Planes Intersecting Space
Date: 11/24/2001 at 14:52:24 From: Gafur Taskin Subject: Intersection of space and planes Hi there, I have a problem. A plane divides space into at most two parts. Two planes divide space into at most four parts. Three planes divide space into at most eight parts. Four planes divide space into at most sixteen parts. Can we say that n planes divide space into at most 2^n parts? Is there a proof of this idea? Thanks, Gafur
Date: 11/25/2001 at 10:33:30 From: Doctor Jubal Subject: Re: Intersection of space and planes Hi Gafur, Thanks for writing to Dr. Math. Actually, 4 planes cannot divide space into 16 parts, but only 15. Let's say you have three planes that divide space into 8 parts. They intersect at a single point. Let's say you take a fourth plane that is not parallel to any of the other three, but also passes through the point. It will intersect each of the other three planes in a line, and these three lines will intersect at a single point. You can represent this on a sheet of paper by drawing three lines that intersect at a single point. The sheet of paper is the fourth plane. The three lines are the lines where it intersects the other three. You will see that the three lines divide the plane into 6 regions. The first three planes divided space into eight regions. Of these, six of them are divided by the fourth one, if the fourth one passes through the point of intersection of the the other three planes. Of the other two regions, one is entirely "above" the sheet of paper, and the other is "below" it. Since the fourth plane doesn't have to pass through the point of intersection of the other three planes, it is possible to divide either one of these regions in addition to the original six. Make a new drawing on your sheet of paper. This time, draw three non-parallel lines that don't intersect in a common point. Instead, each pair of lines intersects each other at one of three different points. You will see they divide the piece of paper into seven regions. This is the case when the fourth plane does not pass through the point of intersection of the other three. It is not parallel to any of the three original planes, so it intersects each of them in a line. Since it does not contain the point of intersection of the other three planes, these three lines do not intersect each other in a single point. You can see that this plane manages to divide seven out of the eight regions the first three planes divide space into. By not passing exactly through the point of intersection, the plane can divide one of the two regions it "missed" in the first case, while still dividing the other six. It is impossible for the plane to divide all eight regions, however, because it is impossible for the plane to pass to both "sides" of the point of intersection of the first three planes. As a result, four planes can divide space into at most 15 regions. Three planes divide space into eight regions, and a fourth plane can split all of these regions except one that it must miss because it can only pass to one side or another of the point of intersection of the first three. In these four planes, there are four points where three planes intersect to make a point. For each one of these points, a fifth plane must pass to one side or the other of it, and so it must miss four out of the 15 regions, and can only split 11 of the 15 into two. So five planes can only divide space into 15+11 = 26 regions. In general, if you have n planes in space arranged that that no two planes are parallel and no two planes intersect each other in a line that is parallel to any other such line, those n planes will meet at n(n-1)(n-2)/6 points. (This formula comes about because you have n choices of the first plane, n-1 choices the second plane, and n-2 choices of the third plane, but the order of the planes doesn't matter, and there are six ways you could have chosen the same three planes.) At each of these points, the three planes that intersect at that point divide space into eight regions, only seven of which can be split by any single plane. So for each point of intersection of three planes, a new plane must "miss" one regions. So if n planes divide space into at most r regions, the (n+1)st plane can only intersect r - n(n-1)(n-2)/6 of them, so n+1 planes can divide space into at most 2r - n(n-1)(n-2)/6 regions. Thus, 4 planes can divide space into 15 regions, 5 planes can divide space into 26 regions, 6 planes into 42 regions, 7 planes into 64 regions, 8 planes into 93 regions, 9 planes into 130 regions, 10 planes into 176 regions, and so on. Does all this make sense? If there's any part of it that you would like explained in more detail, or would like a more thorough proof of, feel free to write back. - Doctor Jubal, The Math Forum http://mathforum.org/dr.math/
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