Pretty Stochastic Patterns
I got the bus home from work today, which gave me lots of time for musing (while listening to good music, rar!).
Bus stops are among my least favourite places. I always get cold. There's no other situation I can think of where I stand around outside doing essentially nothing for a prolonged period apart from stargazing. (That's colder - it's at night as well - but more bearable because I know I can get warm whenever I've had enough). And they smell of stale and often fresh cigarettes, which I don't like. London bus stops are Ok - you're never there long - but ones closer to home are awful.
While I was waiting for it and having my standard "Well is it coming or have they cancelled it?" fret, I got to thinking that bus services have a critical frequency which transforms them from crap choice to reasonable travel option. The same's true of trains - I was stunned to roll up at MK station the other day and have to wait nearly 30 minutes for the next train to London. An hour is too long; ten minutes is Ok. I suspect the right time is around fifteen to twenty minutes.
That got me in to how long you can expect to wait for a bus if you just roll up, and the counterintuitive result that it's the mean interval between buses, not half of it, and the assumptions you have to make to get that result. Which then got me thinking about pseudo-random events and Poisson distributions and constrained random walks.
The bus did finally arrive, and we drove past two fields of sheep. In one, the sheep looked like they'd been fairly randomly spread across the field, but with a very slight preference for being near each other. In the other, they were all up one corner. That got me thinking about how sheep arrange themselves and stochastic processes. The ones up one corner really looked like they'd just been wandering randomly but had hit up against the fence. If I was making Sim Sheep, my first guess at an algorithm for moving them around would be something like "wander pseudo-randomly, with a bias to move in the same direction and speed as any other sheep you can see moving". With luck that might capture the distribution and that phenomenon where a whole flock runs away from a clump of grass. I wanted to make a field infinite like you can do on a computer game, so if you wander off the left hand edge you come in again on the right hand one, and see how sheep would arrange themselves then.
I got off the bus, and walked along a path through some grass under trees. There were lots of fresh dry leaves around. The concrete path was mostly clear of leaves, but the grass either side was almost completely covered with leaves. Another stochastic process! The leaves clearly drifted about in light breezes, but when they landed on the grass it was harder for a slight gust to blow them off than when they landed on the smooth path. So over time the leaves gathered on the grass without anyone doing any work.
The end point of this semi-pointless meandering is that the combination of the beautiful autumnal shades of the leaves and the beautiful mechanism by which they were distributed was truly lovely. The world is a fascinating, wonderful place sometimes.
Bus stops are among my least favourite places. I always get cold. There's no other situation I can think of where I stand around outside doing essentially nothing for a prolonged period apart from stargazing. (That's colder - it's at night as well - but more bearable because I know I can get warm whenever I've had enough). And they smell of stale and often fresh cigarettes, which I don't like. London bus stops are Ok - you're never there long - but ones closer to home are awful.
While I was waiting for it and having my standard "Well is it coming or have they cancelled it?" fret, I got to thinking that bus services have a critical frequency which transforms them from crap choice to reasonable travel option. The same's true of trains - I was stunned to roll up at MK station the other day and have to wait nearly 30 minutes for the next train to London. An hour is too long; ten minutes is Ok. I suspect the right time is around fifteen to twenty minutes.
That got me in to how long you can expect to wait for a bus if you just roll up, and the counterintuitive result that it's the mean interval between buses, not half of it, and the assumptions you have to make to get that result. Which then got me thinking about pseudo-random events and Poisson distributions and constrained random walks.
The bus did finally arrive, and we drove past two fields of sheep. In one, the sheep looked like they'd been fairly randomly spread across the field, but with a very slight preference for being near each other. In the other, they were all up one corner. That got me thinking about how sheep arrange themselves and stochastic processes. The ones up one corner really looked like they'd just been wandering randomly but had hit up against the fence. If I was making Sim Sheep, my first guess at an algorithm for moving them around would be something like "wander pseudo-randomly, with a bias to move in the same direction and speed as any other sheep you can see moving". With luck that might capture the distribution and that phenomenon where a whole flock runs away from a clump of grass. I wanted to make a field infinite like you can do on a computer game, so if you wander off the left hand edge you come in again on the right hand one, and see how sheep would arrange themselves then.
I got off the bus, and walked along a path through some grass under trees. There were lots of fresh dry leaves around. The concrete path was mostly clear of leaves, but the grass either side was almost completely covered with leaves. Another stochastic process! The leaves clearly drifted about in light breezes, but when they landed on the grass it was harder for a slight gust to blow them off than when they landed on the smooth path. So over time the leaves gathered on the grass without anyone doing any work.
The end point of this semi-pointless meandering is that the combination of the beautiful autumnal shades of the leaves and the beautiful mechanism by which they were distributed was truly lovely. The world is a fascinating, wonderful place sometimes.