Center of mass of 2D and 3D objects and the golden ratio

Abstract

Abstract This study delves into the problem of finding the centre of mass for excised polygons in two and three dimensions, revealing intriguing connections and occurrences. We begin with rectangle and rhombus with sides in the ratio of the golden ratio, showcasing certain excised polygons with their centre of mass on the edge. Furthermore, we extend this analysis to 3D shapes like cuboids, pyramids, and cylinders, discovering three ways of excision, resulting in balanced objects on the edge. Exploring physics related to the centre of mass uncovers a fascinating mathematical coincidence between the golden ratio and the Fibonacci series. Additionally, we demonstrate and verify the center of mass location on the edges through 3D-printed objects.