My question was inspired by the Saudi plan to build a city in the desert, on a line.
I tried to calculate the implications on travel.
So I compared a grid of 5x5 homes to a line of 25 homes, and made the following observation:
The worst case distance to the centre-cell for the grid is √8 and for the line this distance is 12. There is a factor 4.242640687119285 between them, which I think is √18.
So I thought, is this a constant factor? And it turns out it is not:
For 11x11 homes in a grid, it is √200 versus 60 which has the exact same factor. √50 vs 60, which is a different factor.
How do these relate?
Let's assume that you have a square with a side of $2n+1$. Then the distance from the center to the corner is $n\sqrt 2$. You have $(2n+1)^2$ houses, so the distance along the line from one edge to the center is $$\frac{(2n+1)^2-1}2=2n^2+2n$$ So the ratio of the two does not tend to a constant, but to $n\sqrt 2$.