On this diagram the original figure and the hole have the same shape and orientation and are in the Golden Ratio proportion (φ):
The hole also touches the edge of the original figure, in the direction the figure was displaced, but is otherwise completely enclosed.
C
H = centroid of hole
C
T = centroid of total figure
C
C = centroid of carved figure
a = distance between C
C and C
TFrom the Golden Hole rule (reply 132) we know that the distance between C
T and C
H is: aφ
b = distance between C
H and outside edge (along line through centroids)
therefore, the distance between C
T and the same edge is: bφ
x = distance from C
C to the edge
We have:
aφ + b = bφ => a = b (φ-1)/φ = b (1-1/φ) = b (1 - (φ -1)) => a = b (2 - φ)
Also:
x = a + bφ
Combining them:
x = b (2 - φ) + b φ = 2b
Therefore the new center of mass, C
c is at the same distance as the edge from C
H, but on the opposite side.
Conclusion: if the centroid of the figure is halfway from the edges, along the line the figure was displaced, the new center of mass is at the edge of the carved figure.