Associated Topics || Dr. Math Home || Search Dr. Math

### Planes Intersecting Space

```
Date: 11/24/2001 at 14:52:24
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/
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search