Questions tagged [space-filling-curves]

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3 votes
1 answer
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Space filling curves

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 votes
2 answers
448 views

Orthogonal Hamiltonian cycles in (n x n x n) grids

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
99 views

Hemispherical space filling hilbert curve

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 ...
  • 131
4 votes
2 answers
444 views

Domino tiling obtained from space-filling curves, is possible to predict basic properties?

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 votes
1 answer
868 views

Is the Hilbert space-filling curve bijective over computable numbers?

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 ...
5 votes
3 answers
814 views

Topological dimension of the image of continuous surjective functions

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 ...
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8 votes
1 answer
576 views

Space filling curve whose all level sets are finite (countable)

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 votes
0 answers
101 views

Quantitative estimates on space filling curves

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 votes
5 answers
3k views

Applications of space filling curves

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.
3 votes
1 answer
627 views

Do Peano curves provide a counterargument to Grothendieck's critique?

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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26 votes
5 answers
6k views

Proof that no differentiable space-filling curve exists

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 votes
3 answers
292 views

Space filling curve to simplify vector addition? [closed]

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 $b$,...
  • 387
19 votes
1 answer
724 views

convexity of images of space-filling curves

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 \...
7 votes
2 answers
1k views

Unusual space-filling curve

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 votes
2 answers
1k views

Are there space filling curves for the Hilbert cube?

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: ...