Fogbreaker
There is a peculiar thing about living in a gridded city. The layout is the most legible street pattern ever devised, you can navigate it on your first day, you can give directions by counting, and somehow the people who live in one tend to know a smaller share of it than people who live in a tangle.
The grid does not hide anything. That turns out to be exactly the problem.
The property nobody tells you about
Take a perfect grid. You are at one corner and you want to reach another corner five blocks east and five blocks north. Walk five east, then five north, and you have covered ten blocks. Walk five north, then five east: ten blocks. Alternate, east-north-east-north all the way: ten blocks.
Every route is ten blocks. Not approximately, exactly. As long as you never double back, every possible path between those two corners is precisely the same length, because you have to make five eastward moves and five northward moves regardless of the order you make them in.
Mathematicians call this taxicab geometry, and it has a consequence that walking apps never mention: in a grid, variety is free. Not cheap, free. The detour does not exist. In a city with organic streets, going a different way home costs you real distance and real minutes, so you have to want it. In a grid you can take a route you have never walked, at zero cost, and arrive at the same time.
How many routes is that, exactly
The count is worth seeing, because it is much larger than intuition suggests.
| Grid to cross | Distinct same-length routes |
|---|---|
| 3 by 3 blocks | 20 |
| 5 by 5 blocks | 252 |
| 10 by 10 blocks | 184,756 |
Ten blocks east and ten blocks north is a walk of maybe half an hour in Manhattan. There are 184,756 different ways to make it, all identical in distance, and most people who make that walk daily have used four or five of them in their life.
That gap between 5 and 184,756 is the whole subject of this post.
Why grid cities get explored less
If variety is free, why does nobody take it?
Because the grid removes the reason to choose. In a city that grew without a plan, streets have character: this one is the fast one, that one avoids the hill, the other one is prettier but adds four minutes. You are forced into a decision every time you leave the house, and decisions produce variety as a side effect.
A grid answers the question before you ask it. Every route being equal means no route is better, and when no route is better you take the one you took last time. The grid's great virtue, that you never have to think about the way, is precisely what stops you from ever thinking about the way.
The second reason is that grids are legible from above and boring from inside. Look at a map of a gridded city and you feel you have seen it, because the pattern is fully described by two numbers. But a pattern is not a place. The fourteen blocks you have never entered look exactly like the ones you have on the map, and are completely unknown in every way that matters.
Block size decides how much this is worth
Not all grids offer the same deal. What matters is not the grid but the size of its blocks, because block size sets how often you get to make a choice.
Portland's downtown blocks are 200 feet on a side. Salt Lake City's are 660. Walk the same mile in each and Portland hands you more than three times as many intersections, which is to say more than three times as many free decisions, and three times as many distinct routes to reach the same corner.
Manhattan sits in between and is lopsided, which most residents feel without ever articulating. The 1811 Commissioners' Plan laid out 12 avenues and 155 cross streets, so the short blocks run about 264 feet north to south, roughly twenty to a mile, while the avenue blocks run 750 to 920 feet. Walking north gives you a decision every ninety seconds. Walking east gives you one every five minutes. That asymmetry is why crosstown walks feel monotonous and uptown walks do not, and it is also why the highest-yield exploring in Manhattan is done heading east or west, where the unfamiliar stretches are.
The rule generalises: the smaller the block, the more your city rewards wandering, and the less it costs you.
Four ways to actually use it
None of these are trips. They are edits to trips you were already making.
- Alternate at every corner. The purest use of the free-zigzag property. Turn at each intersection where turning still points you at your destination. Distance identical, novelty maximised, no planning required.
- Change one leg, not the whole route. Take your usual route out and a parallel street back. On a grid the parallel street is always exactly one block away and always the same length, so this is the cheapest possible change and it doubles the ground you cover.
- Walk the long axis deliberately. In a lopsided grid like Manhattan's, the direction with long blocks is the direction you know least. Point yourself down it on purpose.
- Serpentine a single neighbourhood. If you want to actually finish an area rather than sample it, walk it like a field being mowed: up one street, along one block, down the next. It looks obsessive and it is the only method that leaves nothing out. A ten-by-ten block neighbourhood done this way is about 10 km in Portland and 30 km in Salt Lake City, which tells you something about both cities.
What a map does to this
The argument above is correct and almost nobody acts on it, because free variety with no visible reward is still a decision you have to keep making. The reason people repeat routes is not that they prefer them. It is that nothing registers when they do not.
That is the part a fog map changes. Walking covers a corridor roughly 30 metres wide, so a street you take clears a strip and the parallel street one block over stays dark. On a grid this produces a distinctive and slightly embarrassing picture: a few bright lines running the length of the city, and between them a comb of untouched blocks that were never more than sixty seconds out of your way.
Fogbreaker's Territories turns that into a percentage per neighbourhood, which is the number that makes the zigzag worth doing, because the alternating route and the repeated route now score differently. They did not before, and that is the entire reason you never took the alternating one.
The honest limits
Three things this post is not claiming.
Real grids are imperfect. Highways, rivers, rail cuts, gated blocks and parks break them, and where the grid breaks the free-zigzag property breaks with it. In practice you get the free variety within neighbourhoods and pay normal detour costs when crossing between them.
Hills do not care about taxicab geometry. San Francisco is a grid on paper and a set of very strong opinions in the legs. If your city has terrain, the equal-length routes are not equal-effort routes, and you should let the terrain choose.
And a grid is not always more interesting. Organic cities put a surprise around corners because the corner hides it. A grid shows you the next four blocks before you commit, which is safer, easier and less thrilling. The compensation is that you can cover far more of a grid in the same time, and coverage is its own kind of surprise once you are deliberately going where you have not been.
Fogbreaker is free on iOS and Android. If your interest is ticking off every street in your city by name rather than sweeping area, we would point you at the street-completion apps instead; that is a different goal and they are built for it.
