3 ms·

Yeah, let me try to be more precise, though maybe it's worth writing down in a more formal setting because comments are awkward. Say we're looking at the Hilber

Yeah, let me try to be more precise, though maybe it's worth writing down in a more formal setting because comments are awkward. Say we're looking at the Hilbert curve. I claim that for almost all physical positions, a sufficiently small change in position yields a correspondingly small change in coordinate. In the purely mathematical sense, this means something like "continuous except at points with at least one rational coordinate". In the programming sense, where everything must be approximated and you're mapping a 2-d lattice to a 1-d lattice, then it's a bit tricky to formulate, but something like: if you take the maximum distance in 1-d space between a 2-d point and its eight "neighbors", then the probability of this distance being > N should be O(1/sqrt(N)). Concretely, P(point has an adjacent point whose Hilbert coordinate disagrees past the highest n bits) == P(d > 4^n) < 4*(1/2)^n

ltbarcly3 8y ago
I took a 256 square hilbert curve and took random points in it (discarding points where the euclidian distance was > 8). Here is the plot: https://gist.github.com/justinvanwinkle/22a15bb33536095b05dc9f68a87e2754 https://gist.github.com/justinvanwinkle/22a15bb33536095b05dc... Notice the Log y axis. I don't you think you could consider this even roughly continuous. This is only even bounded because I only took points within the 256x256 grid, if I let it run over say a 1000x1000 grid the hilbert distances wouldn't plot well even on a log plot. The second plot I took any random point within euclidian distance 8 from a random first point, and didn't restrict it to stay in the same 256x256 grid. 10k points total. Even at distance 1, you get hilbert distances like: 4469269309980865699858008332735282042493121940197464459241187147907254676691182325600624766055634475896677286062162016951251294270648856648811216335320387
- kevinventullo 8y ago
  I'm not sure why that first plot wouldn't be considered continuous. There's a pretty clear positive correlation, which should be more pronounced with a non-log scale y-axis, and more pronounced as you increase the order of the curve. I'm confused about the second plot; what order is the curve? It seems weird that there are no points between 10^12 and 10^156.
  - ltbarcly3 8y ago
    The second graph has a huge jump because the left side of the area I'm taking points from is a boundary between two very distant parts of the curve. In any space filling curve you will be able to find arbitrarily large jumps like this. The difference would be more pronounced on a non-log scale, except the huge numbers at the top of the range would squish the graph down so you couldn't see what was going on. The first graph isn't just not continuous, it's not even 1-1. (I'm ignoring the fact that is also discrete so that any formal sense of 'continuity' can't exist. Even if you tried to make it continuous by filling in the gaps it would be impossible.)
    - kevinventullo 8y ago
      Unfortunately I think we're talking past each other and this is not the best format for technical discussion.

ltbarcly3 8y ago

I took a 256 square hilbert curve and took random points in it (discarding points where the euclidian distance was > 8). Here is the plot: https://gist.github.com/justinvanwinkle/22a15bb33536095b05dc9f68a87e2754 https://gist.github.com/justinvanwinkle/22a15bb33536095b05dc... Notice the Log y axis. I don't you think you could consider this even roughly continuous. This is only even bounded because I only took points within the 256x256 grid, if I let it run over say a 1000x1000 grid the hilbert distances wouldn't plot well even on a log plot. The second plot I took any random point within euclidian distance 8 from a random first point, and didn't restrict it to stay in the same 256x256 grid. 10k points total. Even at distance 1, you get hilbert distances like: 4469269309980865699858008332735282042493121940197464459241187147907254676691182325600624766055634475896677286062162016951251294270648856648811216335320387

kevinventullo 8y ago
I'm not sure why that first plot wouldn't be considered continuous. There's a pretty clear positive correlation, which should be more pronounced with a non-log scale y-axis, and more pronounced as you increase the order of the curve. I'm confused about the second plot; what order is the curve? It seems weird that there are no points between 10^12 and 10^156.
- ltbarcly3 8y ago
  The second graph has a huge jump because the left side of the area I'm taking points from is a boundary between two very distant parts of the curve. In any space filling curve you will be able to find arbitrarily large jumps like this. The difference would be more pronounced on a non-log scale, except the huge numbers at the top of the range would squish the graph down so you couldn't see what was going on. The first graph isn't just not continuous, it's not even 1-1. (I'm ignoring the fact that is also discrete so that any formal sense of 'continuity' can't exist. Even if you tried to make it continuous by filling in the gaps it would be impossible.)
  - kevinventullo 8y ago
    Unfortunately I think we're talking past each other and this is not the best format for technical discussion.