Find minimun number of points on the plane

yma16 · 发表于 2005-9-10 19:28:09

such that:

1 They are vertices of concave polygons.

2 All the concave polygons have more than one inner angle with degree > 180.

You can either show a picture or list the coordinates of the points.

If you are inetrested, you can change (2) to be

2' All the concave polygons have exactly two inner angles with degree > 180.

yma16 · 发表于 2005-9-11 07:04:57

Yes.

yma16 · 发表于 2005-9-11 07:19:54

I think we can find a pattern about the minimum number of the points and how to draw the points.

yma16 · 发表于 2005-9-11 23:54:01

each polygon uses all the point as its vertices. It is not the subset.

yma16 · 发表于 2005-9-11 23:56:47

(0,2), (-1, 1), (1,1), (-4,0), and (4,0). It may work.

yma16 · 发表于 2005-9-12 03:21:27

I have to try more. - eom

yma16 · 发表于 2005-9-13 06:39:38

4 vertices of a larger square. Like

. .

You can get exactly 2 angles >180. I could not get only one such angle. Although you can get more than 2 such angles. Try to reduce the number of points until we can't get 2 angles >180.

This is for (1) and (2), not (2'). FZY said (2') is impossible. He also said 6 points (vertices of 2 triangles) are enough. Please show a picture, someone.

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Find minimun number of points on the plane

Yes.

After this, please try 3, 4...

What I mean is

guys and mms, could you try

I have to try more. - eom

How about 4 vertices of a square within