Various Geometric Infinities
In geometry, infinity is a great unifying concept. Three points form a triangle, right? Well, yes and no. Three collinear points that lie on the same straight line do not exactly look like the vertices of a triangle. Nonetheless, it is often a convenience to look at them as such. Then straight lines become circles of an infinite radius and parallel lines intersect at a point at infinity. (It is easy to be flippant, but a caution should be exercised. Straight lines intersect at at most one point; common circles that intersect have to points in common. Is that a logical contradiction?) On other occasions, like, for example, in inversive geometry, all possible points at infinity map onto the center of inversion and, for this reason, thought of as a single point.
Limits - a calculus concept - also appear in geometry and here, too, prove a powerful tool. In geometry, limits may explain some paradoxes, Zeno's paradoxes, for example. On the other hand, they may, unless care is exercised, lead to paradoxical situations as, for example, in the appearance of curves that have non-zero area. Limits of curves exist under various circumstances and, subject to various definitions, have different properties. The definitions that guarantee the continuity of area enclosed by the curves are not those that guarantee the continuity of their length!
Copyright © 1996-2009 Alexander Bogomolny
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