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Nov. 29, 2006 Which has more points, a line segment two inches long or a line segment one inch long? Suppose we draw two such segments on a piece of paper, parallel to each other. Then we draw two more lines, one through the left end of both segments and another through the right end of both, extending the two new lines till they intersect. Now we have a triangle with a base two inches long and an interior line parallel to the base and one inch long. We may choose any point in either of the original line segments and draw another line from the apex passing through that point and intersecting the other line. We then see that for every point in the long line, there is a corresponding point in the short line, and vice versa. Their positions can even be calculated. If we have points at .5, 1 and 1.5 inches from the left end in the long line, they correspond to points .25, .5 and .75 from the left end of the short line. Thus the points are in perfect one-to-one correspondence, and two equations may be written to locate the points in either line, given the points in the other. Moreover, these equations are mutually inverse. Therefore the two line segments are said to be homeomorphic or topologically equivalent. All closed line segments are topologically equivalent, regardless of their length in inches, miles or light-years. Even if we begin to curve one of the lines so that it forms the arc of a circle, the lines are still homeomorphic. However, if we complete the circle, making it perfectly continuous, it means we have lost a point. The two end points of the original straight line have been merged into a single point, and the two figures are no longer homeomorphic. No amount of reasoning will enable you to prove that a line segment and a circle have the same number of points. They are not topologically equivalent. Are a hollow sphere and a hollow cube homeomorphic? Suppose we put a basketball in a cubical cardboard box with a lid, disregarding the thickness of the material in the two objects. Supposing that they are concentric, let us choose any point on the surface of the box, or on the surface of the ball, and produce a line from their common center, extending it through the point, till it passes through the surface of both objects. We then see that each and every point on either the box or the ball corresponds to a single unique point on the other object. If we know the dimensions we can also calculate these points to any degree of accuracy. The points are in a one-to-one correspondence and the location of each may be calculated by formula from the location of the other. Therefore the box and the ball are topologically equivalent or homeomorphic. A little reflection tells us that a whole variety of other shapes—parallelepipeds, prisms, tetrahedrons, spheroids, ellipsoids and closed-end cylinders are all topologically equivalent. However, if we put a torus, like an inflated inner tube, or two basketballs, in a box, in no way will we be able to demonstrate homeomorphism. Generally, if an object consists of two or more unconnected pieces or has one or more holes, it is not topologically equivalent to another object without these features. Can the surface of a sphere be shown to be topologically equivalent to an entire plane? It can if we remove a single point from the surface of the sphere and specify that only points in the plane that have finite coordinates be considered. Imagine a horizontal plane extending indefinitely in all directions. Upon it we rest the south pole of a sphere, while removing a single point from the surface of the sphere, at the north pole. Now if from any point on the plane, regardless of its left-right or forward-backward distance from the sphere, we draw a line to the missing point at the north pole, the line will puncture the surface of the sphere at a unique point. Likewise, producing a line from the north pole through any point on the surface of the sphere till it passes through the plane, we will see that the point on the sphere corresponds to one point and one point only on the place. Thus, this is a homeomorphism or topological equivalence. This sphere is called the Riemann sphere, invented by the famous mathematician Bernhard Riemann in conjunction with his work on complex numbers. This diagram is a cross-section through a Riemann Sphere: http://en.wikipedia.org/wiki/Image:Stereographic.png ------------ About the author Thomas Keyes: I have written two books: A SOJOURN IN ASIA (non-fiction) and A TALE OF UNG (fiction), neither published so far. I have studied languages for years and traveled extensively on five continents. Email: udikeyes@yahoo.com Comment on this article here! ------------ All articles are EXCLUSIVE to Useless-Knowledge.com and are not allowed to be posted on other websites. ARTICLE THIEVES WILL BE PROSECUTED! |
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