Happy Ending Problem - Numberphile
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If you have n points and they set up so that all the points are being used up in an n-gon and you move one of the points away from the center of the 'gon then it shouldn't effect the concavity of the 'gon uless it crosses a line formed by two other points in the 'gon.
The reason would be this, if you take any four points in your 'gon, they form at least two triangles. If three points are in a line--this could happen if you move one of the two points shared by both triangles towards the other--then there are two right triangles. If you continue to bring those two triangles together, they both become obtuse and then you have to take that point out of your 'gon to maintain concavity and the 'gon becomes an n-1-gon.
You can keep doing this and at some point, doing this brings enough points into the inside of the 'gon then you have the situation where you can take one point on the outside of the 'gon out of the connection and instead connect two of the points in the interior to the convex 'gon. because you have a potential acute triangle that does not cut though a potential line in the 'gon.
But where does the 2-(n-2) come from? I don't know. If you start with an 8-gon and move one point into the middle, you have a 7-gon and an extra point. Say point one (they're numbered sequentially around the rim. if you move #3 into the middle, you have an accute triangle with the to moved points and #2, so you can take #2 out of your 'gon and restore #1 and #3 to your 'gon. But only if they have not crossed another potential line between two points still in 'gon.
But this is intersting. If you move #1 in and it crosses the 8-2 line, then we go from 8 to 7 sides. if we bring #2 in it may become a 6-gon, but only if we also moved #1 past the 8-2 line. If we move #3 in past the 8-4 line, it becomes a 5-gon but only if both #1 and #2 have also been moved past that line as well. If we move the #'s 1-4 past the 8-5, then the the formula suggests that there must now be two points in the middle of the 4-gon that form an accute triangle with #5, #6 or #7 and does not cut through the 8-6 or the 7-5 potential lines. It has to be acute be cause an obtuse potential line in one of these potential triangles would violate concavity.
And I'm liable either wrong or just figured out something already know. I bet wrong.
Date: 2022-04-08
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Comments and reviews: 9
Timothy
I'm having a go at trying to solve the problem.
I have tried my best to try and focus the problem down by working with how the various positions of the points on the plane, which I worked on by scribbling out the equations of lines, and the two coordinates the points would take up, and the ordering of the end points of which the next line would be drawn.
I'm going to see if this line of thinking is going to work. maybe
I am trying to see where I can go with looking at the various possible combinations that could exist between the points.
reply
I'm having a go at trying to solve the problem.
I have tried my best to try and focus the problem down by working with how the various positions of the points on the plane, which I worked on by scribbling out the equations of lines, and the two coordinates the points would take up, and the ordering of the end points of which the next line would be drawn.
I'm going to see if this line of thinking is going to work. maybe
I am trying to see where I can go with looking at the various possible combinations that could exist between the points.
reply
LambOfLucifer
Draw a huge grid, plot out 33 points, check each point and make sure it does not line up with any of the other points, start with one point, place a red marker or button on top of it, and using pencil, start making polygons with the other dots one at a time, if none can be made, rule that first dot out. 32 to go! NOTE: make sure your either a lucky bastard or your immortal cuz its gonna be lonnnnnnnnnnnnnnnng!
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Draw a huge grid, plot out 33 points, check each point and make sure it does not line up with any of the other points, start with one point, place a red marker or button on top of it, and using pencil, start making polygons with the other dots one at a time, if none can be made, rule that first dot out. 32 to go! NOTE: make sure your either a lucky bastard or your immortal cuz its gonna be lonnnnnnnnnnnnnnnng!
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Timothy
I think I may have managed to prove the conjecture, took some days of work, and 11 full A4 pages of endless maths, but I think I've got the proof: )
I'll need to get someone who knows mathematics deeper than me to check it and make sure it is solid before I hand it out or message anyone about it though.
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I think I may have managed to prove the conjecture, took some days of work, and 11 full A4 pages of endless maths, but I think I've got the proof: )
I'll need to get someone who knows mathematics deeper than me to check it and make sure it is solid before I hand it out or message anyone about it though.
reply
James
I'm pretty sure the solution to this is to determine all of the different ways that you can form a concave n-gon from convex (n - k)-gons, where n - k > 2 through indenting vertices. And by pretty sure, I mean I'll have to play around a bit more.
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I'm pretty sure the solution to this is to determine all of the different ways that you can form a concave n-gon from convex (n - k)-gons, where n - k > 2 through indenting vertices. And by pretty sure, I mean I'll have to play around a bit more.
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education
Given the backstory with the two mathematicians suddenly getting married and sticking together for 90 years, plus the name of the problem, I was half-expecting this to be some kind of secret mathematical proof about happiness: P
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Given the backstory with the two mathematicians suddenly getting married and sticking together for 90 years, plus the name of the problem, I was half-expecting this to be some kind of secret mathematical proof about happiness: P
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Lucas
is there some part of the conjecture that states that three points or more points in plane can't form a line because otherwise you could just have a line with 33 points and it would be impossible to create a 7-gone with that?
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is there some part of the conjecture that states that three points or more points in plane can't form a line because otherwise you could just have a line with 33 points and it would be impossible to create a 7-gone with that?
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Venkatesh
If you take any three points you can have a wave structure. Two for half wave. A straight line not touching more than two is a collapsed half wave. Seven sides have 14 collapses. So 28 plus 5 for structure.
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If you take any three points you can have a wave structure. Two for half wave. A straight line not touching more than two is a collapsed half wave. Seven sides have 14 collapses. So 28 plus 5 for structure.
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Frank
O. K, if you have a triangle with two points inside of it, you can make one convex 4-gon.
but if you make a convex 4-gon with two points in it, you don't have to make a convex 5-gon.
This is hard.
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O. K, if you have a triangle with two points inside of it, you can make one convex 4-gon.
but if you make a convex 4-gon with two points in it, you don't have to make a convex 5-gon.
This is hard.
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Grayson
I'm pretty sure the definition of convex and concave polygons is convex: all exterior angles of a polygon are greater than 180. and concave: one or more exterior angles are less than 180
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I'm pretty sure the definition of convex and concave polygons is convex: all exterior angles of a polygon are greater than 180. and concave: one or more exterior angles are less than 180
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