# Questions tagged [space-filling-curves]

The space-filling-curves tag has no usage guidance.

The space-filling-curves tag has no usage guidance.

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The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$.
It is ...

8
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2
answers

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Let $C_n$ be a cubical $n \times n \times n$ subset of the integer lattice,
so consisting of $n^3$ vertices.
I am interested in special Hamiltonian cycles in $C_n$, special in the
sense that (a) each ...

2
votes

1
answer

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First question here, sorry for any posting infractions.
I need to create/find/buy a hemispherical space-filling Hilbert(or similar) curve.
something similar to Cube hilbert
but only filling a ...

4
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2
answers

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Periodic and aperiodic domino tiling systems can be obtained by the following construction rules:
Draw a regular square grid n×n of n2 cells.
Select a space-filling curve that is consistent with ...

3
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1
answer

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The Hilbert curve is a continuous space-filling curve that maps $I$ to $I^n$ where $I$ denotes the unit interval [1]. Like all other space-filling curves, it is not one-to-one. I am wondering if the ...

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3
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Consider two topological spaces $(X,\tau)$ and $(Y,\omega)$ and a continuous surjective function $f\colon X\to Y$.
Let $\mathrm{dim}(X)$ and $\mathrm{dim}(Y)$ denote the Lebesgue covering dimension ...

8
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1
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Is there a continuous surjective function $f:[0,1] \to [0,1]^2$ such that
every level set $f^{-1}(y)$ is a finite set? If the answer is no, what about if we replace the finiteness of level sets by "...

2
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To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed ...

15
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I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.

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This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be ...

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Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists?
Or piecewise differentiable?
Must every continuous space-filling curve be nowhere ...

3
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Since points on a euclidean plane can be represented by one coordinate on a space-filling curve, is there any curve such that if two vectors $(x_0,y_0)$ and $(x_1,y_1)$ were represented by $a$ and $b$,...

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Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \...

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Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.
Consider a foliation as a collection of continuous nonintersecting curves that start at $(0,0)$ and ...

17
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There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...