Polygon Diagonal


A polygonal diagonal is a line segment connecting two nonadjacent polygon vertices of a polygon. The number of ways a fixed convex n-gon can be divided into triangles by nonintersecting diagonals is C_(n-2) (with C_(n-3) diagonals), where C_n is a Catalan number. This is Euler's polygon division problem. Counting the number of regions determined by drawing the diagonals of a regular n-gon is a more difficult problem, as is determining the number of n-tuples of concurrent diagonals (Kok 1972).

The number of regions which the diagonals of a convex polygon divide its center if no three are concurrent in its interior is

N=(n; 4)+(n-1; 2)

The first few values are 0, 0, 1, 4, 11, 25, 50, 91, 154, 246, ... (OEIS A006522).

See also

Catalan Number, Euler's Polygon Division Problem, Polygon, Polygon Diameter, Polyhedron Vertex, Polyhedron Diagonal, Polygon Diagonal Intersection Graph, Regular Polygon Division by Diagonals

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Kok, J. Item 2 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 3, Feb. 1972., N. J. A. Sequence A006522/M3413 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Polygon Diagonal

Cite this as:

Weisstein, Eric W. "Polygon Diagonal." From MathWorld--A Wolfram Web Resource.

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