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Ryan O’Connor's avatar

"The final Hilbert curve itself is the limit of these approximations as they get finer and finer."

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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?

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Ruiting Jiang's avatar

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?)

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