January 29th, 2018, 03:57 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 335 Thanks: 26 Math Focus: Number theory  Fill D space with D1 object?
Is there a finite object of dimension D1 that can fill a space of dimension D?

January 29th, 2018, 04:54 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,850 Thanks: 509 
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. Last edited by Maschke; January 29th, 2018 at 05:02 PM. 
January 29th, 2018, 09:46 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 335 Thanks: 26 Math Focus: Number theory 
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 wellbehaved 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(D1)=1. In general, can any finite curve of D=1 fill a space of D=2? 
January 29th, 2018, 11:40 PM  #4 
Senior Member Joined: Oct 2009 Posts: 366 Thanks: 122 
OK, what do you mean with a finite curve? A curve with finite arclength? No, that's impossible.

January 30th, 2018, 09:03 PM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 335 Thanks: 26 Math Focus: Number theory 
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? 
January 30th, 2018, 09:10 PM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,579 Thanks: 541 Math Focus: Yet to find out.  
January 31st, 2018, 06:32 PM  #7  
Senior Member Joined: Sep 2016 From: USA Posts: 356 Thanks: 194 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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.  
February 6th, 2018, 11:15 PM  #8 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 335 Thanks: 26 Math Focus: Number theory 
Do fractals have definable symmetries greater than those of their selfsimilarities? Is the limit of a radius as it approaches zero in a fractal of dimension D at all definable? 

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