We designate as an empire a state that stands out by area and population, as compared to most neighbors. The same population–technology interaction that enables world population to grow also enables states and then empires to form and expand, mostly by brutal force. Empires form where people are. The joint population share of the top five empires reached 50% of the world population by the year +1, and it did so on just 10% of the dry land area. The areas of top empires expanded in three phases, dependent on message speeds but also skills in delegation of power: Runner, Rider and Engineer Empires. The Rider age produced a new type of nomadic “area empires,” with low population density, in contrast with standard high-density “people empires.” The areas and populations of people empires tend to follow a square root relationship: Their share of world population is the square root of their share of dry land area.

The size of states, including empires, is measured by two crude measures: their area and their population. Areas graphed over time represent the growth–decline curves of empires. These are the basic portraits of empires, showing at a glance how rapidly they formed, how large they became, and how long they lasted. Measuring areas in historical atlases supplies most data. Rome, Ottoman, and Manchu Qing offer examples of completed life cycles. Russia and the USA show curves still incomplete. The beginnings of these curves often approximate the “simple logistic” pattern typical of growth of bacteria colonies when they have plenty of food but space is limited. A table lists the following characteristic of 98 major empires during the last 5000 years: maximum size, duration at least at one-half of maximum size, and rise time from 20−80% of maximum size. Of these, 20 are Runner Empires (3000 to 600 BCE), 66 are Rider Empires (600 BCE to 1800 CE), and 12 are Engineer Empires (from 1800 on).

A graph superimposes the growth–decline curves of major Runner Empires, from 3000 to 600 BCE. The Egyptian Old, Middle,and New Kingdoms dominate this period. Egypt’s sheltered location ensured stability, then suffocation. Mesopotamian states remained small, forming a mutual fighting community, until the brief rise of Akkad and Assyria. The Xia–Shang–Zhou empire took off only after 1500 BCE, and in a surprisingly northern location. It’s as if state formation in the west had begun on the Danube instead of Nile. Egypt came closest to the oikos model (state as the ruler’s household), with 90% of produce flown in and out of state storage. It might have been 83% in Sumerian city-states. This aspect of human self-domestication may have begun to retreat with the Hittite empire, and even Assyria. While the Hyksos surprised Egypt with chariots as early as around 1650 BCE, the Assyrians complemented chariots with cavalry only by 665 BCE. The era of Rider Empires was approaching.