Neat post.

Think of it another way. If you consider the Earth and the Moon as a system, orbiting each other, you will realize that there is a point where both objects rotate about each other. If the two masses are similar in magnitude that point where they revolve about lies outside the physical boundaries of the two objects.

A more earthly example might be a boomerang. When it is thrown it rotates about a point that lies outside the boundaries of its physical shape. This is a little easier to visualize because you can see it when you throw the object. Looking at an L shaped virtual object on a sheet of paper that is represented mathematically is much harder for the imagination to grasp. This is a limitation of the human brain and why we "model" objects and concepts in ways that we can visualize.

I find modeling things a helpful tool for me to grasp the larger picture. I back up my model with mathematic constructs so that I can satisfy my visualizations and thought experiments with a proof.

Think about some other objects in daily life. How about a basket ball? Here is a hollow sphere where the center of gravity obviously lies outside the physical walls of the ball. In this case it is the epicenter of the ball, but the center of the ball has no physical matter because it is hollow. However, when the ball is tossed spinning into the air it never spins about a point that lies along its physical surface (outside perimeter of the sphere). So this is an example where the center of gravity is logically outside the physical boundary of the object (its shell).

This is not truly a center of gravity but the point where the total mass is equal in opposing directions. If I draw a line through the "center of gravity" in any direction, I will have the same amount of mass on both sides of the line. There is no Gravity force pulling towards that point. It should more accurately be called the 3D balance point. It is called the center of gravity only because; if the object is spun, it will spin around this balance point.

True gravity on a large object such as a planet does not have a single point that is considered to be the center of gravity. Gravity will be stronger towards more massive objects, thus gravity forces will be in different directions around the planet. However it will have a balance point or center of spin.