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A262244
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Number of concave equilateral n-gons with corner angles of m*Pi/n (0 < m < 2n), where m and n are integers.
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3
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OFFSET
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3,4
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COMMENTS
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An n-gon is a polygon with n corners and n sides, each of which is a straight line segment joining two corners. A polygon P is said to be a simple polygon (or a Jordan polygon) if the only points of the plane belonging to multiple edges of P are the corners of P. Such a polygon has a well-defined interior and exterior. Simple polygons are topologically equivalent to a disk, hence zero angles are not allowed; allowable angles are m*Pi/n (where m and n are integers and 0 < m < 2n). A simple n-gon is concave iff at least one of its internal angles is greater than Pi, or equivalently m > n for at least one of the corners. The sum of the m-numbers (called angle factors) for the n-gon has to be n*(n-2). They are partitions of n*(n-2) into n parts with largest part n < k < 2n, and as the edges of a polygon form a closed path, the sum of unit vectors defined by the angle coordinates m/Pi is zero. The reason the m-numbers sum to n*(n-2) is that the sum of the interior angles of any n-gon is Pi*(n-2), and as angles are m*Pi/n, n = Pi.
Observation: when n is prime, m is odd and m != n.
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LINKS
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Stuart E Anderson, n-gon, produces postscript images of convex, concave and intersecting polygons for n, along with an unsorted list of interior angle multiples m for each polygon. One rotationally invariant representative polygon is produced in postscript.
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EXAMPLE
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For n = 5, the a(5) = 1 solution is (1 3 3 1 7) in m angle factors.
For n = 7, the a(7) = 11 solutions in m angle factors are as follows: (1 11 5 3 5 5 5), (1 5 3 9 1 5 11), (1 5 5 1 11 1 11), (1 5 5 5 1 9 9), (1 5 5 5 3 5 11), (1 9 1 9 3 3 9), (1 9 3 5 1 11 5), (1 9 3 5 5 3 9), (3 3 5 5 3 3 13), (3 3 9 3 5 3 9), (3 5 5 5 5 3 9).
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CROSSREFS
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Cf. A262181 (convex equilateral polygons).
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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