They showed how the matrix chain multiplication problem can be transformed (or reduced) into the problem of triangulation of a regular polygon. The polygon is oriented such that there is a horizontal bottom side, called the base, which represents the final result. The other "n" sides of the polygon, in the clockwise direction, represent the matrices. The vertices on each end of a side are the dimensions of the matrix represented by that side. With "n" matrices in the multiplication chain there are "n"−1 binary operations and "C""n"−1 ways of placing parentheses, where "C""n"−1 is the ("n"−1)-th Catalan number. The algorithm exploits that there are also "C""n"−1 possible triangulations of a polygon with "n"+1 sides.