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 world’s top empire area has grown in three distinct phases, reflecting shifts in message speeds. But the millennial trend has been exponential growth. If this 5000-year trend continued, a single world state would form around 4400. The most populous state’s share of world population also has increased exponentially, pointing to a single world state around 5300. The combined date is around 5000. Projections are not predictions, but still, if some people worry that the United Nations is trying to become a world government, while some others hope for it, they need not hold their breath. If the past offers any guidance whatsoever toward the future, a single world state is highly unlikely much ahead of 3000. Empires form where people are. Hence, the top shares of world population exceed the top shares of world dry land area. They do so in a logically predictable way: the square root law of people empires. The most populous state’s share of the world population tends to be the square root of its share of world dry land area. The law does not apply to “area empires” – those that are the largest but not the most populous.

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).

While worldwide population density increased, that of the of most populous states stubbornly remained between 5 and 10 persons per square kilometer, from 3000 BCE to 500 CE. Expansion into less densely inhabited regions may not have paid off. The range has widened since. In hemmed-in India, density approaches 500 per square kilometer, while the largest empire, Russia, remains at 9, far below the world average of 60. It is not certain that imperial peace outweighs imperial taxes so as to boost population and hence, presumably, wellbeing. The evidence is mixed. Inexplicably, the population share of the most populous empire kept increasing proportional to the square root of world population from 3000 BCE to 1800 CE, but since then it has been decreasing. At the same time, the area share of the most populous empire kept increasing proportional to world population, and it also has been decreasing since 1800. In a crowded world, some five-millennia regularities are breaking down. We live in interesting times.