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Not if your neighbourhood is a grid.
A grid is two-dimensional, not one-dimensional. The number of walkable destinations still increases with the second power of distance, not with the first power.
(The abstraction could break down if the city's blocks are gigantic—but Wikipedia suggests that a typical block size is 1/8 mile × 1/16 mile, which is much smaller than the distance of 1/2 mile that ASTM uses as the limit of walkability, so I think the abstraction remains valid.)
The number of walkable destinations does not increase with the square of the distance. It increases linearly until you're halfway to the maximum distance and then it decreases linearly.
Imagine a grid of grocery stores each half a mile apart. For each grocery store, there is a half mile by half mile square area that is closer to it than to any other grocery store. Each corner of this area is a quarter mile in each direction from the grocery store.
Now we can divide this area into four squares and use the average distance from the grocery store to a point within one of the squares as the average distance from the grocery store to a point within the larger area, because it's symmetrical.
Now, split the square into two triangles, with one triangle containing the grocery store and the other containing the point farthest from the grocery store. The square is symmetrical with respect to the diagonal line dividing the two triangles, so we know that the distance from the grocery store to any point on that diagonal line is equal to the average distance from the grocery store to any point within the square.
Since you can only travel in a direction that is parallel or perpendicular to the lines connecting the points of the grid, the distance to any point on this line is a quarter mile. So the average distance from a grocery store to any point closer to that grocery store than any other is a quarter mile.
This rough sketch appears to vindicate me.
You're assuming the grid of grocery stores is rotated 45 degrees relative to the grid of city blocks, when it would make more sense to for them to be aligned.
I am assuming, not a grid of grocery stores, but a uniform distribution of grocery stores (or, alternatively, of all the myriad establishments that count toward the house's ASTM E2843 walkability score). The average distance (of approximately 72 % of 1/2 mile) is measured to all points (on the road network) that are within 1/2 mile of the house (as measured on the road network), not just the ones that I explicitly marked with big black dots on the edge of the 1/2-mile catchment area.
If you assume the grocery on your map is at (0,0), then your diamond shape only makes sense if there are grocery stores at (1/2, 1/2, (1/2, -1/2), (-1/2, 1/2), and (-1/2, 1/2) with the unit distance being a mile. I take back what I said about this not making sense. It is the distribution of grocery stores with the lowest density of grocery stores that are still reachable from anywhere without travelling more than 1/2 mile.
But I was assuming that the grocery stores would be at (1/2, 0), (1/2, 1/2), (0, 1/2), etc. So, instead of taking the average distance to the points within your diamond shape, you would take the average distance to the points within a 1/2 mile x 1/2 mile square. The average distance to a grocery store is less because this gives you 4 grocery stores per square mile, whereas your arrangement gives you 2 grocery stores per square mile.
I will explicitly abandon the focus on grocery stores. Rather, I will state that I am thinking of all 48 of the different establishments of which ASTM E2843's walkability score requires four to six examples within the 1/2-mile catchment area. Those 48 establishments need not be arranged in evenly spaced grids. Rather, even if they are arranged haphazardly, in total they still can result in the entire city's being considered walkable.
One house may be within range of a supermarket, a movie theater, a public park, and a bank.
A different house may be within range of a post office, a library, a hardware store, a liquor store, a farmers' market, and a zoo.
A third house may be within range of a dentist's office, a day care, a pharmacy, a book store, a bar, and a church.
All three of those houses count as being "in walkable proximity to neighborhood assets" under the requirements of ASTM E2843. Maybe you disagree, and think that: (1) only grocery stores (or perhaps even only supermarkets, not just smaller grocery stores that some would consider convenience stores) should count toward walkability; and therefore (2) a city interested in being walkable (a) would subsidize at least one grocery store within a 1/2-mile distance of every point in the city (if the interest is felt politically by the government), or (b) would result in the free market's doing the same thing (if the interest is felt culturally by the people); (3) causing grocery stores to be present in a grid pattern. The activists who decry "food deserts" (1 2) might agree with you. But ASTM disagrees with you. And I think that, as long as all 48 establishments taken together are uniformly distributed across the network of roads (or walking paths), then my rough sketch is an accurate representation of the situation.
On what point are you saing that ASTM disagrees with me?
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