When some states are huge and some others tiny, an “effective number” widely used for parties in political science can be applied both to state areas and populations. These numbers decrease exponentially over 5000 years, pointing to a single world state around 3700 by area and 5000 by population. Combination with estimates from top state area and population points toward a single world state by 4600, if the 5000-year trends continue. But projections are not predictions!

A square root law of effective numbers of states applies: The population-based effective number of states and tribes tends to be the square root of the area-based number. The zone of variation around the average trends is wide. Thus, even when the average millennial trends keep holding, the immediate future is wide open. The zigzags in the past curves remind us of how fleeting human history has been. Often a mere hundred years has thoroughly altered the number and size of states. We can expect similar reversals in future.

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.

Cities seem to form when state population surpasses 10,000. Over 5000 years, top city populations have remained 0.1−0.5% of world population. The square root law of city population: The largest city population tends to be hundred times the square root of state population. This law held from 3000 BCE to 1800 CE. Constantinople exemplifies it from 400 to 1800. It no longer holds, as the economic reach of major cities surpasses international borders.

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.