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 Skip Navigation LinksMath Help > Geometry > Polygons and Triangles > Diagonals of an n-gon

 

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.

When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.

When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.

I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet, and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points, and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

Do more than 7 diagonals of a regular polygon ever meet in a single interior point other than the center?

 

Work some more on this web page . . . . . .  see spreadsheet / VBA program PuzzleNgonPolygonIntersections4.xls 

It would be nice to understand the Poonen-Rubinstein paper a little better to see if the conjecture, above, is corroborated by that.

 

Internet references

OEIS Sequences related to Poonen-Rubinstein paper on sequences formed by drawing all diagonals in regular polygon (B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Math., v.11 (1998), p. 135�156.)  And, in addition, these sequences that I worked on or referenced...

A000332 -- C(n,4) = number of intersection points of diagonals of convex n-gon.

A006561 -- number of intersections of diagonals in the interior of regular n-gon

A101363 -- number of 3-way intersections in the interior of a regular 2n-gon

A101364 -- number of 4-way intersections in the interior of a regular n-gon

A101365 -- number of 5-way intersections in the interior of a regular n-gon

A137938 -- number of 4-way intersections in the interior of a regular 6n-gon

A137939 -- number of 5-way intersections in the interior of a regular 6n-gon

www.math.rutgers.edu/~erowland/polygons.html

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