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

_{T}From 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.