Geometry: Spheres
Spheres
Thanks to early explorers, people have lost the fear of falling off the edge of the earth. It took quite awhile for people to accept the fact that the earth is round, but not circular. I mean that the earth is not the two-dimensional kind of round that CDs and plates are made of, but the three-dimensional kind of round that basketballs and oranges are made of. Even though the sun and the moon look like two-dimensional round objects up in the sky, you know that they have three dimensions of roundness. They are spherical: the three-dimensional equivalent of two-dimensional circles.
Tangent Line
Spheres are appealing for a variety of reasons. You can rotate a sphere or look at it from any vantage point and it will look the same. A sphere is one of the most symmetric shapes in nature. It is interesting that very few sports involve a perfectly symmetrical ball (Ping-Pong is one that immediately comes to mind). Asymmetries are introduced to change the features of the ball. For example, a golf ball has dimples and a baseball has stitches. But the dimples of a golf ball and the stitches of a baseball are strategically placed. New designs promise that your ball will travel farther and faster, and curve more or less (bending to your will, of course).
As with most three-dimensional objects, spheres are very similar to their two-dimensional counterparts. A circle is defined as the set of points in a plane that are a fixed distance r from a certain point, which is called the center of the circle. A sphere is defined in a similar manner, without the restriction that the points have to lie in one plane. A sphere is the set of all points (in three-dimensional space) that are a fixed distance r from a certain point, which is called the center of the sphere. The fixed distance r is the radius of the sphere.
Solid Facts
A sphere is the set of all points (in three-dimensional space) that are a fixed distance r from a certain point.
The fixed distance is called the radius of the sphere.
The center of the sphere is the point that is a fixed distance from each point on the sphere.
Spheres have diameters just like circles do. A diameter is a line segment connecting two points on the sphere that passes through the center. Circles can be cut by chords, so when this idea is generalized in three-dimensional space, planes will slice spheres. The intersection of a sphere and a plane is a circle.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can also purchase this book at Amazon.com and Barnes & Noble.
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