14
votes
Accepted
Space filling curve whose all level sets are finite (countable)
Recall the definition of the Peano square-filling curve $f:[0,1]\to[0,1]^2$, which is given in terms infinite ternary strings. If $a\in [0,1]$ has a base $3$ representation of the form $0,a_1a_2a_3\...
- 60.5k
8
votes
Orthogonal Hamiltonian cycles in (n x n x n) grids
In the recent book "Bicycle or Unicycle?" by Velleman and Wagon, this is problem #16 "Wiggle Room." (Actually, there it's generalized to computing the maximum length path, with a ...
- 133
6
votes
Accepted
Is the Hilbert space-filling curve bijective over computable numbers?
The Hilbert curve, due to its fractal nature, is mapping certain subintervals of the unit interval to certain squares in the unit square. On any given resolution, we have a bijection between the ...
- 4,667
5
votes
Accepted
Orthogonal Hamiltonian cycles in (n x n x n) grids
Partial answer: It is possible for $C_n$ if $n$ is a power of two.
$C_2$ and $C_4$ are shown in the question. For larger $n$ the idea is to take a three-dimensional Moore curve (a recursive ...
- 4,219
4
votes
Accepted
Space filling curves
I would call such $f_\epsilon$ coarsely $\epsilon$-dense, or more simply $\epsilon$-dense. These exist in the compact case as follows. Tile $M$ by $n$-balls (these may overlap, but only finitely). ...
- 28.1k
3
votes
Accepted
Domino tiling obtained from space-filling curves, is possible to predict basic properties?
I'm not sure I understood the question and answer given by the asker, but I gather that they are interested in decidability questions about some substitutive subshifts.
As far as I can tell, the ...
- 6,652
1
vote
Hemispherical space filling hilbert curve
Well, I still havent found a solution for a hemisphere, but for a full sphere the solution was normalizing my cube hilbert curve around the zero and then mapping it with the equations from "...
- 131
1
vote
Topological dimension of the image of continuous surjective functions
I did some more reading on the topic and found out a few results in Dimension Theory, R. Engelking, Ch. 1, § 12, that I want to share in case anyone else is interested. They apply for separable ...
- 71
1
vote
Do Peano curves provide a counterargument to Grothendieck's critique?
Here I am expanding Doug Zare's comment into an answer.
The original poster asks:
Can the paper Group Invariant Peano Curves by Cannon and Thurston be regarded as exhibiting the naturality of ...
- 28.1k
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