N lines separate the plane into...

In summary, the conversation discusses the problem of showing that n lines divide a plane into (n^2+n+2)/2 regions if no two lines are parallel and no three pass through a common point. The solution involves extending the lines to infinity and noting that the number of bounded regions is (1/2)(n-1)(n-2), which is also the genus formula for a complex projective plane curve of degree n.
  • #1
r0bHadz
194
17

Homework Statement


Show that n lines separate the plane into (n^2+n+2)/2 regions if no two of these lines are parallel and no three pass through a common point

Homework Equations

The Attempt at a Solution


Look at my picture. The one on the left separates into 4 reigions, the one on the right separates into 3. Yet using the same n=2 lines. I don't understand
 

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  • #2
Your drawing is finite. The plane is not.

Extend the lines to infinity.
 
  • #3
Orodruin said:
Your drawing is finite. The plane is not.

Extend the lines to infinity.

gahh I am an idiot. thank you lol
 
  • #4
r0bHadz said:
gahh I am an idiot. thank you lol
Don’t worry. I have seen hundreds of posts like this so you are not alone. Sometimes you need to be pointed to the trees in the forest.
 
  • #5
moreover it seems that exactly 2n of the regions are unbounded, and (1/2)(n-1)(n-2) are bounded. The number of bounded regions, or "holes" in the figure, interestingly gives the genus formula for a complex projective plane curve of degree n. This is the argument by degenerating a curve of degree n into n general lines. At first I was puzzled as to why the number of regions was larger than the genus, since, knowing the genus formula, I thought I it would be also the answer to your problem.
 
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1. How do N lines separate the plane into regions?

When N lines intersect in a plane, they create N+1 regions. For example, if there are 3 lines, they will create 4 regions. This is because each line intersects with every other line, creating a new region.

2. What is the maximum number of regions that N lines can create in a plane?

The maximum number of regions that N lines can create in a plane is given by the formula N(N+1)/2 + 1. For example, if there are 5 lines, they can create a maximum of 16 regions.

3. How does the number of regions change as N lines are added to the plane?

As N lines are added to the plane, the number of regions increases by N+1. This is because each new line intersects with every other line, creating a new region.

4. Can N lines create an infinite number of regions in a plane?

No, N lines can only create a finite number of regions in a plane. This is because the lines must intersect with each other to create new regions, and there is a limit to how many intersections can occur in a finite plane.

5. How does the orientation of N lines affect the number of regions in a plane?

The orientation of N lines does not affect the number of regions in a plane. As long as the lines intersect, the number of regions will be the same regardless of their orientation.

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