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January 29th, 2018, 03:57 PM   #1
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Fill D space with D-1 object?

Is there a finite object of dimension D-1 that can fill a space of dimension D?
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January 29th, 2018, 04:54 PM   #2
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There's the standard example of the space filling curve, which fills the unit cube in $n$-dimensions for any positive integer $n$.

According to the Wiki article, this procedure can be slightly modified to produce a continuous curve that fills the entire $n$-dimensional space.

Note that these curves are surjections but not bijections. They hit some points more than once. So they are not homeomorphisms (topological isomorphisms) because they don't have continuous inverses. That's good, because dimension is a topological invariant so a line could not be homeomorphic to a square.
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Last edited by Maschke; January 29th, 2018 at 05:02 PM.
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January 29th, 2018, 09:46 PM   #3
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What I was trying to avoid by imposing a finite limit constraint (e.g. for the length of an arbitrary curve), is that if its length did approach infinity, it might fill any well-behaved space.

As for the Peano curve (and most curves, for that matter), the dimension of the space itself minus the fractal dimension of the curve does not obey my condition D-(D-1)=1.

In general, can any finite curve of D=1 fill a space of D=2?
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January 29th, 2018, 11:40 PM   #4
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OK, what do you mean with a finite curve? A curve with finite arclength? No, that's impossible.
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January 30th, 2018, 09:03 PM   #5
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Thank you, mu mass; my initial conjecture is not doable.

However, if the difference between dimensions yields 0<Δ<1 -- that is, relatively fractal -- could a finite curve possibly fill a finite space as long as their relative dimension is between one and zero?
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January 30th, 2018, 09:10 PM   #6
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Quote:
Originally Posted by Loren View Post
Thank you, mu mass;
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January 31st, 2018, 06:32 PM   #7
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Originally Posted by Loren View Post
Thank you, mu mass; my initial conjecture is not doable.

However, if the difference between dimensions yields 0<Δ<1 -- that is, relatively fractal -- could a finite curve possibly fill a finite space as long as their relative dimension is between one and zero?
This question is far too vague. Typically fractal curves aren't even rectifiable so its meaningless to ask about finite arc length.

If you want to ask a question in this area you must be much more precise about what is meant by fractal and what you mean by a finite curve.
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February 6th, 2018, 11:15 PM   #8
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Do fractals have definable symmetries greater than those of their self-similarities?

Is the limit of a radius as it approaches zero in a fractal of dimension D at all definable?
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