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A homotopy from a circle around a sphere down to a single point. Any continuous mapping from a circle to an ordinary sphere can be continuously deformed to a one-point mapping, and so its homotopy class is trivial. One way to visualize this is to imagine a rubber-band wrapped around a frictionless ball: the band can always be slid off the ball.
A hole in X is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point. [1]: 78 Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally, A d-dimensional sphere in X is a continuous function:.
They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.
The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometry in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp. spheres, planes and points) are treated on an equal footing.
If "line" is taken to mean great circle, spherical geometry only obeys two of Euclid's five postulates: the second postulate ("to produce [extend] a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three.
Familiar shapes, such as the surface of a ball (which is known in mathematics as the two-dimensional sphere) or of a torus, are two-dimensional. The surface of a ball has trivial fundamental group, meaning that any loop drawn on the surface can be continuously deformed to a single point.
A light is fixed in one port, pointing into the sphere, which collects all the light, allowing none to escape, so that it can be measured with a sensor in a second port. ... in a line, every 5 ...
A continuous map : [,] is a deformation retraction of a space X onto a subspace A if, for every x in X and a in A, (,) =, (,), (,) =.In other words, a deformation retraction is a homotopy between a retraction and the identity map on X.