So in a previous post I had come up with the labels of strong walkable and strong bikeable, the idea that "walkable" means one should be able to walk across a whole city in reasonable time. It's an unreasonable ideal now, but still fun to think about. And the same numbers can apply somewhat to a neighborhood or a 'walkshed' within a large city. Or to a still-utopian idea a la Garden Cities, of urban pods surrounded by greenspace.
Assumptions:
A fine grained grid, at least for active transport (walk and bike etc.)
A diamond shape, with 'radius' s from the center to a corner, and thus area of 2*s^2.
Walking speed of 5 km/hour (3 mph), bike speed of 15 km/hour, or sometimes 20.
'Bike' includes various alternatives to conventional bikes, including handcycles or tricycles for various forms of disability, or higher-speed electric mobility scooters (which on a quick check, can range from 4-15 MPH)
Average population density of 20,000 people/km2. This is a big variable and requires some discussion. I want to envision 'complete' cities, so can't just extrapolate from high density residential neighborhoods. OTOH a strong walkable city can exile any cars to garages at the corners or edge, thereby saving a lot of space internally. For real cities, Osaka is 12,000/km2, Tokyo's 23 Wards 15,000, Brooklyn 14,500, Paris 21,000. My own models suggest easily being able to hit 20,000 or more, but I'm inclined to trust reality over models; OTOH again, cars really do eat up space. But hey, Paris hits 21,000 while still having lots of cars, it just also has lots of 6-8 story buildings, rather than houses.
So, some cities:
very strong walkable. s = 1 km. 12 minute walk to the center, 24 minutes across. Area of 2 km2, population 40,000. In medieval times, pretty big. Now, not so much. Around 2000 people of high school age, likewise of 4-year college age (but not everyone goes to college). Pretty good as an area around a metro stop.
strong walkable. s = 1.5 km. 18 minutes to center, 36 across. I feel this is the outer edge of non-painful walkability. Area 4.5 km2, pop 90,000.
Center walkable. s = 2 km. 24 minutes to center, 48 across. Area 8 km2, pop 160,000. Decent for a small modern city. 8000 people of college age, so as a self-contained area it still only supports a small college. Walking across the whole thing would be painful, but you could walk to meet friends downtown, and many trips would be shorter, especially if density were non-uniform and clustered toward the center.
Now, how about biking?
very strong bikeable. s = 3 km. 12 minute ride to center, 24 across. Area 18 km2, pop 360,000.
strong bikeable. s = 4.5 km. 18 minute ride to center. Area 40.5 km2, pop 810,000. 40,500 people of college age. This is as many people as San Francisco proper, and more than Seattle, Vancouver or Boston, though those all have suburbs and larger metro areas which I'm assuming away. Pretty good! Though with a 54 minute walk from corner to center, you need some public transit for people who can't bike for some reason. Car owners would probably grumble about their cars being stored on the edge of town, but tough.
center bikeable. s = 6 km, 24 minute to center. Area 72 km2, pop 1.44 million. That's definitely a City. About the population of Manhattan (not counting weekday visitors), though much more compact.
center bikeable, bike 20 kph. s = 8 km, assuming everyone exerts themselves just a bit more. Still 24 minutes to center, 48 across. Area 128 km2, pop 2.56 million. That's about as many people as many actual metro areas, such as Denver or Vancouver. More than Paris proper.
ditto, just a bit larger. s = 10 km. 30 minutes to center, 60 across. Area 200 km2, 4 million. That's on the order of the Seattle, Montreal, or SF metro areas. So it is at least possible to have a 4 million person city where the average person, at what it is still fairly low biking speed, can go anywhere in an hour or less, on a bike that costs a few hundred dollars. (Also where such a person can go from anywhere to an edge of town, and whatever nature or parks are there, in under 30 minutes.) Seems silly to have car cities, where you spend 10-20x as much private money and even more public money, just to accomplish the same thing...
One caveat would be whether at these sizes and density, bikes start getting in each other's way, the way cars do at much lower size and density. I don't know. Paris exists, at this density and with cars, but also with a major metro system to get people off of any vehicle.