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While we can drape a string on a doughnut so that it can be shrunk to a point, there are ways we can loop strings around a doughnut so that they cannot-through the hole in the center, for example. For example, it is clear that any string we tie around a sphere, a basketball, for example, can be contracted until it is very small, even a point. Strings, surfaces, and looping around For more information please see: Fundamental group and Homotopy groupīy strings and surfaces, we mean the possible ways of embedding and into other spaces.įor the embedding of strings ( ) into two-dimensional spaces, it is easiest to imagine holding strings on these surfaces. The three sphere, or, refers to the set of all points in a four-dimensional Euclidean space which are the same distance from a single point. Note that the interior of a sphere is not, and that like, a deflated beach ball stretched into some strange shape is still, so long as it does not self-intersect. The defining quality of a sphere is that it is the set of all points in a Euclidean (i.e., normal) three-dimensional space which are the same distance from a set point. The surface of an ordinary sphere, like the Earth, is called a two sphere, or. Any deformation (though not every rejoining) of this circle is an −an object does not need to be a perfect circle to be called a one sphere by topologists: any closed loop which does not intersect itself is an example of an.
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The defining quality of a circle is that it is the set of all points in a plane which are the same distance from a specific point. Mathematicians call the ordinary circle the "one sphere," which is written. Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three-dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself. The Poincaré conjecture is this: the set of points at an equal radius from the center in four-dimensional space (known as a "three-dimensional sphere") is the only finite, bounded, simply connected object that locally looks like three-dimensional space and has the property that all loops and spheres within it can be shrunk to points. A doughnut is an example of an object that is not simply connected. 1.2 Strings, surfaces, and looping aroundĪn ordinary sphere (such as the surface of orange) is " simply connected" because a stretchable loop on it can be reduced to a single point without tearing it.