You are the only commenter quoting the horizontal development. So maybe my idea relates to you.
Are you familar with the concept of the Big O notation [1], which computer scientists use to (more or less) describe the "growth" of their algorithms? E.g. when you compare something that growth quadratically O(n²) to something that only growth linear O(n) you will see, that at some point the quadratically growth will grow faster as the linear growth. Forever! The linear growth will never win.
Comparing this to the horizontal sprawl of a city is easy. The area of a city is planar which is mathematically described as quadratic O(n²) but the average growth of the domestic product is just linear O(n). Assuming the maintainance cost of a planar sprawling city growth quadratic while the domestic product only growth linear then the costs will win over the income. Forever!
The conclusion is to limit the horizontal sprawl of a city to linear growth. This means every new house, street, rail road, pipe system, power line and so on must be build inside a diameter to the city's center with a distance which is logaritmic O(lg(n)) on average. The rest must growth vertically.
Are you familar with the concept of the Big O notation [1], which computer scientists use to (more or less) describe the "growth" of their algorithms? E.g. when you compare something that growth quadratically O(n²) to something that only growth linear O(n) you will see, that at some point the quadratically growth will grow faster as the linear growth. Forever! The linear growth will never win.
Comparing this to the horizontal sprawl of a city is easy. The area of a city is planar which is mathematically described as quadratic O(n²) but the average growth of the domestic product is just linear O(n). Assuming the maintainance cost of a planar sprawling city growth quadratic while the domestic product only growth linear then the costs will win over the income. Forever!
The conclusion is to limit the horizontal sprawl of a city to linear growth. This means every new house, street, rail road, pipe system, power line and so on must be build inside a diameter to the city's center with a distance which is logaritmic O(lg(n)) on average. The rest must growth vertically.
[1] https://en.wikipedia.org/wiki/Big_O_notation [2] https://www.statista.com/graphic/1/268750/global-gross-domes...