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/