Re: [math-fun] Doin' the Hilbert walk
It may be worth mentioning that for any given dimension = d the d-dimensional Hilbert polygonal curves [0,1] -> [0,1]^d approach a limit as the number of vertices approaches infinity.
Which is, of course, a continuous surjection
H_d : [0,1] -> [0,1]^d.
Although this is old hat to some of us, when it was first discovered (around 1900) it stunned the world of math, forcing people to figure out what the concept of dimension really meant.
By making adjustments to the Hilbert polygons, one can easily arrange that the limit function is one-to-one (but of course no longer onto). So the limit function is a homeomorphism of [0,1] onto its image.
I suspect the Hausdorff dimension of such a one-one curve can be any real number in [1,d).
(Can you confirm this, rwg?)
Unfortunately, no! This is why I detest the misleading term "spacefilling curve". A curve is a set--the graph of a function. A spacefilling curve is therefore a solid area! How do you define the limit of a sequence of polygonal curves? Is it a set or a function? The only coherent notion is spacefilling function-- a continuous map of a closed interval onto a closed 2(or more)D set. All those different patterns in http://www.tweedledum.com/rwg/sampeano.htm are polygonal subsequences of the smae function! You cannot change the function by tweaking the curve and then taking some nebulous limit. A spacefilling function (D>=2) *cannot* be 1-1. Incredibly, every spacefilling function must overfill by visiting infinitely many points at least three times! (And uncountably many points twice.) --rwg PS to my Hilbert walk msg: Besides rational, it is frequently possible to give the exact image of regularly formed binary constants like Thue's and the parity constant. ANTAGONIST STAGNATION ENTHUSIASTIC UNCHASTITIES
A spacefilling curve is one in which the plot thickens! -- Mark Miller On Thu, Nov 13, 2008 at 10:02 PM, <rwg@sdf.lonestar.org> wrote:
It may be worth mentioning that for any given dimension = d the d-dimensional Hilbert polygonal curves [0,1] -> [0,1]^d approach a limit as the number of vertices approaches infinity.
Which is, of course, a continuous surjection
H_d : [0,1] -> [0,1]^d.
Although this is old hat to some of us, when it was first discovered (around 1900) it stunned the world of math, forcing people to figure out what the concept of dimension really meant.
By making adjustments to the Hilbert polygons, one can easily arrange that the limit function is one-to-one (but of course no longer onto). So the limit function is a homeomorphism of [0,1] onto its image.
I suspect the Hausdorff dimension of such a one-one curve can be any real number in [1,d).
(Can you confirm this, rwg?)
Unfortunately, no! This is why I detest the misleading term "spacefilling curve". A curve is a set--the graph of a function. A spacefilling curve is therefore a solid area! How do you define the limit of a sequence of polygonal curves? Is it a set or a function? The only coherent notion is spacefilling function-- a continuous map of a closed interval onto a closed 2(or more)D set. All those different patterns in http://www.tweedledum.com/rwg/sampeano.htm are polygonal subsequences of the smae function! You cannot change the function by tweaking the curve and then taking some nebulous limit. A spacefilling function (D>=2) *cannot* be 1-1. Incredibly, every spacefilling function must overfill by visiting infinitely many points at least three times! (And uncountably many points twice.) --rwg PS to my Hilbert walk msg: Besides rational, it is frequently possible to give the exact image of regularly formed binary constants like Thue's and the parity constant.
ANTAGONIST STAGNATION ENTHUSIASTIC UNCHASTITIES
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