Originally Posted by
hrugner
It's much simpler to look at the amount of land required to support one person in an inhospitable environment and look at the speed of communication required to govern over long distances and come up with a decent idea of the level of tech required to create an empire in places that don't accommodate people. Pox Americana has a good description of how the introduction of the horse to America changed the reach of several Native cultures changing their hunting and raiding practices into something more unsustainably imperialistic. The book is looking at the spread of disease by horse, but it touches on the spread of other terrors by horse well enough to get the meaning across. You can't govern without force and communication, and the amount of force and speed of communication need to increase with the size of your holdings.
So, with P being population, A being area needed to sustain an individual and D being distance of communication and use of force, you need
PA/πD ≤ D
I think, math isn't my strong suit.