Center of Mass, Center of Geometry

Center of Geometry

Place one square on top of a row of two squares, on top of a row of three squares. These rows are left justified, so that the 6 squares look like steps going up and to the left. Let's find the center of geometry.

If it exists, the center of geometry is a point such that any line drawn through the point cuts the shape in half. The horizontal line that bisects the shape is at y = 1, between the first and second rows of squares. There are 3 squares below and 3 above. Similarly, the vertical line that bisects the shape is at x = 1. Therefore the center of geometry must be at 1,1. Draw a line through this point with slope -1, down and to the right. The line cuts the shape into pieces of areas 2 and 4, hence there is no center of geometry. However, there is always a center of mass, as we shall see.