the parametrization: x ( u , v ) = ( 1 + v 2 cos u 2 ) cos u {\displaystyle x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u} y ( u , v ) = ( 1 + v 2 cos u 2 ) sin u {\displaystyle y(u,v)=\lef ...

... One way to represent the Möbius strip as a subset of threedimensional Euclidean space is using the parametrization: x ( u , v ) = ( 1 + v 2 cos u 2 ) cos u {\displaystyle x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u} y ( u , v ) = ( 1 + v 2 cos u 2 ) sin u {\displaystyle y(u,v)=\lef ...

# Möbius strip

The Möbius strip or Möbius band (/ˈmɜrbiəs/ (non-rhotic) or US /ˈmoʊbiəs/; German: [ˈmøːbi̯ʊs]), also Mobius or Moebius, is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a loop. However, the Möbius strip is not a surface of only one exact size and shape, such as the half-twisted paper strip depicted in the illustration. Rather, mathematicians refer to the closed Möbius band as any surface that is homeomorphic to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle. This allows for a very wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape. For example, any rectangle can be glued to itself (by identifying one edge with the opposite edge after a reversal of orientation) to make a Möbius band. Some of these can be smoothly modeled in Euclidean space, and others cannot.A half-twist clockwise will give a different embedding of the Möbius strip than a half-twist counterclockwise – that is, as an embedded object in Euclidean space the Möbius strip is a chiral object with right- or left-handedness. However, the underlying topological spaces within the Möbius strip are homeomorphic in each case. There are an infinite number of topologically different embeddings of the same topological space into three-dimensional space, as the Möbius strip can also be formed by twisting the strip an odd number of times greater than one, or by knotting and twisting the strip, before joining its ends. The complete open Möbius band is an example of a topological surface that is closely related to the standard Möbius strip but that is not homeomorphic to it.It is straightforward to find algebraic equations, the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above. In particular, the twisted paper model is a developable surface, having zero Gaussian curvature. A system of differential-algebraic equations that describes models of this type was published in 2007 together with its numerical solution.The Euler characteristic of the Möbius strip is zero.