Center of Gravity of an L-Shaped Sheet and the Golden Ratio

An L-shaped sheet is constructed by cutting out a square piece from a larger square. The center of gravity (marked as the green point) of the L-shaped sheet shifts as the size of the removed square increases. The center of gravity is outside the L shape if the side of the removed square is greater than the reciprocal of the golden ratio (i.e. ) relative to the side of the larger square.

In this Demonstration, the L shape is divided into two rectangles in two different ways. The center of gravity is then at the intersection of two lines connecting the centers of each pair of rectangles. A proof is given in [3].