N lines separate the plane into...

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Homework Help Overview

The problem involves determining how many regions n lines can separate a plane into, given that no two lines are parallel and no three lines intersect at a single point.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • One participant attempts to visualize the problem with drawings but expresses confusion regarding the separation of regions. Another participant suggests extending the lines to infinity to better understand the problem. There is also a mention of bounded and unbounded regions in relation to the number of lines.

Discussion Status

The discussion includes attempts to clarify the problem through visual representation and the implications of extending lines. Some participants share their experiences with similar confusion, indicating a supportive environment. There is exploration of the relationship between the number of regions and concepts from topology, but no consensus has been reached.

Contextual Notes

Participants are discussing the implications of finite versus infinite representations of lines and the resulting regions, as well as the relationship between the problem and concepts from complex projective geometry.

r0bHadz
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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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Your drawing is finite. The plane is not.

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

Extend the lines to infinity.

gahh I am an idiot. thank you lol
 
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.
 
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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