Spaces Formed by Intersecting Planes

Date: 07/19/98 at 22:20:10
From: Kirste Brooks
Subject: Planar geometry

Do you know of a proof that would be used to show how many subspaces 
can be formed by the intersecting of five planes in space? This should 
be the largest number, of course. Pascal's triangle and other patterns 
can lead to the conclusion that there is 1 space divided by 0 planes, 
2 spaces divided by 1 plane, 4 spaces divided by 2 planes, 8 spaces 
divided by 3, 15 spaces divided by 4, and 26 spaces divided by 26 
planes. But is there a proof? 

Date: 07/20/98 at 20:22:31
From: Doctor Tom
Subject: Re: Planar geometry

Hi Kirste,

There's a nice way to look at this by looking at the problem in lower 
dimensions.

In the easiest case, 1 dimension, suppose you have a line and you ask 
how many regions it can be divided into by n points. That's easy: each 
point divides one of the existing segments into 2, so each new point 
adds 1, so the total is n+1. (When n is zero, there's one line; adding 
a point makes 2 half-lines, and so on.)

Now look at 2 dimensions - a plane divided up by n lines. By messing 
around, you can see that for n = 0, 1, 2, 3, the answers are: 1, 2, 4, 
7 regions, right?  Well, suppose you've worked it out for some number 
n of lines, and you add the next line. In general, it will hit all n 
lines, so if you look at the intersections of the old lines with your 
new one, it gets hit n times, right? So the new line is divided into 
n+1 segments. (We just worked this out in the previous paragraph).

So each of those n+1 segments will divide an existing region into two 
regions, so there will be n+1 new regions created, so you can work out 
the number of 2-D regions you get with n lines:

   1 + 1 + 2 + ... + (n+1)

(The initial "1" is because there's already one region when you start.) 
So the values from the formula above are:

   lines   regions
   0       1 = 1
   1       1+1 = 2
   2       1+1+2 = 4
   3       1+1+2+3 = 7
   4       1+1+2+3+4 = 11
   5       1+1+2+3+4+5 = 16
   ...

Now go to three dimensions. Assume you know the answer for n planes, 
and you want the answer for n+1. Well, the (n+1)st plane will hit all 
the n planes in one line each, so that plane is hacked into the number 
of regions that n lines will create, which we just worked out as the 
sum above.

Each of those plane regions will divide a volumetric region into 2 
pieces, so the answers for 3 dimensions are:

   planes   regions
   0        1 = 1
   1        1+1 = 2
   2        1+1+2 = 4
   3        1+1+2+4 = 8
   4        1+1+2+4+7 = 15
   5        1+1+2+4+7+11 = 26
   6        1+1+2+4+7+11+16 = 42
   ...

Let me list all the results above:

          number of points, lines, planes:
          0  1  2  3  4  5  6  7
   dim    --------------------------
    1     1  2  3  4  5  6  7  8 ...
    2     1  2  4  7 11 16 22 29 ...
    3     1  2  4  8 15 26 42 64 ...

The first row is obvious - to get any other number in the chart, add 
together the number to it's left to the number above it.

You can actually find a formula for it if you like. For dimension 1, 
it's linear: n+1. For dimension 2, it'll be quadratic, and for 
dimension 3, cubic. You can find a good reference for finding the 
formula for dimension 2 at:

  http://mathforum.org/dr.math/problems/regimbald3.9.98.html   

And by the way, this works fine for hyperspace of still higher 
dimensions.

- Doctor Tom, The Math Forum
Check out our web site! http://mathforum.org/dr.math/