Spaces Formed by Intersecting PlanesDate: 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/ |
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