In a previous post I identified the size of the pedestrian city as on the order of 50,000, let's do this a bit more systematically.
Let's illustrate with some assumptions:
- One-way Travel time budget (B) = 0.5 h
- Walking speed (S) = 5 km/h
- Walking network radius (Rn = S/B) = 2.5 km
- Network circuity (C) = 1.25
- Walking euclidean radius (Re = Rn/C) = 2 km
- Walking euclidean area (potential) (Ae=Pr*Re^2) = 12.56 km^2
- Population density (D) = 5,000 persons per km^2 [As a point of reference, the current population density of Manhattan is 27,485/km^2, which I would argue is only enabled by 19th century technologies like elevators and transit. Rome currently has a population density of 2,101/km^2]
- Population within TTB threshold (P=D*Ae) 62,800
Obviously you can construct a spreadsheet and play with population densities, which are highly disputed in ancient times. One sees claims that the City of Rome in ancient times had a population of 1 million people, but it is unclear over what area that was measured, and some estimates of those densities far exceed the densities of modern elevator cities (like Manhattan). I believe it is possible that high crowding occurred, but I think it unlikely that such crowding extended over large areas.
Also one can have a pedestrian city that exceeds the one-way walking travel time budget, but not a city one interacts with on a daily basis. This is more the equivalent of adjacent and overlapping cities, and likely have multiple cores.