How unexpected difficulties in a practical measurement problem challenged fundamental conceptions of length and led to a new theory of dimension, while revealing monsters lurking at small scales.
If we create a computer program to iterate and output successive Hilbert curves, it will never output the final Hilbert curve since there is no last iteration. Does it exist? Is it fair to say that there is a final curve but not a final iteration? I think It might be reasonable to say that the final Hilbert curve is a 'mirage' (as opposed to an output) of the program. This may be a subtle distinction, but I think it's important because it puts the final Hilbert curve in a different class of existence than all the other Hilbert curves, one that doesn't rely on completing an infinite process. What do you think?
The Hilbert curve is computable according to the usual accounts of what this should mean. Namely, a real number is computable when there is a computable procedure to compute approximations to it within any given accuracy. And similarly, a curve gamma is computable if there is a uniform computable procedure which, when given approximations to a real number t, it computes a corresponding approximation to gamma(t). For the Hilbert curve, gamma(0) starts at the lower left and gamma(1) ends at the lower right, and gamma(1/2) is exactly in the middle---the approximations are exactly the values that cross the bridge I had mentioned from left to right. We can compute any particular gamma(t) as accurately as we please, for any input t for which we can provide close enough approximations.
Thanks! I agree entirely with your comment - the Hilbert curve is computable in the conventional sense as you describe. However, my issue is that the computation breaks down when looking for perfect accuracy. If we can only ever deal with approximations, who is to say that the ideal (which is being approximated) actually exists. To me it seems like a mirage (a very useful one, no doubt).
Another perspective is that the perfect accuracy consists precisely of the infinite sequence of successive and increasingly accurate approximations. That is, after all, what a decimal expansion like 3.14159265... ultimately means.
Obviously the Hilbert curve is not a curve within such a sequence. Are you saying that the Hilbert curve is the sequence itself (in contrast to the limit of the sequence)?
Thanks for the essay!!! I enjoyed reading it. In the space filling curve, it is claimed that the skater will visit "every point on the ice at some point during her performance”. I am wondering if this can give another characterisation of the dimension. For example, can I say that in Sierpinski gasket approximations, the skater will never visit certain points on the ice (which is the outer triangle), and the "proportion of these points on the ice" depends only on the dimension of the curve? (Is it measure theory that I need to study to be able to understand more about these results?)
A truly space-filling curve will always have dimension 2. For the other fractals I present, there are points you can identify that are never visited by the limit curve. But speaking of the "proportion" of points visited won't quite work, since if the set has dimension less than 2, it will have zero area with respect to the usual Lebesgue measure. Indeed, the fractal dimension d can be defined as the dimension that is the boundary between infinite d'-dimensional measure and zero d''-dimensionsional measure, where d'<d<d''.
"The final Hilbert curve itself is the limit of these approximations as they get finer and finer."
-----------------------------------------------------------------------------------------------------------------
If we create a computer program to iterate and output successive Hilbert curves, it will never output the final Hilbert curve since there is no last iteration. Does it exist? Is it fair to say that there is a final curve but not a final iteration? I think It might be reasonable to say that the final Hilbert curve is a 'mirage' (as opposed to an output) of the program. This may be a subtle distinction, but I think it's important because it puts the final Hilbert curve in a different class of existence than all the other Hilbert curves, one that doesn't rely on completing an infinite process. What do you think?
The Hilbert curve is computable according to the usual accounts of what this should mean. Namely, a real number is computable when there is a computable procedure to compute approximations to it within any given accuracy. And similarly, a curve gamma is computable if there is a uniform computable procedure which, when given approximations to a real number t, it computes a corresponding approximation to gamma(t). For the Hilbert curve, gamma(0) starts at the lower left and gamma(1) ends at the lower right, and gamma(1/2) is exactly in the middle---the approximations are exactly the values that cross the bridge I had mentioned from left to right. We can compute any particular gamma(t) as accurately as we please, for any input t for which we can provide close enough approximations.
Thanks! I agree entirely with your comment - the Hilbert curve is computable in the conventional sense as you describe. However, my issue is that the computation breaks down when looking for perfect accuracy. If we can only ever deal with approximations, who is to say that the ideal (which is being approximated) actually exists. To me it seems like a mirage (a very useful one, no doubt).
Another perspective is that the perfect accuracy consists precisely of the infinite sequence of successive and increasingly accurate approximations. That is, after all, what a decimal expansion like 3.14159265... ultimately means.
Obviously the Hilbert curve is not a curve within such a sequence. Are you saying that the Hilbert curve is the sequence itself (in contrast to the limit of the sequence)?
No, I would say that the Hilbert curve is the limit of the sequence. It is a computable curve, in virtue of the computability of that sequence.
Thanks for the essay!!! I enjoyed reading it. In the space filling curve, it is claimed that the skater will visit "every point on the ice at some point during her performance”. I am wondering if this can give another characterisation of the dimension. For example, can I say that in Sierpinski gasket approximations, the skater will never visit certain points on the ice (which is the outer triangle), and the "proportion of these points on the ice" depends only on the dimension of the curve? (Is it measure theory that I need to study to be able to understand more about these results?)
A truly space-filling curve will always have dimension 2. For the other fractals I present, there are points you can identify that are never visited by the limit curve. But speaking of the "proportion" of points visited won't quite work, since if the set has dimension less than 2, it will have zero area with respect to the usual Lebesgue measure. Indeed, the fractal dimension d can be defined as the dimension that is the boundary between infinite d'-dimensional measure and zero d''-dimensionsional measure, where d'<d<d''.
I see, thanks for the elaboration!
Thanks for the write-up, that was a very interesting read!